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This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} A thorough Montague and Lambek oriented monograph exploiting a ``logic of signs'' for formal grammar; signs are associations of a symbol with a meaning, which is handled in a parallel manner (i.e., not first autonomous syntax, then semantics). The author attempts to refine the methods already used in categorial grammar; especially Curry-Howard correspondence between (Gentzen-like) intuitionistic logic and typed lambda calculus is exploited. The first four chapters are of general importance, reproducing Montague grammar and Lambek calculus (calculi). The author shares, of course, with all Mantagovians illusions about Montague's ``cup'' and ``cap'', but discussing this topic is not what this review should involve. Special problems are tackled or solved in Chapters 6-8 (Polymorphism, Structural control, Extraction). A general summing up (containing, e.g., a comparison of the categorial/computational approach with the transformational one) is given in Chapter 9 (Conclusion). Three useful appendices follow: Interpretation of types; Gentzen sequent rules; Summary grammar. A bibliography (more than 100 titles), and an Index complete the work. Erratum: p. 72, (5) a., (6) a.: In the conclusions of these rules, read \(B\) instead of \(A\).

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} Let \(G=NS\) be a solvable Lie group which is a semidirect product of simply connected abelian groups \(N\) and \(S\), where \(N\) is normal in \(G\) and \(S\) is one dimensional. Let \(L\) be a left invariant differential operator on \(G\). By definition, \(L\) is semiglobally solvable if for every \(f\in C_ c^{\infty}(G)\) and for every relatively compact open set \(U\) there exists \(u\in C_ c^{\infty}(G)\) such that \(Lu=f\) on \(U\). This paper deals with the question of whether a given differential operator on such a Lie group is semiglobally solvable. Using the partial Fourier transform over \(N\), \(L\) is equivalent to a direct integral over \(\xi\) in the dual of the Lie algebra of \(N\) of operators \(\pi^{\xi}(L)\) acting on \(L^ 2(\mathbb{R})\). The idea is that the operators \(\pi^{\xi}(L)\) are more basic than \(L\) and the question is reduced to the solvability of these component operators. Theorem. Suppose there exists a polynomial \(p\) such that for every \(f\in C_ c^{\infty}(G)\), \(\pi^{\xi}(L)u_{\xi}=f_{\xi}\) is solvable with \(\xi\to p(\xi)u_{\xi}(t)\) a tempered distribution for a.e. \(t\in\mathbb{R}\). Then \(L\) is semiglobally solvable. The author discusses the history of the problem and the relationship between this result and other similar results in the literature. A significant part of the paper is devoted to applications of the theorem above. Specific elliptic, sub-elliptic and parabolic operators are shown to satisfy the hypotheses of the theorem.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} This paper is an attempt to answer what is the notion of a morphism of rings. While exploring bimodules as a potential candidate, the author first studies the double category of rings, denoted \(\mathbb{R}\mathrm{ing}\), where horizontal morphisms are unital ring homomorphisms, vertical morphisms are bimodules with double cells as bilinear maps between them; the vertical structure is not a strict bicategory since the composition of bimodules, i.e., their tensor product, is associative only up to isomorphism. Modelled on the relationship between functions and their graphs, the author introduces the categorical notion of companions (as well as the dual notion of conjoints) of horizontal morphisms, and characterizes companions in \(\mathbb{R}\mathrm{ing}\) as bimodules \(_SM_R\) where \(_SM\cong\,_SS\). While investigating the generalization of the double category \(\mathbb{R}\mathrm{ing}\) where bimodules \(_SM_R\) where \(_SM\cong\,_SS^{(p)}\) for \(p\geq1\) are conjoints, the author introduces the graded double category of rings, denoted \(\mathbb{R}\mathrm{ing_{Mat}}\), where horizontal morphisms are matrix valued homomorphisms \(f:R\to \mathrm{Mat}_p(S)\) (these are so-called amplifying morphisms in the quantum field theory literature) while giving up unitality of horizontal morphisms, and where each horizontal arrow has a companion as well as a conjoint. If a horizontal arrow \(f\) in a double category has a companion \(f^*\) and a conjoint \(f_*\), then \(f_*\) is left adjoint to \(f^*\). The double categories where the converse holds are called ``Cauchy''. Modifying the vertical structure of the graded double category \(\mathbb{R}\mathrm{ing_{Mat}}\), the author obtains a Cauchy double category \(\mathbb{A}\mathrm{mpli}\) that is vertically self-dual and that enjoys the desirable property of the existence of a companion as well as a conjoint for each horizontal morphism.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We propose a new potential renormalization method for a semiclassical model of the Hartree-Fock-Slater type. The method allows easily constructing the wave functions and, in contrast to most of the similar methods, does not require an explicit form of the potential.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} The tensor product of a list of vector spaces is commonly introduced after defining multilinear functions. Many naturally occurring tensor products on categories also represent some kind of multimorphism. A multicategory [\textit{J. Lambek}, ``Deductive systems and categories. II'', Lect. Notes Math. 86, 76-122 (1969; Zbl 0198.33701)] is a category with an abstract notion of multimorphism and substitution operations. This paper defines a precise 2-category of multicategories by making efficient use of the bicategory of spans. There is an obvious sense in which the multimorphisms may be represented by a multiple tensor product. The author points out that, besides this, one should also demand that the universal morphisms should be closed under composition; this is equivalent to a strong representability condition. The consequential concept of representable multicategory is then compared with monoidal category. A form of coherence is obtained.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} The test sphere method is described to study the global stability of nonlinear autonomous systems. Some interesting examples are considered.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We prove the Chomsky conjecture (all languages recognized by the Lambek calculus are context-free) for both the full Lambek calculus and its product-free fragment. For the latter case, we present a construction of context-free grammars involving only product-free types.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We consider an object calculus in which open terms interact with the environment through interfaces. The calculus is intended to capture the essence of contextual interactions of Middleweight Java code. Using game semantics, we provide fully abstract models for the induced notions of contextual approximation and equivalence. These are the first denotational models of this kind.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} The Zbl relevant articles will be reviewed individually, see ``Language and information''.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} A simple algorithm is suggested for the construction of a polynomial divisible by the denominator of any rational solution of the linear difference equation \[ a_n(x)y(x+ n)+\cdots+ a_0(x) y(x)= b(x), \] with polynomial coefficients and a polynomial right-hand side. Then, the same problem is solved for \(q\)-difference equations. Nonhomogeneous equations with hypergeometric right-hand sides are considered as well.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} Verf. formalisiert die einfache Typentheorie, wobei er einen Teil des \(\lambda\)-Konversionskalküls mitnimmt. Er bemerkt, daß es nicht möglich ist, den ganzen \(\lambda\)-Kalkül mitzunehmen, falls \(\lambda\) die Bedeutung eines Abstraktionsoperators und Nebeneinanderstellung ihre Bedeutung als die Beziehung von Funktion zu Argument behalten sollen. Die Symbole ı\ und \(\circ\) stellen die Grundtypen, Individuum und Aussage, dar. Sind \(\alpha\) und \(\beta\) Typen, so ist \((\alpha\beta)\) der Typus einer Funktion mit Argument vom Typus \(\beta\) und Funktionswert vom Typus \(\alpha\). Funktionen mehrerer Variablen sind im Sinne \textit{Schönfinkel}s (Math. Ann., Berlin, 92 (1924), 305-316; F. d. M. 50, 23) zu verstehen. Die natürlichen Zahlen werden als Abkürzungen für gewisse \(\lambda\)-Ausdrücke eingeführt. Das System hat sechs Schlußregeln. Die drei ersten stellen die Regeln der \(\lambda\)-Konversion dar; die anderen sind die Abtrennungs-, die Einsetzungs- und eine Generalisationsregel. Weiter hat das System 11 formale Axiome, von denen die vier ersten die Axiome des klassischen Aussagenkalküls sind (\textit{Hilbert-Ackermann}, Grundzüge der theoretischen Logik (2. Aufl. 1938; F. d. M. \(64_{\text I}\), 26), S. 23), die sechs ersten für den Prädikatenkalkül genügen, und die neun ersten für die elementare Zahlentheorie hinreichend sind. Drei der Peanoschen Axiome für die Arithmetik folgen bereits aus den sechs ersten Axiomen des Systems. Einige Theoreme für beliebige Typen werden bewiesen und zum Schluß die Formalisierung der Definition durch primitive Rekursion innerhalb des Systems gezeigt.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using \textit{O. V. Lepski} et al.'s method [Ann. Stat. 25, No. 3, 929--947 (1997; Zbl 0885.62044)]. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We prove strong completeness of the Lambek syntactic calculus (with product) with respect to an algebraic semantics consisting of residuated semigroups spread over arbitrary semigroups. If the latter are free semigroups, strong completeness fails, but it is an open problem whether weak completeness holds. We provide several partial results in this direction. Similar questions are considered for some related calculi.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces with the aid of some recently proven continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of \(m\)-accretive multivalued differential equation, the nonlocal Cauchy problem for semilinear differential inclusion, abstract integral inclusion of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusion.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} This article presents five styles of deductive systems, paying particular attention to equality of deductions within them. The five are: (1) Hilbert-style, with deductions of the form \(f:\;\to B\) (\(f\) is a proof of \(B\) without hypotheses); (2) Lawvere-style, \(f: A\to B\) (\(f\) is a proof of \(B\) from hypothesis \(A\)); (3) Gentzen-intuitionistic-style, \(f: A_ 1,\dots, A_ m\to B\) (as (2), but with \(m\geq 0\)); (4) Gentzen- classical-style, \(f: A_ 1, \dots, A_ m\to B_ 1,\dots, B_ n\) (\(f\) is a proof of alternatives \(B_ 1, \dots, B_ n\) from hypotheses \(A_ 1,\dots,A_ m\), \(m,n\geq 0\)); and (5) Schütte-style, \(f: \;\to B_ 1,\dots, B_ n\) (\(f\) is a proof of alternatives \(B_ 1, \dots, B_ n\) from no hypotheses). The interest in equality of deductions, and comparing different forms of systems in terms of such equality, leads to an examination of particular instances of these styles. E.g., the Hilbert-style formulation of intuitionist implication is singled out since this enables proof of a deduction theorem, which may be taken as a form of functional completeness, especially as the system is analyzed as a set of equations. Similarly for the other, more general forms of deductive systems. In addition, as part of the examination of Gentzen- intuitionistic systems, attention is given to systems lacking one or more of Gentzen's structural rules, which leads to relevance logic, linear logic, and a pure syntactic or intuitionistic bilinear calculus.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} In this paper, we study the minimal submanifolds in a Riemannian manifolds with parallel Ricci curvature. We obtain an integral inequality of Simons' type. Their result for minimal submanifolds in a locally symmetric \(\delta\)-pinching Riemannian manifold is generalized.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} In [\textit{S. Lack} and \textit{R. Street}, Adv. Math. 258, 351--396 (2014; Zbl 1350.18012)], the free skew monoidal category Fsk on a single generating object was described. We sharpen this by giving a completely explicit description of Fsk, and so of the free skew monoidal category on any category. As an application we describe adjunctions between the operad for skew monoidal categories and various simpler operads. For a particular such operad \(\mathcal{L}\), we identify skew monoidal categories with certain colax \(\mathcal{L}\)-algebras.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We generalize several studies in the area of extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finite-sample and the limiting behavior of the estimators.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} This is an exposition of some applications of the calculus of binary relations to anthropology, linguistics, computer science and mathematics. Binary relations appear in the rewrite systems used by anthropologists to study kinship terminologies of primitive societies. They also serve to model the syntactic calculus, a form of categorial grammar once proposed by the author. Moreover, if we introduce an operation perp as the complement of the converse, we may take the calculus of binary relations as a model for cyclic linear logic, studied by Yetter in a first attempt to remove commutativity from linear logic, recently of interest in theoretical computer science. Partial recursive functions may best be viewed as recursively enumerable binary relations which happen to be single-valued; they are of the form \(f\) composed with \(g\) converse, where \(f\) and \(g\) are primitive recursive functions satisfying the condition: \(g\) converse composed with \(g\) is contained in \(f\) converse composed with \(f\). In mathematics, binary relations were pushed into the background when it was decided that functions be single-valued and universally defined. However, they made a comeback in homological algebra, where they provide the easiest construction for the so-callcd connecting homomorphism. While such constructions were originally confined to abelian categories, it is now clear that they can be extended to arbitrary categories, provided one chooses appropriate generalizations of `exactness' and other relevant concepts.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} The author presents a part of the elementary algebra (numbers) treating the contents on more wider and deeper way then the usual teaching matter. In this chapter the author gives an excellent survey of the main ideas concerning the integers and irrational numbers without insisting on the logical foundations. These topics are enlightened from the historical point of view from the creation of some ideas and concepts to their further developments. It means from the knowledge formed in Ancient Egypt or Babylon than to the development started in Ancient Greece (the Euclid's ``Elements'') and is continued in Western Europe after Renaissance up to nowadays. For example, from the notion of division of integers, the Euclid algorithm to the factorization in primes and an application of that in today's the theory of coding. The second very important aspect of this text is connection between different branches of mathematics, specially of algebra and geometry. The text is provided by a few figures and by a nice collection of selected problems. This is translation of a text from Russian journal ``Matematicheskoe obrazovanie'' April--June 1997.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We present a type theoretic analysis of Latin grammar, which pays as much attention to inflectional morphology as to syntax. We assign different types to the finite forms of Latin verbs as well as to their infinitives. The rich repertory of agreement information exhibited by Latin is accounted for by a system of numerical indexes (superscripts and subscripts) attached to the types. Agreement coherence and control of sentencehood for strings of words is to be guaranteed by calculations performed on the corresponding strings of types, in accordance with the ``pregroup'' grammar developed as a refinement of classical bilinear logic.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} In this paper, a particular inhomogeneous inclusion problem is studied. In the problem, Eshelby's eigenstrain takes the type \(x^my^n\), where \(m+n=2\), and the remote loadings \(\sigma^\infty_x\), \(\sigma^\infty_y\) are applied. In the solution, the complex variable method is used. The continuity conditions along the interface of the matrix and the inclusion are formulated exactly. Because the stress field is no longer uniform in inclusion in this case, the studied problem has an inherent difficulty. After some manipulation, the final result for stress components \(\sigma^{\mathrm{in}}_x\), \(\sigma^{\mathrm{in}}_y\) and \(\sigma^{\mathrm{in}}_{xy}\) in inclusion are obtainable. In the present study, \(\sigma^{\mathrm{in}}_x\), \(\sigma^{\mathrm{in}}_y\) and \(\sigma^{\mathrm{in}}_{xy}\) are no longer uniform.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} Skew monoidal categories are a variation of monoidal categories in which the unit and associativity transformations are no longer required to be natural isomorphisms but merely natural transformations in a certain direction. They first arose in [\textit{K. Szlachányi}, Adv. Math. 231, No. 3--4, 1694--1730 (2012; Zbl 1283.18006)], being introduced to deal with bialgebroids. The authors construct free skew monoidal categories using an appropriate Gentzen sequent calculus, analyzing the decision word problem. The techniques they use are inspired by linear logic proof search [\textit{J.-M. Andreoli}, J. Log. Comput. 2, No. 3, 297--347 (1992; Zbl 0764.03020)], based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. The theory of skew monoidal categories has been investigated in an extensive series of papers by \textit{J. Bourke} [J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001)], \textit{J. Bourke} and \textit{S. Lack} [J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019); Theory Appl. Categ. 35, 19--63 (2020; Zbl 1431.18012)], \textit{S. Lack} and \textit{R. Street} [Theory Appl. Categ. 26, 385--402 (2012; Zbl 1252.18016); Adv. Math. 258, 351--396 (2014; Zbl 1350.18012); Appl. Categ. Struct. 22, No. 5--6, 789--803 (2014; Zbl 1317.18012); Theory Appl. Categ. 30, 985--1000 (2015; Zbl 1331.18007)], and \textit{R. Street} [J. Pure Appl. Algebra 217, No. 6, 973--988 (2013; Zbl 1365.18008)], with which the authors of this paper compare their work, discussing the sense in which Lambek's language of multicategories provides a better understanding of the proof-theoretic analysis given here. Finally, the authors formalize this development in the dependently typed programming language Agda.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} We consider a complex strongly harmonizable stationary symmetric stable process in continuous time. In this paper we give an unbiased consistent estimate of the spectral density for this process sampling at periodic instants. To avoid the aliasing phenomenon, it is assumed that the spectral density has a compact support.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} In previous work [Linguistic Analysis 36, 345--384 (2010)] the author developed a formal system, referred to as DisCoCat (\underline{Cat}egorical \underline{Co}mpositional \underline{Dis}tributional -- in reverse order), for analyzing how words interact in a sentence in order to produce the meaning of that sentence, exploiting the structural match between grammar and categories of meaning spaces. The present work extends the system DisCoCat of the basic interactive syntax and semantics at the sentence level to the new system DisCoCirc, in which the question of textual meaning is addressed. Sentences are gates within a circuit which updates the variable meanings of those words. The compositional structure is given by string diagrams representing information flows, and a text yields a single string diagram in which word meanings lift to the meaning of the entire text. The publication of \textit{J. Lambek}'s seminal paper [Am. Math. Mon. 65, 154--170 (1958; Zbl 0080.00702)] was monumental. Lambek himself replaced \textit{Lambek grammar} with \textit{pregroup grammar} [\textit{J. Lambek}, Lect. Notes Comput. Sci. 1582, 1--27 (1999; Zbl 0934.03043); J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305); From word to sentence. A computational algebraic approach to grammar. Monza: Polimetrica (2008; Zbl 1166.03315)]. Around 2004 the author no sooner gave a talk at the McGill category seminar about his then new diagrammatic description of quantum teleportation [\textit{B. Coecke}, AIP Conf. Proc. 810, 81--98 (2006; Zbl 1116.81011); Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 415--425 (2004)] than Lambek pointed out that those were pregroups. It was this connection, between grammar and teleportation diagrams, that inspired the DisCoCat model, making it look as if word meaning being teleported around in sentences by means of the channels provided by the pregroup grammar [Malaysian J. Math. Sci. 8, 15--25 (2014)]. Lambek himself explicitly stressed this connection between language and physics [\textit{J. Lambek}, Lect. Notes Phys. 813, 467--487 (2011; Zbl 1253.81013)]. While the developments in this paper is independent of a physical embodiment, both the compositional formalism and the model of meaning are highly quantum inspired, allowing for implementation on a quantum computer. The sentence type used in this paper was also used in the recent DisCoCat paper [Electoric Proceedings in Theoretical Computer Science 283, 75--88 (2018)]. Also within the context of DisCoCat, the papers [Electoric Proceedings in Theoretical Computer Science 283, 1--12 (2018); \textit{A. Toumi}, Categorical compositional distributional questions, answers \& discourse analysis. University of Oxford (Master's thesis) (2018)] involve multi-sentence interaction by relying on discourse representation structures [\textit{H. Kamp} and \textit{U. Reyle}, From discourse to logic. Introduction to modeltheoretic semantics of natural language, formal logic and discourse representation theory. Dordrecht: Springer (1993)] at the cost of reducing meaning to a number.

This volume gathers together noted experts to address the state of the art of various works by Joachim Lambek (1922--2014). Although Lambek was a highly original mathematician with a wide range of scientific interests, there is a single thread running through every work of his. Therefore let us overlook his research career from scratch to his final days. Joachim Lambek was born in Leipzig, Germany in 1922. He got a PhD in Mathematics at McGill University under the supervision of a new McGill professor, Zassenhaus, presenting [\textit{J. Lambek}, Can. J. Math. 3, 34--43 (1951; Zbl 0042.01701)] as his PhD theses. During the 1950s, Lambek primarily pursued research in elementary number theory and algebra. In the period 1955--1965, Lambek was primarily an algebraist, publishing two famous books [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London-Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403); \textit{N. J. Fine} et al., Rings of quotients of rings of functions. Montreal: Distributed by McGill University Press (1966; Zbl 0143.35704)]. Lambek's interest in category theory dates from his sabbatical year 1965--1966 at the ETH in Zurich, where Bruno Eckmann gathered together an influential group of mathematicians to study algebraic topology, homological algebra, and category theory. This sabbatical year turned Lambek's research interest increasingly towards the newly developing subjects of category theory and categorical logic, along with his interesting activity in linguistics and formal grammar. His published work in category theory began with [\textit{J. Lambek}, Completions of categories. Seminar lectures given 1966 in Zürich. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.02802)]. Lambek was fascinated by MacLane's coherence problems, observing that such questions are often to be formulated as the decision problem for the hom-sets of certain \textit{freely generated categories}. Lambek turned this into a proof-theoretical problem [\textit{J. Lambek}, Math. Syst. Theory 2, 287--318 (1968; Zbl 0176.28901); Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701); Lect. Notes Math. 274, 57--82 (1972; Zbl 0244.18006); Stud. Log. Comput. 4, 141--159 (1994; Zbl 0821.03003); Trends Log. Stud. Log. Libr. 5, 279--294 (1999; Zbl 0960.03022)] by developing a Gentzen-style logical system, whose \textit{formulas} are the objects of a free category and whose \textit{equivalence classes} of formal sequent calculus proofs are the arrows. The decision problem is then resolved by an appropriate \textit{cut-elimination theorem}. \textit{J. Lambek} [Lect. Notes Math. 86, 76--122 (1969; Zbl 0198.33701)] presented deductive systems for a host of interesting mathematical structures, as well as novel approaches, called \textit{multicategories}, for developing deductive systems for various kinds of monoidal categories and associated \textit{substructural} logics. Lambek's key observation is that instead of defining monoidal categories, closed categories, etc. in the traditional way, it is often possible to to equationally introduce these structures more directly into a multicategory. Lambek studied a host of structured multicategories, often stemming from his work in linguistics, such as \textit{right-closed}, \textit{left-closed} and \textit{residuated} multicategories. A fundamental idea in [\textit{J. Lambek}, Contemp. Math. 92, 217--239 (1989; Zbl 0685.18003)] is that of generating multicategories via an \textit{internal language}. For an abstract treatment of multicategories, so-called \textit{representable multicategories} and coherence theory, one is referred to [\textit{C. Hermida}, Adv. Math. 151, No. 2, 164--225 (2000; Zbl 0960.18004)]. Lambek had a long-time interest in the calculus of relations and its applications [Theoretical Linguistics 10, 227--245 (1983); Theoretical Linguistics 13, 19--36 (1986); Theoretical Linguistics 21, 241--253 (1995); \textit{J. Lambek}, Stud. Log. Comput. 2, 207--237 (1994; Zbl 0941.03518); Lond. Math. Soc. Lect. Note Ser. 222, 43--59 (1995; Zbl 0823.03038); J. Pure Appl. Algebra 116, No. 1--3, 221--248 (1997; Zbl 0870.18004); Synth. Libr. 280, 155--163 (1999; Zbl 0947.03049); Stud. Fuzziness Soft Comput. 65, 135--147 (2001; Zbl 1009.68172); Lect. Notes Pure Appl. Math. 180, 161--179 (1996; Zbl 0868.08002)]. \textit{Categorical grammars} were first introduced by \textit{K. Ajdukiewicz} [Stud. Philos. 1, 1--27 (1935; Zbl 0015.33702)], who traced back the idea to Lesniewski and Husserl. His idea of \textit{syntactic connection} inspired \textit{Y. Bar-Hillel} [Language and information. Selected essays on their theory and application. Jerusalem, Israel: The Jerusalem Academic Press, Ltd (1964; Zbl 0158.24102); in: Language and Information 61--74 (1964); Language 29, 47--58 (1953; Zbl 0156.25402)] to formalize the distinction between \textit{left-looking} and \textit{right-looking} categories. Lambek arrived at a similar idea while studying homological algebra with George Findlay, and introduced sucssh module operations as \(\oslash\)\ in [\textit{J. Lambek}, Lectures on rings and modules. Waltham/Mass.-London- Toronto: Blaisdell Publishing Company (1966; Zbl 0143.26403)]. By considering Church's type theory [\textit{A. Church}, J. Symb. Log. 5, 56--68 (1940; JFM 66.1192.06)], Lambek arrived at the idea of \textit{syntactic calculus}, which is a logical system with three binary operations \(\otimes\), \(/\) (over) and \(\backslash\) (under), resulting in a \textit{residuated semigroup}. Later, Lambek added an identity element to the syntactic calculus, the resulting ordered algebraic system becoming a \textit{residuated monoid}. In 1961, Lambek was invited to a symposium of the American Mathematical Society on the applications of mathematics to linguistics, where he discussed a non-associative version of the syntactic calculus, pointing out that Bourbaki's introduction of the tensor product was essentially a categorical version of a Gentzen style introduction. At the symposium Lambek learned of a parallel development by H. H. Curry, who had similarly explored a positive intuitionistic propositional logic for types in linguistics, which later gave rise to Montague semantics in view of what is called Curry-Howard isomorphism between proofs in intuitionistic logic and the lambda terms of Church. The period around 1980 witnessed a revival of interest of categorical grammars and particularly Lambek's syntactic calculus [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 32, 13--28 (1986; Zbl 0594.03015); Z. Math. Logik Grundlagen Math. 28, 539--548 (1982; Zbl 0499.03010)]]. 1n 1985 Lambek was present at a conference in Tuscon, Arizona, contributing with a paper concerning the connections between the syntactic calculus and category theory [``Categorial and categorical grammars'', in: Categorical Grammars and Natural Languages Structures. Reidel: Dordrecht. 297--317 (1988)]. In the following years a number of theoretical questions concerning the syntactic calculus were answered [\textit{M. Pentus}, J. Symb. Log. 62, No. 2, 648--660 (1997; Zbl 0882.68084)], while its connection to Montague semantics was observed in [\textit{G. V. Morrill}, Type logical grammar. Categorial logic of signs. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0848.03007)]. A new stream of ideas flooded the scene with \textit{J.-Y. Girard}'s [Theor. Comput. Sci. 50, 1--102 (1987; Zbl 0625.03037)], which differed from the syntactic calculus in generalizing classical rather than intuitionistic logic as well as in retaining Gentzen's structural rules. This led \textit{V. M. Abrusci} [Z. Math. Logik Grundlagen Math. 36, No. 4, 297--318 (1990; Zbl 0810.03005); J. Symb. Log. 56, No. 4, 1403--1451 (1991; Zbl 0746.03044)] and \textit{J. Lambek} [``From categorical logic to bilinear logic'', in: Substructural logics. Oxford. 207--237 (1993)] to study a classical version of the syntactic calculus, a noncommutative version of Girard's linear logic, which was named \textit{classical bilinear logic} by Lambek. In 1997, after a lecture on applications of classical bilinear logic to linguistics by Claudia Casadio at McGill University, Lambek arrived at the idea of \textit{compact bilinear logic}, in which the tensor product and its de Morgan dual are identifed. The calculus of \textit{pregroups} is the algebraic incarnation of compact bilinear logic, being based on the categorical tradition as well as being a natural generalization of partially ordered groups. The underlinkg in the pregroup analysis of a sentence can be viewed as degenerate instances of what linear logicians call \textit{proof nets}, representing the way the type of a phase or a sentence is obtained starting from the types of its constituents while being considered in representing the \textit{deep structure} of the context under analysis from a viewpoint of linguistics. The final dash after a string of types represents what Lambek called a Chomskyan trace, it turning out that \textit{double adjoints} occur wherever modern European languages would require traces [\textit{J. Lambek}, J. Logic Lang. Inf. 16, No. 3, 303--323 (2007; Zbl 1160.03305)]. \textit{J. Lambek} presented his algebraic approach to grammar via free groups at several conferences and in a number of papers written in coorporation with \textit{C. Casadio} and [Res. Lang. Comput. 3, No. 1, 45--60 (2006; Zbl 1083.68129); Stud. Log. 71, No. 3, 315--329 (2002; Zbl 1011.03016); Lect. Notes Comput. Sci. 2099, 110--124 (2001; Zbl 0990.03506)], \textit{A. Preller} [Math. Struct. Comput. Sci. 17, No. 2, 309--340 (2007; Zbl 1151.18007)], \textit{D. Bargelli} [Lect. Notes Comput. Sci. 2099, 62--78 (2001; Zbl 0990.03507)] and others. Theoretically, Buszkowski established cut-elimination for compact bilinear logic [\textit{W. Buszkowski}, Lect. Notes Comput. Sci. 10054, 68--84 (2016; Zbl 1480.03011)], showing that pregrout grammars are context-free [\textit{W. Buszkowski} and \textit{M. Farulewski}, Lect. Notes Comput. Sci. 5533, 45--58 (2009; Zbl 1239.03016); J. Comput. Syst. Sci. 78, No. 6, 1899--1909 (2012; Zbl 1250.68138)]. The volume consists of the following chapters with an introduction as Chapter 0 by Claudia Casadio and Philip Scott. \begin{itemize} \item[1.] Lambek's syntactic calculus and noncommutative variants of linear logic: Laws and proof-nets by Michele Abrusci and Claudia Casadio \item[2.] Sheaf representations and duality in logic by Steve Awodey \item[3.] On the naturalness of Maltsev categories by D. Bourn, M. Gran and P. A. Jacqmin \item[4.] Extensions of Lambek calculi by Wojciech Buszkowski \item[5.] Categories with families: Untyped, simply typed, and dependently types by Simon Castellan, Pierre Clairambault and Peter Dybjer \item[6.] The mathematics of text structure by Bob Coecke \item[7.] Aspects of categorical recursion theory by Pieter Hofstra and Philip Scott \item[8.] Morphisms of rings by Robert Paré \item[9.] Pomset logic: The other approach to noncommutativity in logic by Christian Retoré \item[10.] Pregroup grammars, their syntax and semantics by Mehrnoosh Sadzadeh \item[11.] The sequent calculi of skew monoidal categories by Tarmo Uustalu, Niccoloò Veltri and Noam Zeilberger \end{itemize} The content of each chapter goes as follows. \begin{itemize} \item[Chapter 1] is devoted to the relationship between Lambek's Syntactic Calculus (LC) and Cyclic Multiplicativ Linear Logic (CyMLL), proposing a geometric representation of a well-known laws of LC (Residuation, Monotomicity, Application, Expansion, Type-raising, Composition, Geach and Switching laws) by means of cyclic-multiplicative proof-nets (CyM-PN). The definitions and detailed representations of the CyM-PNs for each of the mentioned laws offer a better understanding of their internal structure, allowing for their classification into three main families of graphs. A detailed review of this chapter can be seen in [\textit{V. M. Abrusci} and \textit{C. Casadio}, Outst. Contrib. Log. 20, 1--37 (2021; Zbl 1495.03035)]. \item[Chapter 2] elegantly introduces the point of view, often noted by Lambek, of connections of this work with the grandted themes of Duality Theory such as Gelfand Duality in 20th-century mathematics. A detailed review of this chapter can be seen in [\textit{S. Awodey}, Outst. Contrib. Log. 20, 39--57 (2021; Zbl 07440903)]. \item[Chapter 3] surveys the history and current research on Mal'tsev categories, discussing the many reasons why such categories continue to play a fundamental role in contemporary research. A detailed review of this chapter can be seen in [\textit{D. Bourn} et al., Outst. Contrib. Log. 20, 59--104 (2021; Zbl 1482.18003)]. \item[Chapter 4] gives a comprehensive survey of Lambek calculi (associative and nonassociative) as formal logics, both as categorical grammars, aiming to emphasize the role of the Lambek calculus in the arena of nonclassical logics. A detailed review of this chapter can be seen in [\textit{W. Buszkowski}, Outst. Contrib. Log. 20, 105--134 (2021; Zbl 1500.03008)]. \item[Chapter 5] enhances a uniform modern treatment of a strong categorical equivalence between cartesian closed categories and typed lambda calculi, based upon the increasingly important notion of \textit{categories with families}, to various higher biequivalence theorems between Martin-Löf's dependent type theories and (by way of example) cartesian operads, Lawvere theories, and cartesian and locally cartesian closed categories. A detailed review of this chapter can be seen in [\textit{S. Castellan} et al., Outst. Contrib. Log. 20, 135--180 (2021; Zbl 1478.18001)]. \item[Chapter 6] aims to extend a mathematical foundation for sentence-meaning composition, in which word meanings interact to generate sentence meaning, to text-meaning composition, in which sentence meanings interact to generate possible text meanings. A detailed review of this chapter can be seen in [\textit{B. Coecke}, Outst. Contrib. Log. 20, 181--217 (2021; Zbl 1482.81006)]. \item[Chapter 7] examines three particular questions that have occupied Lambek for many years. \begin{itemize} \item[1.] Are there natural recursion theories? \item[2.] What are the computable functions and functionals in various concrete categorical structures? \item[3.] Are there intrinsic algebraic/categorical approaches to recursion theory? \end{itemize} Questions 1 and 2 were often asserted by Lambek that natural theories of computable numerical functions should correspond to numerical functions or functionals definable in various familiar \textit{free} categories, including free monoidal cartesina and cartesian closed categories, free topos, and so on. This is the very viewpoint of Part III in [\textit{J. Lambek} and \textit{P. J. Scott}, Introduction to higher order categorical logic. Cambridge: Cambridge University Press (1986; Zbl 0596.03002)], where standard logicians' techniques are employed to classify such functions. The chapter surveys the background and literature of Lambek's ideas concerning computability theory. Related to question 3, the authors address several categorical approaches to \textit{abstract} computability such as computation by normalization, realizability toposes, Turing categories, a co-algebraic approach to computable partial functions, categorical approaches to complexity theory, and so on. A detailed review of this chapter can be seen in [\textit{P. Hofstra} and \textit{P. Scott}, Outst. Contrib. Log. 20, 219--269 (2021; Zbl 1486.81060)]. \item[Chapter 8] discusses the seemingly naive question of what is a morphism of rings. As a surprising application of the notion of double categories [\textit{C. Ehresmann}, Ann. Sci. Éc. Norm. Supér. (3) 80, 349--426 (1963; Zbl 0128.02002)], the author shows that a natural class of morphisms is the \textit{amplimorphisms}, which were also discovered independently by mathematical physicists in quantum field theory. A detailed review of this chapter can be seen in [\textit{R. Paré}, Outst. Contrib. Log. 20, 271--298 (2021; Zbl 1490.18027)]. \item[Chapter 9] gives an up-to-date and detailed analysis of the author's own approach to noncommutativity in logic called \textit{pomset logic}, a variant of classical linear logic motivated by Girard's coherence semantics for linear logic. The author describes the fine details of his approach versus Lambek's, Lambek's calculus being basically a noncommutaive \textit{restriction} of intuitionistic linear logic while the author's one being basically a noncommutative \textit{extension} of classical linear logic. The author addresses sequentialization and the quest for a complete sequent calculus and the quest for a complete sequent calculus for his system, providing a surprising example of a proof net that does not derive from any simple sequent calculus. The author finishes with applications of pomset logic to linguistics. A detailed review of this chapter can be seen in [\textit{C. Retoré}, Outst. Contrib. Log. 20, 299--345 (2021; Zbl 1495.03038)]. \item[Chapter 10] studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the chapter presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about. A detailed review of this chapter can be seen in [\textit{M. Sadrzadeh}, Outst. Contrib. Log. 20, 347--376 (2021; Zbl 1491.68252)]. \item[Chapter 11] presents the proof theory of skew monoidal categories as an \ instance of a Lambek-style analysis of proofs, solving the coherence problem. The authors construct free skew monoidal categories putting an appropriate Gentzen sequent calculus to use. The techniques they exploit involve use of machinery inspired by linear logic proof search, based on the technique of \textit{focusing}, having a distinguished \textit{stoup} position in the antecedents of sequents and adaption of sequent calculus rules of inference to accommodate stoups. The resultant calculus is shown to be sound and complete with respect to the existence of maps in the free skew monoidal category. By setting up an appropriate equivalence relation on proofs and associated rewriting machinery, the authors can pick canonical representatives of each equivalence class of proofs, solving the coherence problem. Finally, the authors compare their work with [\textit{J. Bourke}, J. Homotopy Relat. Struct. 12, No. 1, 31--81 (2017; Zbl 1417.18001); \textit{J. Bourke} and \textit{S. Lack}, J. Pure Appl. Algebra 222, No. 10, 3255--3281 (2018; Zbl 1428.18025); J. Algebra 506, 237--266 (2018; Zbl 1401.18019)]. A detailed review of this chapter can be seen in [\textit{T. Uustalu} et al., Outst. Contrib. Log. 20, 377--406 (2021; Zbl 1495.03073)]. \end{itemize} This research examines the combined influence of three causal mechanisms shaping interpersonal dynamics. Sharing a demographic characteristic has been shown to strengthen a relationship between people. The initial explanation for this positive effect emphasized the importance of individual preferences for social contact. Similar people share common interests, which makes them more attractive as exchange partners. More recent explanations have highlighted the important role of identification and competition. Identification increases the tendency for strong ties to develop among similar people, while competition reduces that tendency. I argue that tie strength is a function of all three mechanisms: a baseline level of interpersonal attraction, a positive identification effect, and a negative competition adjustment. Identification and competition vary with the number of people sharing the focal attribute to define a predictable association between sharing the focal attribute and the strength of a network connection. Predictions are tested with tenure and tie-strength data from a small research and development firm. Analysis indicates that having the same tenure has the predicted effect on communication frequency. The implications of the findings are discussed for homophily research in particular and demography research in general.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Der Verfasser beweist in der vorliegenden Abhandlung den folgenden, für die Theorie der Gleichungen principiell wichtigen Satz: ``Wenn \(F\) eine irreducible ganze ganzzahlige Function der Variabeln \(x, y, \ldots, w, t, r, \ldots, q\) bezeichnet, so ist es stets auf unendlich viele Weisen möglich, für die Variabeln \(t, r, \ldots, q\) ganze rationale Zahlen einzusetzen, so dass dadurch \(F\) in eine irreducible Function von \(x, y, \ldots, w\) übergeht''. Der Beweis wird zunächst für den besonderen Fall durchgeführt, dass \(F\) nur von zwei Variabeln \(x\) und \(t\) abhängt, und sodann wird der allgemeine Fall auf diesen besonderen zurückgeführt. Die wesentlichen Mittel für den Beweis bilden einerseits die Puiseuxschen Entwickelungen, andererseits ein Hülfssatz, der seines interessanten, eigentümlichen Charakters wegen hier erwähnt werden möge. Derselbe lautet: Wenn jedes Element \(a_s\) der unbegrenzten Reihe \(a_1, a_2, a_3, \ldots\) einen der Werte \(1, 2, 3, \ldots, a\) besitzt, wenn ferner \(m\) irgend eine ganze positive Zahl bezeichnet, so lassen sich die positiven ganzen Zahlen \(\mu^{(1)}, \mu^{(2)},\ldots, \mu^{(m)}\) stets so bestimmen, dass die \(2^{\text{m}}\) Elemente \[ a_{\mu + \varepsilon^{(1)} \mu^{(1)} + \varepsilon^{(2)} \mu^{(2)} + \cdots + \varepsilon^{(m)} \mu^{(m)}} \quad \quad (\varepsilon^{(i)} = 0, 1; \; i = 1, 2, \ldots, m) \] für unendlich viele Indices \(\mu\) gleich der nämlichen Zahl (aus der Reihe \(1, 2, \ldots, a\)) sind. Nachdem der Verfasser den Satz auf solche ganze rationale Functionen ausgedehnt hat, deren Coefficienten einem beliebigen algebraischen Zahlenkörper angehören (wobei die Irreducibilität eben auf diesen Zahlenkörper zu beziehen ist), geht er dazu über, die gewonnenen Resultate auf die Theorie der Gleichungen anzuwenden. Von den sich hierbei ergebenden Sätzen mögen beispielsweise die folgenden herausgehoben werden, deren principielle Bedeutung unmittelbar einleuchtet: ``Es giebt unbegrenzt viele Gleichungen \(n^{\text{ten}}\) Grades mit ganzzahligen Coefficienten, deren Gruppe im Bereiche der rationalen Zahlen die symmetrische Gruppe ist''. Der entsprechende Satz wird für die alternirende Gruppe nachgewiesen. Ferner: ``Es giebt unbegrenzt viele Bereiche von bestimmtem Grade \(n\), in denen (abgesehen vom Bereiche aller rationalen Zahlen) kein Bereich niederen Grades enthalten ist''. Schliesslich bemerkt der Verfasser noch, dass mit den in der vorliegenden Abhandlung benutzten Hülfsmitteln auch der folgende Satz bewiesen werden kann: ``Wenn eine algebraische Function von \(t\) für alle rationalen, in einem beliebig kleinen Intervalle gelegenen Werte stets selber rationale Werte annimmt, so ist sie notwendig eine rationale Function''.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. A multi-level adaptive numerical technique is applied to a nonlinear formulation of the mild-slope equation, to obtain the nearshore wave field, where the dominant processes of wave transformation are shoaling, refraction and diffraction. The advantage of this formulation over the traditional elliptic, parabolic and hyperbolic formulations is to require a lower minimum number of grid nodes per wavelength, thus, its capacity to predict the wave field for larger coastal areas. The efficiency of the interactions between the grid mesh levels, where two robust Krylov subspace iterative methods, the Bi-CGSTAB and the GMRES, are applied to solve the governing equation, is tested, for several hierarchies of grid mesh levels. The results show that the multi-level adaptive technique is efficient only if the GMRES iterative method is applied, and that for six grid mesh levels good results can be achieved for a residual as low as \(10^{-3}\) for the finest grid.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Since the original works of A. Robinson many approaches to nonstandard analysis were elaborated. The authors investigate several different constructions in algebra, set theory, and topology where ``hyper-methods'' are used. Nonstandard extensions naturally arise in several quite different domains of mathematics. The article consists of two parts. The first part contains an introduction to the basic notions of nonstandard analysis. The three fundamental tools are presented here, namely, the star-map, the transfer principle and the saturation property, and several examples illustrate their use. The material is presented in an elementary and semi-formal way, so it may serve as a quick presentation of nonstandard analysis for beginners. Then the authors discuss connections between the hyper-methods of nonstandard analysis and ultrapower techniques. The second part presents a survey of the authors' contribution to nonstandard analysis during the last decade. Eight different paths to nonstandard analysis are presented here: superstructure approach, a purely algebraic approach, so-called Alpha Theory, and so on. These approaches are not strictly equivalent to each other, but all of them explicitly or implicitly include the fundamental hyper-tools introduced in the first part. The authors stick to the so-called external viewpoint of nonstandard methods, based on the existence of a star-map providing a hyper-extension \(^*A\) for each standard object \(A\).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of \textit{J.-Y. Cai} [``Parallel computation over hyperbolic groups'', Proc. STOC 92, 106-115 (1992)].

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We provide primitive recursive bounds for the finite version of Gowers' \(c_{0}\) theorem for both the positive and the general case. We also provide multidimensional versions of these results.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. A regression model \(Y=\beta X+\varepsilon\) is considered where the CDF \(F_0\) of the error term \(\varepsilon\) is completely unknown. The response \(Y\) is not observed, but a random interval \((L_j,R_j)\) to which \(Y_j\) belongs is known. (The mixed case interval censorship model is in force.) The authors describe a semi-parametric maximum likelihood estimator for \((\beta,F_0)\) and demonstrate its consistency.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. By using nonstandard analysis, we prove embeddability properties of differences \(A-B\) of sets of integers. (\(A\) set \(A\) is ``embeddable'' into \(B\) if every finite configuration of \(A\) has shifted copies in \(B\).) As corollaries of our main theorem, we obtain improvements of results by I. Z. Ruzsa about intersections of difference sets, and of \textit{R. Jin}'s theorem [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)] (as refined by \textit{V. Bergelson} et al. [in: Topics in discrete mathematics. Dedicated to Jarik Nešetřil on the occasion of his 60th birthday. Berlin: Springer. 13--37 (2006; Zbl 1114.37008)]), where a precise bound is given on the number of shifts of \(A-B\) which are needed to cover arbitrarily large intervals.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We consider the \(k\)-traveling repairman problem, a generalization of the metric traveling repairman problem, also known as the minimum latency problem, to multiple repairmen. We give an \(8.497\alpha\)-approximation algorithm for this generalization, where \(\alpha\) denotes the best achievable approximation factor for the problem of finding the least cost rooted tree spanning \(i\) vertices (\(i\)-MST) problem. This can be compared with the best known approximation algorithm for the case \(k = 1\), which is \(3.59\alpha\). We are aware of no previous work on the approximability of the present problem.In addition, we give a simple proof of the \(3.59\alpha\) approximation result which can be extended to the case of multiple repairmen.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Ein Dicksonsches Theorem [\textit{L. E. Dickson}, J. Reine Angew. Math. 135, 134--141 (1908; JFM 39.0260.02); ibid. 135, 181--188 (1909; JFM 40.0254.04)] wird hier elementar und äußerst einfach hergeleitet. Der Beweis stützt sich auf folgenden Hilfssatz: Verteilt man die Zahlen \(1, 2, \ldots, N\) irgendwie auf \(m\) Zeilen, so müssen, sobald \(N>m!e\) wird, in mindestens einer Zeile zwei Zahlen vorkommen, deren Differenz in derselben Zeile enthalten ist.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Kinetische und potentielle Energie werden durch Entwicklung der Gestalt des Seils in eine \textit{Fourier}reihe dargestellt und demnach die Bewegung bzw. wechselnde Belastung durch eine Reihe von Sinuslinien.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Teilt man die ganzen Zahlen \(1, 2, \ldots, N\) in \(k\) Klassen ein, so enthält für jedes \(l\) mindestens eine der Klassen eine arithmetische Progression von \(l\) Gliedern, wenn nur \(N\) oberhalb einer angebbaren, nur von \(k\) und \(l\) abhängenden Schranke liegt. Der Beweis wird rein kombinatorisch geführt.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. In this paper, we studied the approximate scattering state solutions of the Dirac equation with the hyperbolical potential with pseudospin and spin symmetries. By applying an improved Greene-Aldrich approximation scheme within the formalism of functional analytical method, we obtained the spin-orbit quantum numbers dependent scattering phase shifts for the spin and pseudospin symmetries. The normalization constants, lower and upper radial spinor for the two symmetries, and the relativistic energy spectra were presented. Our results reveal that both the symmetry constants (\(C_{\mathrm{ps}}\) and \(C_{\mathrm{s}}\)) and the spin-orbit quantum number \(\kappa\) affect scattering phase shifts significantly.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We prove a generalization of Gowers' theorem for \(\operatorname{FIN}_k\) where, instead of the single tetris operation \(T : \operatorname{FIN}_k \rightarrow \operatorname{FIN}_{k - 1}\), one considers all maps from \(\operatorname{FIN}_k\) to \(\operatorname{FIN}_j\) for \(0 \leq j \leq k\) arising from nondecreasing surjections \(f : \{0, 1, \ldots, k \} \rightarrow \{0, 1, \ldots, j \}\). This answers a question of \textit{D. Bartošová} and \textit{A. Kwiatkowska} [``The universal minimal flow of the automorphism group of the Lelek fan'', Preprint]. We also describe how to prove a common generalization of such a result and the Galvin-Glazer-Hindman theorem on finite products, in the setting of layered partial semigroups introduced by \textit{I. Farah} et al. [ibid. 98, No. 2, 268--311 (2002; Zbl 0991.05100)].

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The multi-choice goal programming allows the decision maker to set multi-choice aspiration levels for each goal to avoid underestimation of the decision. In this paper, we propose an alternative multi-choice goal programming formulation based on the conic scalarizing function with three contributions: (1) the alternative formulation allows the decision maker to set multi-choice aspiration levels for each goal to obtain an efficient solution in the global region, (2) the proposed formulation reduces auxiliary constraints and additional variables, and (3) the proposed model guarantees to obtain a properly efficient (in the sense of Benson) point. Finally, to demonstrate the usefulness of the proposed formulation, illustrative examples and test problems are included.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Let \(A\) be an infinite subset of the set of natural numbers \(\mathbb N\), denote \(A(x,y)=\# A\cap[x,y]\) and put \(A(x)=A(1,x)\). After basic approach, in the main part of this expository paper the author illustrates nonstandard methods deriving a theorem about upper Banach density from an existing theorem about Shnirel'man density or lower asymptotic density. Here Shnirel'man density \(\sigma(A)=\inf_{x\geq1}\frac{A(x)}{x}\), lower asymptotic density \(\underline{d}(A)=\liminf_{x\to\infty}\frac{A(x)}{x}\), and upper Banach density \(BD(A)=\lim_{x\to\infty}\sup_{k\in\mathbb N}\frac{A(k,k+x)}{x+1}\). The author starts with Mann's theorem which says that if \(0\in A\cap B\), then \(\sigma(A+B)\geq\sigma(A)+\sigma(B)\). \(BD\)-version is \(BD(A+B+\{0,1\})\geq BD(A)+BD(B)\). It follows Plünnecke's theorem which says that if \(B\) is a basis of order \(h\) and \(A\subset\mathbb N\), then \(\sigma(A+B)\geq\sigma(A)^{1-\frac{1}{h}}\). In the \(BD\)-version \(BD(A+B)\geq BD(A)^{1-\frac{1}{h}}\). The author also quotes Kneser's theorem which is saying that if \(\underline{d}(A+B)<\underline{d}(A)+\underline{d}(B)\), then there are integer \(g>0\) and \(G\subset[0,g-1]\) such that (i) \(\underline{d}(A+B)\geq\underline{d}(A)+\underline{d}(B)-\frac{1}{g}\), (ii) \(A+B\subset G+g\mathbb N\), (iii) \(G+g\mathbb N\setminus(A+B)\) is finite. A weaker \(BD\)-version was proved by \textit{P. Bihani} and the author in [J. Théor. Nombres Bordx. 18, No. 2, 323--343 (2006; Zbl 1138.11044)] and the author proved a full \(BD\)-version in URL [http://www.cofc.edu/math/jin/research/publication.html]. He concludes the paper with Freiman's little theorem and conjecture and he gives a briefs review of recent results. All proofs of this paper are informative and gives only a sketch of the nonstandard ideas.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. This note reexamines the single-profile approach to social-choice theory. If an alternative is interpreted as a social state of affairs or a history of the world, it can be argued that a multi-profile approach is inappropriate because the information profile is determined by the set of alternatives. However, single-profile approaches are criticized because of the limitations they impose on the possibility of formulating properties such as anonymity. We suggest an alternative definition of anonymity that applies in a single-profile setting and characterize anonymous single-profile welfarism under a richness assumption.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Let \(\{T_i\}_{i=1}^m\) be a collection of commuting, invertible measure preserving transformations. Recently T. Tao has proved the convergence in \(L^2\) of multiple ergodic averages \(\frac1n\sum_{k=1}^n \prod_{i=1}^m f_i(T_i^k x)\) for \(L^\infty\) test-function \(f_i\). Tao's argument translates the problem into a statement about the combinatorics of finite spaces. The present paper gives an ergodic proof of this theorem.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. In the past two decades, one of the main endeavours in nonlinear functional analysis was focused on minimax methods for semilinear differential equations. The roots of this machinery are the Lyusternik- Schnirelman and the Morse theory. To the R. Palais and S. Smale extensions for mappings on Banach manifolds, given in the 60s, there have been added in the 70s a dual least action principle by D. C. Clark, the A. Ambrosetti and P. H. Rabinowitz dual variational methods and later I. Ekeland's variational principle. The subject received a tremendous impetus in the last decade from the development of a general theory of periodic solutions of Hamiltonian systems \(J\dot u(t)+\nabla H(t,u(t))=0\). The book is an outgrowth of the authors' outstanding contributions and is organized in ten chapters. After prerequisites of the convex analysis and periodic solutions, the Fenchel transform and Clarke duality in the first two chapters, the authors consider the circumstances where \(H(t,\cdot)\) is convex. In this case the dual least action principle of Chapter 3 seems to provide the best results in the simplest way. In the absence of convexity, the existence of critical point of the saddle point type can be proved by using minimax techniques. For indefinite functionals, Chapter 4 describes the mountain pass lemma, which is derived from Ekeland's variational principle. Based on the Borsuk-Ulam theorem, the index is introduced as a ``size'' measure of some sets, symmetric with respect to the origin. The Lyusternik- Schnirelman theory and corresponding equivariant deformations are presented in Chapter 6. This allows a local description of periodic orbits with prescribed energy. In Chapter 7 one finds the Morse-Ekeland index for linear positive definite and convex asymptotically linear autonomous Hamiltonian systems. A more difficult part of the critical point theory is its connection to algebraic topology concepts. The properties of the critical points of a smooth function \(f\) on a manifold \(M\) are related to the topology of \(M\). The Morse theory derives homotopy properties of \(M\) from an analysis of the second derivative of \(f\) at its critical points. Chapter 8 deals with critical groups at a point of mountain pass type or a saddle point and connections with bifurcation phenomena. Applications of the Morse theory are made to asymptotically linear nonautonomous systems. The final chapter is devoted to nondegenerate critical manifolds and periodic solutions of forced superlinear equations. The volume concludes with a rich bibliography of more than 450 titles, most of them from the 80s. Each chapter begins with a heuristic presentation of the basic ideas in the topic and is followed by reference notes and well chosen problems that are invitations to deeper investigations. Although, there are several books in the subject as those of \textit{P. H. Rabinowitz} [Minimax methods in critical point theory with applications to differential equations. Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] or \textit{K. C. Chang} [Infinite dimensional Morse theory and its applications. NATO Advanced Study Institute, 97. Département de Mathématiques et de Statistique, Université de Montréal. Montréal (Québec), Canada: Les Presses de l'Université de Montréal (1985; Zbl 0609.58001)], the clear and progressive exposition of the book under review should be fully appreciated. The tools can provide valuable insight that will stimulate our understanding of analytic mechanics. The book constitutes a self-contained and comprehensive introduction in a top research field and it can also be an excellent text for an up-to-date graduate course.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Ein System \(F\) von Teilmengen einer Menge \(E\) heißt nach Verf. ein Filter, wenn 1) \(F\) nicht leer ist und die leere Menge nicht zu \(F\) gehört; 2) der Durchschnitt je zweier Mengen von \(F\) und 3) jede Obermenge einer Menge von \(F\) wieder zu \(F\) gehört. Den Filtern weist Verf. die Rolle von Umgebungssystemen zu. Beispiel: Bei einer abzählbaren Folge unendlich vieler Elemente bilden die Teilmengen, die aus der Folge durch Streichung endlichvieler Elemente entstehen, ein Filter, ein Elementarfilter (welches als Umgebungssystem eines durch die Folge erklärten idealen Elementes anzusehen ist). (An den Axiomen 2) und 3) ist nichts Neuartiges, sie sind völlig identisch mit dem \textit{Hausdorff}schen Axiom (B) in \textit{Tietze}schen Fassung (Math. Ann. 88 (1923), 290-312 (F. d. M. 49, 397 (JFM 49.0397.*)), insbesondere S. 295), d. Ref.) Gilt für zwei Filter \(F_1, F_2\) die Beziehung \(F_1 \subset F_2\), so heißt \(F_2\) feiner als \(F_1\). Mittels dieses Begriffes läßt sich ein dem Diagonalverfahren bei Folgen entsprechender Satz über Elementarfilter formulieren. Ist jedem Punkt einer Menge \(E\) ein Filter zugeordnet, so wird \(E\) zu einem Raum; es werden dabei zusätzliche Forderungen gestellt, die den \textit{Hausdorff}schen Axiomen (C) und (D) entsprechen. Man kann dann jedem Filter ``Häufungspunkte'' zuordnen, Kompaktheit definieren usw. Verf. betrachtet auch sogenannte privilegierte Familien von Filtern, d. h. solche Filterfamilien, die mit jedem Filter auch jedes dazu feinere, ferner zu jedem beliebigen Filter ein dazu feineres enthalten.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. [For part I, see the authors and \textit{A. Chandna}, ibid. 6, No. 6, 437- 445 (1990; Zbl 0699.76037).] The pure inverse problem of designing a body with a prescribed surface pressure in incompressible axisymmetric potential flow is solved using the streamfunction as a coordinate in the computational plane. The problem is formulated mathematically as a mixed boundary value problem and the solution is obtained using the finite- difference technique. The body shape is computed simultaneously with the entire flow field and does not have to be iteratively adjusted to meet the target surface pressure. Results are presented for elliptical, NACA 0012 and minimum-drag cross-sections.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. An equation $f(x_1, x_2, \dots, x_n)=0$ is called partition regular on the set $\mathbb{N}$ of natural numbers, if for any partition $\mathbb{N} = C_1\cup \dots \cup C_r$, there exists a (monochromatic) solution $f(a_1, \dots, a_n) = 0$, with $a_1, \dots, a_n \in C_i$, for some $i\in\{1, \dots, r\}$.\par Two of the main results identify conditions under which the equation: $$\sum_{l=1}^h\left(\sum_{i=1}^{s_l} a_{l,i}x^{n_l}_{l,i}\right) = 0$$ (with $a_{l,i}$ non-zero integers and $n_1 < \dots <n_h$), is not partition regular. Similar conditions of non-partition regularity are identified for equations of type: $$\sum_{i=1}^s a_ix_i^n = y^{n+1}$$ where $a_i$ are non-zero integers and $n \in \mathbb{N}$.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Let \((\Omega, \Sigma, \mu)\) be a measure space, \(E\) be a Banach space and \(L^ \infty (\mu,E)\) be the set of all essentially bounded \(\mu\)- measurable \(E\)-valued functions on \(\Omega\) (two functions are identified here if they differ only on a set \(A \in \Sigma\) for which \(\mu (A \cap B) = 0\) for all \(B \in \Sigma\) with \(\mu (B) < \infty)\). It is shown that if \(L^ \infty (\mu,E)\) contains a complemented copy of \(c_ 0\), then \(E\) contains a copy of \(c_ 0\). At this, if \((\Omega, \Sigma, \mu)\) is a \(\sigma\)-finite purely atomic (not purely atomic) measure space, then \(L^ \infty (\mu,E)\) contains a complemented copy of \(c_ 0\) if and only if \(E\) contains a complemented copy (respectively a copy) of \(c_ 0\).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The author investigates the \(\omega_ 1\)-saturated model \({}^*{\mathbb{N}}\). For an internal subset A of \({}^*{\mathbb{N}}\) and an interval I he denotes by \(g_ A(I)\) the ratio of the length of the maximal gap of A to the length of I. Let \(y,z\in^*{\mathbb{N}}\) with \((z- y)\in^*{\mathbb{N}}\setminus {\mathbb{N}}\). Consider the image of A under a linear mapping of [y,z] onto [0,1] and denote by \(st_{[y,z]}(A)\) the set of standard parts of elements of the image. A is said to have the IM property on [y,z] iff for every real standard \(\beta >0\) there is an \(\alpha >0\) such that [u,v]\(\subseteq [y,z]\), \((v- u)\in^*{\mathbb{N}}\setminus {\mathbb{N}}\) and \(g_ A([u,v])\leq \alpha\) implies \(\mu (st_{[u,v]}(A))\geq 1-\beta\) (\(\mu\) is the Lebesgue measure). For an infinite sequence A of natural numbers the SIM (for standard internal measure) property is defined. Roughly speaking it indicates that large gaps of A are rare. A combinatorial theorem involving the notion of an IM set is proved by using the Lebesgue density theorem. From this result a standard theorem is obtained: Let \(A=<a_ n>\) be a sequence of natural numbers such that \(\lim (a_{n+1}-a_ n)=\infty\). Then A does not have the SIM property. Denote by D(A) the set of such n that \(n=a_ i-a_ j\) for infinitely many i, j. If \(A_ i\) has the SIM property for \(i=1,...,n\) then \(D(A_ 1)\cap...\cap D(A_ n)\) has bounded gaps.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Der Gang dieses anschaulich gestalteten Berichts ist folgender: Es wird zunächst die spezielle Raumkurve dritter Ordnung \(\xi = ut\), \(\eta = vt^2\), \(\zeta =wt^3\) (\(u\), \(v\), \(w=\) const, \(t =\) Parameter) betrachtet. Sie liegt auf dem parabolischen Zylinder \(\xi ^2=\dfrac {u^2}{v}\,\eta \). Die unendlich ferne Ebene ist ebenso wie die \((\xi,\eta )\)-Ebene Oskulationsebene. Berechnet werden Tangente, Binormale, Krümmung \(\dfrac {1}{r}\) und Windung \(\dfrac {1}{\varrho }\). Dann wird gefragt: Gibt es Raumkurven dritter Ordnung mit obigen Gleichungen, für die \(r : \varrho =\) const? Antwort: Ja. \ Z. B.: \(\xi = t\), \(\eta = vt^2\), \(\zeta =\dfrac {2v^2}{3}\,t^3\). Anderseits sind nach einem Satz von \textit{Lancret-Bertrand} Kurven, für die \(r : \varrho =\) const, Schraubenlinien eines allgemeinen Zylinders, d. h. isogonale Trajektorien der Erzeugenden eines solchen Zylinders. Dieser rektifizierende Zylinder wird für ein Beispiel \(\Bigl(\xi = t\), \(\eta = \dfrac {1}{4}\,t^2\), \(\varrho =\dfrac {1}{24}\,t^3\Bigr)\) bestimmt: Die Richtung der Zylindererzeugenden ist parallel zur Winkelhalbierenden der positiven \((\xi, \zeta )\)-Richtungen. Der Zylinder ist von dritter Ordnung. Nun werden weitere Beispiele solcher Raumkurven mit einem rektifizierenden Zylinder, also von allgemeinen Schraubenlinien, angegeben und ausführlich durchgerechnet. Photographien von Modellen erläutern die gewonnenen Sätze der klassischen Theorie. (V 4.)

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Recently \textit{B. Green} and \textit{T. Tao} proved that the primes contain arbitrarily long arithmetic progressions [``The primes contain arbitrarily long arithmetic progressions. Preprint \url{http://arxiv.org/abs/math.NT/0404188}]. In 1936 \textit{P. Erdős} and \textit{P. Túran} conjectured that if \(\sum_{k=1}^{\infty}\frac{1}{a_k}\) diverges, where \(a_k\in{\mathbb N}\), then the sequence \((a_k)\) contains arbitrarily long arithmetic progressions [``On some sequences of integers'', J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203 and JFM 62.1126.01)]. This conjecture still remains unresolved. In the paper under review the author uses methods of non-standard analysis to investigate conditions that imply that a sequence \(A\) contains intervals in arithmetic progression on which \(A\) is nonempty, and on which various density properties hold. The following elegant lemma is proved by standard methods. If \(E\) is a subset of the reals with positive Lebesgue measure and \(t\) is any natural number, then there exists a real number \(s>0\) such that for all \(d<s,d>0,\) \(E\) contains a \(t\)-term arithmetic progression with difference \(d.\) The author introduces the notion of a \textit{cell} with 4 parameters. These cells are ``near'' arithmetic progressions in some sense. The main result of the paper is Theorem 1. It gives a condition for the existence of ``near'' arithmetic progressions for any sequence on any interval \(I\) in which the density does not drastically increase as the size of the subinterval decreases. The proof of the theorem uses a nonstandard model which involves deep results from analysis and measure theory.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Die Resultate des Herrn Jürgens (Borchardt J. LXXX. p. 150, siehe F. d. M. VII. 180, JFM 07.0180.01) betreffs der Art, wie den Gliedern eines in bestimmten Gruppen geordneten, zu einem singulären Punkt gehörigen Fundamentalsystems von Integralen einer gegebenen linearen Differentialgleichung die Glieder eines ebenso geformten Fundamentalsystems von Integralen der Multiplicatorgleichung der ersteren zugeordnet werden können, werden auf eine einfachere Weise abgeleitet. Daran schliesst sich eine Anwendung seiner Methode auf den Beweis einiger Sätze über reductible Differentialgleichungen, die Herr Frobenius in Borchardt J. LXXVI. (vgl. F. d. M. V. p. 176, JFM 05.0176.01) entwickelt hat, insbesondere des gleichzeitig von Herrn Frobenius und Thomé gefundenen Satzes: Wenn eine lineare Differentialgleichung \(n^{\text{ter}}\) Ordnung mit einer \(m^{\text{ter}}\) Ordnung alle Integrale gemeinsam hat, so die Gleichung, der ihr Multiplicator genügt, mit einer Differentialgleichung \(n-m^{\text{ter}}\) Ordnung alle Integrale gemeinsam. Schliesslich sei noch bemerkt, dass die Eintheilung der zu gleichen Wurzeln der Fundamentalgleichung bezüglich eines singulären Punktes gehörigen Gruppen von Integralen in Untergruppen, welche vom Herrn Verfasser (p.144) irrthümlich Herrn Jürgens zugeschriebern wird, sich bereits in der Abhandlung ''Bemerkung über die Form der Integrale etc.'' Borchardt J. LXXVI. p. 113 entwickelt findet.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. In recent years there is huge activity on how to generalize Szemerédi's Regularity and Removal lemmas for hypergraphs. There are different approaches, different results, even different definitions. (See for example [\textit{V. Rödl} and \textit{J. Skokan}, Random Struct. Algorithms 25, No. 1, 1--42 (2004; Zbl 1046.05042), \textit{B. Nagle}, \textit{V. Rödl}, and \textit{M. Schacht}, ibid. 28, 113--179 (2006; Zbl 1093.05045); \textit{T. Tao}, J. Comb. Theory, Ser. A 113, 1257--1280 (2006; Zbl 1105.05052 ); \textit{G. Elek} and \textit{B. Szegedy}, Limits of hypergraphs, regularity and removal lemmas. A non-standard approach. \url{http://arxiv1.library.cornell.edu/abs/0705.2179v1}]). The paper under review develops its own versions of the Regularity and Removal lemmas for hypergraphs, and as application it gives the first combinatorial proofs for the multidimensional Szemerédi theorem (and also providing an explicit bound).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The problem on canonical representation of \(\delta\)-subharmonic in \(\mathbb R^m\) \((m\geq 2)\) functions of finite \(\lambda\)-type is solved. It is proved that this problem reduces to the study of growth majorant properties of subharmonic functions with restriction on their Riesz measures.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Problems are presented in the following areas: the theorem of van der Waerden and Szemerédi; covering congruences, additive number theory, dense sets of integers, infinite subsets of integers (related to the work of Hindmann), sieve methods and other miscellaneous topics. An attempt is made to describe what has happened to problems mentioned in previous surveys. Some old conjectures still remain open, for example the following one of 45 years standing: if \(1\leq a_1< \dots< a_k\leq x\) is a sequence of integers such that the sums \(\sum_1^k\varepsilon_ia_i\), \(\varepsilon_i=0\text{or}1\), are all different, then \(\max k=\frac{\log x}{\log 2}+0(1)\). 500 dollars is coffered for its solution. A larger survey due to the author and R. L. Graham can be found in ''Old and new problems and results in combinatorial number theory'' [L'Enseignment Math., Monographie 28 (1980; Zbl 0434.10001)].

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The authors present proofs (the first one is an alternative) for the convergence of collocation methods for both an integral equation of the first kind on a 2D rectangle and for a hypersingular pseudo-differential equation corresponding to the Neumann problem for the Laplace equation in 3D. The results are deep and come from elaborate and elegant proofs.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. A set \(A\) of natural numbers is finitely embeddable in another such set \(B\) if every finite subset of \(A\) has a rightward translate that is a subset of \(B\). This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. [For the entire collection see Zbl 0746.00080.] The author mainly gives a summary of Papakyriakopoulos's work on the Poincaré Conjecture. He discusses Papakyriakopoulos's conjectures and their subsequent improvements by Swarup, McCool, and Moran.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We give a constructive proof of the finite version of Gowers' FIN\(_k\) theorem for both the positive and the general case and analyse the corresponding upper bounds provided by the proofs.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. In recent years there is huge activity on how to generalize Szemerédi's Regularity and Removal lemmas for hypergraphs. There are different approaches, different results, even different definitions. (See for example [\textit{V. Rödl} and \textit{J. Skokan}, Random Struct. Algorithms 25, No. 1, 1--42 (2004; Zbl 1046.05042), \textit{B. Nagle}, \textit{V. Rödl}, and \textit{M. Schacht}, Random Struct. Algorithms 28, 113--179 (2006; Zbl 1093.05045), \textit{W.T. Gowers}, Ann. Math. (2) 166, 897--946 (2007; Zbl 1159.05052), and \textit{G. Elek} and \textit{B. Szegedy}, Limits of hypergraphs, removal and regularity lemmas. A nonstandard approach. \url{http://arxiv1.library.cornell.edu/abs/0705.2179v1}]) The paper under review developed an infinitary analogous to the Furstenberg Correspondence Principle, which gives a new proof of the Removal Lemma, and which requires no Regularity Lemma. Then the paper discusses several applications of the method.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. This paper proposes regression estimators based on a partial replacement sampling scheme over two successive occasions and derives the minimum variance of them. PP-SWR, RHC, \(\pi\)PS and PPSWOR schemes are considered to select unequal probability samples on two occasions. Simulation results over four populations are given for comparison of composite estimators and regression estimators.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The author surveys some recent results in additive number theory (mainly Banach density) using the tools of nonstandard analysis. His main concern is to show how the nonstandard techniques help in obtaining new and easier proofs of known results and in proving new ones. Proofs will appear elsewhere.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We present a powerful technique for obtaining natural deduction proof systems for first-order fixpoint logics. The term `fixpoint logics' refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with, e.g., dynamic logic, temporal logic and the \(\mu\)-calculus among them. In this paper we present a technique that allows us to derive automatically natural deduction systems for modal logics from fixpoint equations defining the modalities.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. It is shown that if \(f\) is a Lipschitz function defined on the unit sphere of \(c_ 0\), then for every positive \(\varepsilon\) there is an infinite dimensional subspace of \(c_ 0\) on whose unit sphere \(f\) differs by less than \(\varepsilon\). The theorem is related to the open question if every space isomorphic to \(\ell_ p\) (for \(1<p<\infty\)) contains a subspace that is \((1+\varepsilon)\) isomorphic to \(\ell_ p\).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Following \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Can. J. Math. 57, No. 3, 494--505 (2005; Zbl 1160.11340)], the objective of this note are the coefficients \(a_n\) of the Dirichlet series for \(L(s,\chi_1)L(s,\chi_2) L(s,\chi_3)\) where \(\chi_1, \chi_2, \chi_3\) are primitive Dirichlet characters with modules \(D_1,D_2,D_3\). For \(\sum_{n\leq x}a_n\), with \(x\) large, sharp asymptotics are established which are uniform in \(D_1,D_2,D_3\). To this end, the modern method for the estimation of exponential sums, due to \textit{M. N. Huxley} [Area, lattice points, and exponential sums. Oxford: Clarendon Press (1996; Zbl 0861.11002)] and others, is applied with gain.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Using minimal idempotent ultrafilters a Ramsey theorem is proved for decompositions of words with a variable, extending Carlson's result. Extensions are given for \(k\)-tuples even for analytic decompositions of infinite sequences, and finally for decompositions of the set of words.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equation \[ \begin{cases} \mathbb{D}^\beta_t u(n,t) = - (-\Delta_d)^\alpha u(n,t)+f(n-ct,u(n,t)),\qquad & n\in \mathbb{Z}, \\ \quad t>0,\quad 0< \alpha,\, \beta<1, \\ u(n,0)=\varphi(n),\quad &n\in\mathbb{Z}, \end{cases}\leqno{(*)} \] for some constant \(c\geq 0\) where \(\mathbb{D}^\beta_t\) denotes the Caputo fractional derivative in time of order \(\beta\) and \((-\Delta_D)^\alpha\) denotes the discrete fractional Laplacian of order \(\alpha\in (0,1]\). We also prove a comparison principle. A special case of this equation is the discrete Fisher-KPP equation with and without delay. We show that if \(0\leq \varphi (n)\leq \gamma\) for every \(n\in\mathbb{Z}\) and the function \(f(x,\cdot)\) is concave on \([0,\gamma]\), \(f(x,s)\) is nonnegative for every \( x\in\mathbb{R}\), \(s\in [0,\gamma]\) and satisfies \[ f(x,0) \text{ and } f(x,\gamma) \leq 0,\, \forall x\in\mathbb{R}, \] for some \(\gamma >0\) then the system \((\ast)\) has a nonnegative unique solution \(u\) satisfying \(0\leq u(n,t)\leq\gamma\) for every \(n\in\mathbb{Z}\) and \(t\geq 0\). Our results include cubic nonlinearities and incorporate new results for the discrete Newell-Whitehead-Segel equation. We use Lévy stable processes as well as Mittag-Leffler, Wright and modified Bessel functions to describe the solutions of the linear lattice model, providing a useful framework for further study. For the nonlinear model, we use a generalization of the upper-lower solution method for reaction-diffusion equations in order to prove existence and uniqueness of solutions.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. Let \(a_1 < a_2 < \cdots < a_r \leq n\) be a set of positive integers such that \(a_i-a_j\neq a_j-a_k\) for \(1\leq k < j < i\leq r\). For given \(n\) let \(r(n)\) be the maximum value of \(r\) for which such a set exists. The authors prove that (1) \(r(2n)\leq n\) for \(n\geq 8\), (2) \(\limsup \frac{r(n)}{n} \leq \frac 49\). They conjecture that \(r(n)=o(n)\), and \textit{G. Szekeres} conjectured that \(r(\tfrac12 (3^k+1)) =2^k\).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. The paper deals with the following periodic problem: \[ u_{tt} - u_{xx} = F[u,u_{t},u_{x}], \tag{1} \] \[ u(x+2 \pi , t)=u(x,t),\;u(x,t+T)=u(x,t),\quad (x,t) \in \mathbb{R}^2. \tag{2} \] The operator \( F[u,u_{t},u_{x}] \) in general is nonlinear and maps a smooth function into a scalar continuous one. The author studies the problem of existence of smooth solutions of the problem (1)-(2).

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. An old and still unsolved problem in Ramsey theory asks whether if the positive integers are coloured with finitely many colours, then there are positive integers \(x\) and \(y\) such that \(x,y,x+y\) and \(xy\) all have the same colour. In fact, it is not even known whether it is always possible to find \(x\) and \(y\) such that \(x+y\) and \(xy\) have the same colour. This paper is about the corresponding question when \(\mathbb N\) is replaced by a finite field \(\mathbb F_p\), and gives a positive answer. More than that, it proves that a positive fraction of the quadruples \((x,y,x+y,xy)\) are monochromatic. The result is interesting for several reasons. One is that the standard tools of Ramsey theory appear to be hopelessly inadequate when they are applied to questions that mix addition and multiplication, so the fact that the authors have obtained a positive result of this kind is surprising and may well have further ramifications. Another is that several people have tried, without much success, to apply techniques from additive combinatorics to colouring problems. The techniques work well for many density problems, from which one can of course deduce colouring results. Until now the challenge has been to get them to work for colouring problems when the corresponding density statements are false, as is the case here (since the set of numbers between \(p/3\) and \(2p/3\) does not even contain a triple of the form \((x,y,x+y))\). A third reason, which is almost implied by the previous two, is that the paper introduces some striking new techniques. One of these techniques is to ``smooth'' the colouring in a way that converts the count of quadruples \((x,y,x+y,xy)\) into a count of purely linear configurations, thereby making the problem more amenable to conventional Ramsey-theoretic techniques. It also uses deep character sum estimates from number theory. For all these reasons, the paper will repay careful study by those who work in additive combinatorics.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. It is possible to characterize multivariate distributions by requiring that particular marginal and/or conditional distributions belong to specified parametric families. A spectrum of new results is described relying heavily on Gelman and Speed's recently published theorem on characterizing joint distributions by conditionals.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. In dieser Arbeit behandelt \textit{Ramsey} zwei Aufgaben über die Erfüllbarkeit von Aussagen. Die erste Aufgabe betrifft Aussagen, die mittels der Quantifikation ``alle'' aus Ausdrücken gebildet sind, welche wiederum mittels der drei elementaren logischen Operationen Konjunktion, Disjunktion und Negation, teils aus variablen Satzfunktionen und teils aus der Identitätsbeziehung aufgebaut sind. Wie der Verf. bemerkt, ist diese Aufgabe trivial für jeden einzelnen endlichen Bereich, weil man nur alle Möglichkeiten durchzuprobieren hat. Die Bedeutimg seiner Arbeit liegt deshalb darin, daß er auf einmal die Erfüllbarkeit in beliebig großen endlichen und auch unendlichen Bereichen untersucht. Es gelingt ihm im Abschnitt II, eine notwendige und hinreichende Bedingung für die Erfüllbarkeit in jedem Bereiche zu finden, dessen Elementzahl größer ist als eine gewisse Zahl, die von der Zahl der gegebenen Satzfunktionen und ihrer Argumente abhängt. Für kleinere Bereiche ist die Bedingung hinreichend, aber nicht notwendig. Um dies Hauptergebnis seiner Arbeit beweisen zu können, braucht er gewisse kombinatorische Sätze, die er im Abschnitt I beweist; diese Sätze sind übrigens auch sonst unzweifelhaft von großer Bedeutung. Im Abschnitt III macht er eine Anwendung seines allgemeinen Ergebnisses auf den Fall, wo von der Identität abgesehen bloß eine Funktion von zwei Variablen auftritt. Die zweite Aufgabe behandelt der Verf. im Abschnitt IV. Er betrachtet hier Aussagen, worin auch die Quantification ``es gibt'' vorkommt, aber so, daß alle Existenzzeichen im Aussagenausdruck vor den Allzeichen stehen. Er gibt eine kurze Darstellung davon, wie seine Methode noch anwendbar bleibt, indem er zeigt, wie die Frage der Erfüllbarkeit einer Aussage der letzteren Art in einem Bereiche auf die Frage der Erfüllbarkeit einer Aussage der ersteren Art in einem gewissen kleineren Bereiche zurückgeführt werden kann. (II.)

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. For a given binary BCH code \(C_{n}\) of length \(n= 2^{s}-1\) generated by a polynomial \(g(x)\in \mathbb{F}_2[x]\) of degree \(r\) there is no binary BCH code of length \((n + 1)n\) generated by a generalized polynomial \(g(x^{\frac{1}{2}})\in \mathbb{F}_2[x\frac{1}{2} \mathbb{Z} \geq 0]\) of degree \(2r\). However, it does exist a binary cyclic code \(C_{(n+1)n}\) of length \((n+1)n\) such that the binary BCH code \(C_{n}\) is embedded in \(C_{(n+1)n}\). Accordingly a high code rate is attained through a binary cyclic code \(C_{(n+1)n}\) for a binary BCH code \(C_{n}\). Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code \(C_{n}\) through the decoding of a binary cyclic code \(C_{(n+1)n}\), while the codes \(C_{n}\) and \(C_{(n+1)n}\) have the same minimum hamming distance.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. It is well known and not difficult to prove that if \(C \subseteq \mathbb{Z}\) has positive upper Banach density, the set of differences \(C - C\) is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, \textit{R. Jin} [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)] showed that whenever \(A\) and \(B\) have positive upper Banach density, then \(A - B\) is piecewise syndetic. Jin's result follows trivially from the first statement provided that \(B\) has large intersection with a shifted copy \(A - n\) of \(A\). Of course this will not happen in general if we consider shifts by integers, but the idea can be put to work if we allow ``shifts by ultrafilters''. As a consequence we obtain Jin's theorem.

This book is organized into four parts -- Part I (Preliminaries, Chapters 1--5); Part II (Ramsey theory, Chapters 6--9); Part III (Combinatorial number theory, Chapters 10--15); Part IV (Other topics, Chapters 16--17); and an appendix A (Foundations of nonstandard analysis). Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 777--779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595--598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [\textit{R. Arens}, Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [\textit{P. Civin} and \textit{B. Yood}, Pac. J. Math. 11, 847--870 (1961; Zbl 0119.10903); \textit{M. M. Day}, Ill. J. Math. 1, 509--544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by \textit{R. Ellis} [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)]. Chapter 2 is concerned to nonstandard analysis which was introduced by \textit{A. Robinson} [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix ``hyper''. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, \textit{W. A. J. Luxemburg} [Non-standard analysis. Lectures on A. Robinson's theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38--67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by \textit{R. Goldblatt} [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by \textit{V. Benci} et al. [Lect. Notes Log. 25, 3--44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by \textit{R. Jin} [Bull. Symb. Log. 6, No. 3, 331--341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)]. In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by \textit{M. Di Nasso} [in: \textit{P. A. Loeb} (ed.) and \textit{M. P. H. Wolff} (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443--474]. Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual ``standard'' universe to a different (larger) ``nonstandard'' universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by \textit{L. Luperi Baglini} [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), \url{arXiv:1212.2049}]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado's theorem is discussed by \textit{M. Di Nasso} [Proc. Am. Math. Soc. 143, No. 4, 1749--1761 (2015; Zbl 1386.03057)]. Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure's construction was introduced by \textit{P. A. Loeb} [Trans. Am. Math. Soc. 211, 113--122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by \textit{T. Kamae} [Isr. J. Math. 42, 284--290 (1982; Zbl 0499.28011)] is also discussed. Chapter 6 deals with Ramsey's theorem, which was studied by \textit{F. P. Ramsey} [Proc. Lond. Math. Soc. (2) 30, 264--286 (1929; JFM 55.0032.04)]. Ramsey's theorem was rediscovered in 1950's by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including \textit{R. Rado}'s decomposition theorem [Ann. Discrete Math. 3, 191--194 (1978; Zbl 0388.05031)]. Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that \textit{B. L. van der Waerden}'s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212--216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey's theory, although preceded by the Hindman cube lemma in the paper of \textit{D. Hilbert} [J. Reine Angew. Math. 110, 104--129 (1892; JFM 24.0087.03)] and by \textit{I. Schur}'s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)]. \textit{A. W. Hales} and \textit{R. I. Jewett}'s theorem [Trans. Am. Math. Soc. 106, 222--229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as ``Tic-Tac-Toe'' or ``Go-Moku''. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by \textit{V. Bergelson} et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449--476 (1994; Zbl 0809.04005)]. \textit{R. L. Graham} and \textit{B. L. Rothschild}'s theorem [Trans. Am. Math. Soc. 159, 257--292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey's theorem. Rota's conjecture was established by \textit{R. L. Graham} et al. [Adv. Math. 8, 417--433 (1972; Zbl 0243.18011)]. Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild's theorem [loc. cit.] was first proved by \textit{N. Hindman} [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman's theorem on finite sums. Hindman's theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman's theorem; which is a straightforward consequence of van der Waerden's theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey's theorem (by taking finite sets $F_1, \dots, F_m$, to have cardinality one) and Hindman's theorem (by taking $m=1$). \textit{W. T. Gowers}' theorem [Eur. J. Comb. 13, No. 3, 141--151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space $C_0$. Although Gowers' original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by \textit{D. Ojeda-Aristizabal} [Combinatorica 37, No. 2, 143--155 (2017; Zbl 1399.05221)] and \textit{K. Tyros} [Mathematika 61, No. 3, 501--522 (2015; Zbl 1409.05208)]. The more general version of Gower's theorem, which was already established by \textit{M. Lupini} [J. Comb. Theory, Ser. A 149, 101--114 (2017; Zbl 1358.05299)] is presented in this chapter. Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by \textit{N. Hindman} [Integers 11, No. 4, 431--439, A10 (2011; Zbl 1243.05237)], \textit{M. Di Nasso} and \textit{M. Riggio} [Combinatorica 38, No. 5, 1067--1078 (2018; Zbl 1438.11075)], \textit{P. Csikvári} et al. [Combinatorica 32, No. 4, 425--449 (2012; Zbl 1286.11026)] and \textit{B. Green} and \textit{T. Sanders} [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed. Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be \textit{S. C. Leth}'s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by \textit{T. C. Brown} [Pac. J. Math. 36, 285--289 (1971; Zbl 0211.33504)]. After 6 years, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi's theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to \textit{H. Towsner} [Ergodic Theory Dyn. Syst. 29, No. 4, 1309--1326 (2009; Zbl 1179.37006)]. Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to \textit{L. Luperi Baglini} [Arch. Math. Logic 55, No. 5--6, 705--734 (2016; Zbl 1354.05139)]; also it was jointly studied by \textit{A. Blass} and \textit{M. Di Nasso} [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195--206 (2015; Zbl 1382.03068)]. \textit{R. Jin} also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20--38 (2001; Zbl 1071.11503)]. Chapter 12 is devoted to Jin's sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss \textit{R. Jin}'s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855--861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by \textit{M. Beiglböck} [Isr. J. Math. 185, 369--374 (2011; Zbl 1300.11015)]; and also an alternative proof given by \textit{M. Di Nasso} [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin's sumset theorem are also discussed. Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős' conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [\textit{P. Erdős}, Ann. Discrete Math. 6, 89--115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [\textit{M. B. Nathanson}, J. Comb. Theory, Ser. A 28, 150--155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős' conjecture; and a weak density version of Folkman's theorem are also discussed. Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of \textit{S. C. Leth} [Proc. Am. Math. Soc. 134, No. 6, 1579--1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [\textit{P. Erdős} and \textit{P. Turán}, J. Lond. Math. Soc. 11, 261--264 (1936; Zbl 0015.15203; JFM 62.1126.01)]. Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to \textit{S. C. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to \textit{I. Goldbring} and \textit{S. Leth} [``On supra-SIM sets of natural numbers'', Preprint, \url{arXiv:1805.05933}] is also discussed in brief. Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939--945 (1978; Zbl 0393.05031)]. The valuable findings obtained by \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)], \textit{W. T. Gowers} [Ann. Math. (2) 166, No. 3, 897--946 (2007; Zbl 1159.05052)], \textit{I. Goldbring} and \textit{H. Towsner} [Isr. J. Math. 199, Part B, 867--913 (2014; Zbl 1298.03100)] and \textit{T. Tao} [J. Anal. Math. 103, 1--45 (2007; Zbl 1146.05038)] are also discussed. Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by \textit{T. Tao} [Combinatorica 28, No. 5, 547--594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to \textit{L. van den Dries} [Astérisque 367--368, 79--113, Exp. No. 1077 (2015; Zbl 1358.11024)]. We consider a system of two types of particles called A-particles and B-particles moving on one-dimensional lattice according to simple excluded random walks. A- and B-particles are distinguished by their jump rates. We derive the Stefan free boundary problem in a hydrodynamic scaling limit for a certain class of initial conditions. An extension to \(n\)-component system \((n\geq 3)\) is also discussed.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The starting point to topological quantum field theory was given by \textit{E. Witten} [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] where he explained the geometric meaning of super-symmetry, pointing out that for super-symmetric quantum mechanics the Hamiltonian is just the Hodge- Laplacian. He then outlined the corresponding ideas for super-symmetric quantum field theories viewed as differential geometry of certain infinite dimensional manifolds, including the associated analysis and topology. This shows that there may be interesting topological aspects of quantum field theory and that these should be important for physics. On the other hand one can use these quantum field theories as a conceptual tool to suggest new mathematical results. Indeed this reversed process led to spectacular progress in the understanding of geometry in low dimensions. The author starts with presenting a set of axioms for topological quantum field theories following \textit{G. B. Segal} [The definition of conformal field theory, Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988; Zbl 0657.53060)], then lists examples of theories (known to exist) satisfying such axioms. These include in \(d=1\) the Floer/Gromov theory and holomorphic conformal field theories (Segal); in \(d=2\) the Jones/Witten theory, Casson theory, Johnson theory and Thursten theory and in \(d=3\) the Floer/Donaldson theory.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Many online applications, especially in the financial industries, are running on blockchain technologies in a decentralized manner, without the use of an authoritative entity or a trusted third party. Such systems are only secured by cryptographic protocols and a consensus mechanism. As blockchain-based solutions will continue to revolutionize online applications in a growing digital market in the future, one needs to identify the principal opportunities and potential risks. Hence, it is unavoidable to learn the mathematical and cryptographic procedures behind blockchain technology in order to understand how such systems work and where the weak points are. The book provides an introduction to the mathematical and cryptographic concepts behind blockchain technologies and shows how they are applied in blockchain-based systems. This includes an introduction to the general blockchain technology approaches that are used to build the so-called immutable ledgers, which are based on cryptographic signature schemes. As future quantum computers will break some of the current cryptographic primitive approaches, Andreas Bolfing considers their security and presents the current research results that estimate the impact on blockchain-based systems if some of the cryptographic primitive break. Based on the example of Bitcoin, he shows that weak cryptographic primitives pose a possible danger for the ledger, which can be overcome through the use of the so-called post-quantum cryptographic approaches.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The paper contains the definition, examples and a coherence theorem for braided tensor categories; a category theory over a braided base and a classification theorem for braided categorical groups are also given.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In this paper the problem of the annulus of \(\alpha\)-convexity is considered. Analytically, this problem can be written as follows: find \(\max_{\phi \in S}| \phi (z)|\), \(\min_{\phi \in S}| \phi (z)|\), respectively under the conditions: \[ Re[(1-\alpha)z\phi '(z)/\phi (z)+\alpha (\phi ''(z)/\phi '(z)+1)]=0, \] where S is the class of univalent functions, \(| z| =r\) is given, \(r>r^{\alpha}\), \(r^{\alpha}\) the radius of \(\alpha\)-convexity. The author solved the above problem basing himself on the variational method of Schiffer- Goluzin.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The Zamolodchikov tetrahedra equations appeared in connection with higher-dimensional integrable systems [cf. \textit{A. B. Zamolodchikov}, Zh. Eksperiment. Teor. Fiz. 79, No. 2, 641-661 (1980)] and are a three- dimensional generalization of the Yang-Baxter equation. It is not easy nor illuminating to even write these equations without using some pictorial notations. The paper gives an overview of Zamolodchikov equations and develops a conceptual framework underlying them, provided by the theory of 2- categories. The most important feature of 2-categories is that an algebraic expression in them has the shape of a subdivided plane polygon and the commutative diagrams have therefore the shape of boundaries of convex polytopes in \(\mathbb{R}^ 3\). The authors develop a theory of monoidal structures in 2-categories and of braidings (or quasi- symmetries) of these monoidal structures. The axiomatics of these objects and interpretation of Zamolodchikov equation involves some remarkable convex polytopes: permutohedron, Stasheff polytope, resultohedra which appeared earlier in other contexts. The main result is the bilateral correspondence between braidings on monoidal 2-categories and Zamolodchikov systems. This is a more complicated version of such a relation between Yang-Baxter operators and usual braided monoidal categories. Many others topics are considered on the way to this theorem: Different peoples often mean by tetrahedra (or simplex) equations different things. The authors give a detailed overview of various versions of Zamolodchikov equations, their interrelations, and known examples of solutions. Different 2-categories, analogs of the category of vector spaces, are described in detail. These categories were used later by Crane and Yetter in definition of factorizable TQFT. The present paper is the first more or less systematic and self-contained description of the subject. A separate book is announced by the authors. For more treatment in the higher category theory see also the preprint of \textit{Gordon, Power} and \textit{Street} ``Coherence for tricategories'' (1993). The forthcoming article of \textit{R. Street}, ``Higher categories, strings, cubes and simplex equations'' seems to be closely related with the subject.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Assessing prediction error is a problem which arises in time series analysis. The distinction between the conditional prediction error \(e\) and the unconditional prediction error \(E(e)\) has not received much attention in the literature. Although one can argue that the conditional version is more practical, we show in this article that assessing \(e\) is nearly impossible. In particular, we use the correlation coefficient \(\text{corr} (\widehat {e}, e)\), where \(\widehat {e}\) is an estimate of \(e\), as a measure of performance and show that \(\lim _{T\to \infty} \sqrt{T} \text{ corr} (\widehat {e}, e)=C\) where \(T\) is the sample size and \(C>0\) is some constant. Furthermore, the value of \(C\) is large only when the process is extremely non-Gaussian or nearly nonstationary.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Let \(\mathfrak g\) be a symmetrizable Kac-Moody algebra and \(U_q(\mathfrak g)\) be the corresponding quantized universal enveloping algebra. Let \(U ^-\) denote the negative part of \(U_q(\mathfrak g)\). Using topological methods, \textit{G. Lusztig} [J. Am. Math. Soc. 4, 365-421 (1991; Zbl 0738.17011)] defined the canonical basis for \(U^-\). At the same time, using algebraic methods, \textit{M. Kashiwara} [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] defined the global crystal basis for \(U^-\). The main result of this paper is that these two bases for \(U^-\) coincide.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. This paper is concerned with the problem of exponential stability for a class of impulsive switched nonlinear time-delay systems with delayed impulse effects. By using the multiple Lyapunov-Krasovskii functionals technique, some exponential stability criteria are obtained, respectively, for two kinds of impulsive signals (destabilizing impulsive signal and stabilizing impulsive signal). The derived results not only characterise the effects of delayed impulse, time delay and switching on nonlinear systems, but also remove some restriction conditions. Furthermore, a more precise bound of system state is given. Compared with existing results on related problems, the obtained results are less conservative. Three examples are provided to illustrate the effectiveness and the generality of the proposed results.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. A twisted version of four dimensional supersymmetric gauge theory is formulated. The starting point is the connection between the Floer and Donaldson theories which has led to the conjecture that the ``Morse theory'' interpretation of Floer homology must be an approximation to a relativistic quantum field theory. It is shown that the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The Floer theory is generalized to the relativistic case and then the formula for the supersymmetry current and the energy-momentum tensor are obtained. The most important result of the paper is the assertion that the stress tensor is a ``BRST'' commutator \(T_{\alpha \beta}=\{Q,\lambda_{\alpha \beta}\},\) where Q is a linear transformation of the space of all functionals of the field variables, and \(\lambda_{\alpha \beta}^ a \)tensor field characteristic for the model. Finally, the possible physical interpretation of the model is presented. It is pointed out that the model is in a sense a generally covariant quantum theory in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In this paper, graphical evaluation and review technique is applied to model and analyse the dynamics of a two-unit standby system with two types of repair facilities. Various reliability parameters are derived and illustrated numerically.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Let \(U^+\) be the \(+\) part of the quantized enveloping algebra \(U\) associated to a root system. This is an algebra over the field \({\mathbb Q}(v)\) which for \(v=1\) specializes to the classical enveloping algebra \(U^+_ 1\) of the nilpotent radical of a Borel subalgebra in a semisimple Lie algebra. One of the main results of the paper is the construction of a canonical basis \(B\) of \(U^+\) as a \({\mathbb Q}(v)\)-vector space. There is an analogy between the definition of \(B\) and the definition of a new basis for a Hecke algebra given in \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165--184 (1979; Zbl 0499.20035)]. The canonical basis \(B\) has a number of remarkable properties. One of them is that the product of two elements in \(B\) is a linear combination of elements in \(B\) with coefficients in \({\mathbb N}[v,v^{-1}]\). Another one is that \(B\) is well adapted to finite dimensional representations of \(U\). Namely, let \(L_ d\) be a finite dimensional simple \(U\)-module corresponding to the dominant weight \(d\) and let \(x_ 0\) be a lowest weight vector for it. Then it is shown that the set \(\{\delta x_ 0|\delta\) running through the set of all elements \(\delta\in B\) such that \(\delta x_ 0\neq 0\}\) forms a basis of \(L_ d\). It gives rise to a canonical basis in any finite dimensional simple module of the corresponding semisimple Lie algebra, that for type \(A\) should be closely related to the basis in \textit{C. De Concini} and \textit{D. Kazhdan} [Isr. J. Math. 40, 275--290 (1981; Zbl 0537.20006)]. It is also given a purely combinatorial formula for \(\dim L_ d\).

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. A Tikhonov-regularizing algorithm has been proposed by \textit{O.~M.~Alifanov} and \textit{E.~A.~Artyukhin} in [Teplofiz. Vys. Temp. 16, No. 4 (1978)] for the numerical solution of the nonlinear one-dimensional inverse heat-conduction problem. In the article under review, the author generalizes this algorithm to the two-dimensional case. The influence of a heat flow on the accuracy of determination of temperature and heat flux density is studied by inverse heat-conduction methods. The author also demonstrates the influence of the error of the initial data on the solution to the inverse problem. Numerical calculations show the efficiency and practicability of the computational algorithm proposed.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The staggering array of relatively recent combinatorial-algebraic- topological methods of defining invariants of low-dimensional objects such as links, braids, 3-manifolds etc., is attacked in this big book. Utilizing the basic notion of a category of modules (modular category) equipped with a tensor product and various morphisms, depending on the need, geometric objects are given algebraic realizations. Algebraic invariants can then be associated with topological spaces. The book contains many ideas and the more or less random sampling from each chapter given below may give some idea of the contents: Chapter I. Invariants of graphs in Euclidean 3-space: 1. Ribbon categories. Chapter II. Invariants of closed 3-manifolds: 2. Modular tensor categories; 5. Hermitian and unitary categories. Chapter III. Foundations of topological quantum field theory: 1. Axiomatic definition of TQFT's; 4. Quantum invariants; 5. Hermitian and unitary TQFT's. Chapter IV. Three- dimensional topological quantum field theory: 3. Lagrangian relations and Maslov indices; 5. Action of the modular groupoid; 9. Anomaly-free TQFT; 10. Hermitian TQFT; 11. Unitary TQFT; 12. Verlinde algebra. Chapter V. Two-dimensional modular functors: 3. Weak and mirror modular functors. Chapter VI. \(6j\)-symbols: 3. Symmetrized multiplicity modules; 4. Framed graphs. Chapter VII. Simplicial state sums on 3-manifolds: 3. Simplicial 3-dimensional TQFT. Chapter VIII. Generalities on shadows: 3. Shadow links; 4. Surgeries on shadows; 7. Shadow graphs. Chapter IX. Shadows of manifolds: 1. Shadows of 4-manifolds; 8. Shadows of framed graphs. Chapter X. State sums on shadows: 1. State sum models on shadowed polyhedra; 7. Invariants of framed graphs from the shadow viewpoint; 9. computations for graph manifolds. Chapter XI. An algebraic construction of modular categories: 1. Hopf algebras and categories of representations; 6. Quantum groups at roots of unity. Chapter XII. A geometric construction of modular categories: 1. Skein modules and the Jones polynomial; 3. The Temperley-Lieb algebra; 4. The Jones-Wenzl idempotents; 8. Multiplicity modules. Appendix I. Dimension and trace re- examined. Appendix II. Vertex models on link diagrams. Appendix III. Gluing re-examined. Appendix IV. The signature of closed 4-manifolds from a state sum.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. This paper deals with certain classes of surfaces in the 3-sphere \(S^3 \subset {\mathbb R}^4\). It is shown that any flat canal surface in \(S^3\) is Lie equivalent to the stereographic projection of either a generic cylinder or a generic cone or a surface of revolution in \({\mathbb R}^3\). The non-flat canal surfaces in \(S^3\) are classified up to Lie equivalence. Moreover, a classification of Lie homogeneous surfaces is given. If \(US^3 := \{(x, \xi) \in S^3\times S^3\mid x\cdot \xi = 0\}\), then an immersion \(\lambda = (x, \xi): M \longrightarrow US^3\) is called a Lie geometric surface if \(dx\cdot \xi = 0\) holds. The Lie geometric surfaces which allow a Lie sphere transformation for any prescribed preimage-image pair of points chosen on them are called homogeneous Lie surfaces. The author gives a complete classification of such surfaces with respect to the Lie transformation group of \(S^3\). It is shown that there exist 19 types of such surfaces.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Although the `strict' notions of higher-dimensional category have been well understood for twenty years or more, our understanding of the more `relaxed' notions (in which lower-dimensional compositions are associative up to higher-dimensional equivalences, which in turn are coherent up to still higher-dimensional equivalences, \(\dots)\) has not, until recently, progressed much beyond the notion of bicategory introduced by \textit{J. Bénabou} nearly thirty years ago [Introduction to bicategories, Rep. Midwest Category Semin., Lect. Notes Math. 47, 1--77 (1967; Zbl 0165.33001)]. In the words of the authors of the present monograph, `the two-dimensional notion of bicategory \(\dots\) has sufficed for a whole generation of mathematicians requiring higher-dimensional categories with a weakly associative composition'. There may have been good reasons for this, besides the innate laziness of mathematicians; but undoubtedly an important factor was the comforting nature of the coherence theorem for bicategories, which (by asserting that every bicategory is biequivalent to a 2-category) seemed to be telling us that `it doesn't matter if you assume everything is strict'. In recent years, however, pressures have been building from a number of directions outside category theory (cohomology theory, algebraic homotopy types, quantum field theory and even theoretical computer science) for a serious study of lax higher-dimensional categories (or at least of the three-dimensional notion, tricategories). The authors of this monograph have finally done the job of laying the foundations of tricategory theory. The bad news is that the comforting coherence theorem no longer holds: not every tricategory is triequivalent to a 3-category. (The authors provide an explicit counterexample using algebraic homotopy 3- types.) The good news: every tricategory is triequivalent to a \({\mathcal G}ray\)-category, where \({\mathcal G}ray\) is a symmetric monoidal closed category whose underlying category is 2-\({\mathcal C}at\) but whose tensor product is (not the cartesian product but) an `exotic' structure first introduced by \textit{J. W. Gray} [Formal category theory: Adjointness for 2-categories, Lect. Notes Math. 391 (1974; Zbl 0285.18006)]. The proof of this theorem is the main goal of the present work. Inevitably, the very definitions of the concepts involved take up a lot of space: this is not an easy read for anyone who finds his head spinning when confronted by large pasting diagrams. (For example, a single equation in the definition of a tritransformation occupies the whole of pages 21-22.) But all readers can rejoice that the necessary hard work has been done, and category-theorists (and others) can now go on to apply it in the contexts mentioned above.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. For image filtering applications, it has been observed recently that both diffusion filtering and associated regularization models provide similar filtering properties. The comparison has been performed for regularization functionals with convex penalization functional. In this paper we discuss the relation between non-convex regularization functionals and associated time dependent diffusion filtering techniques (in particular the Mean Curvature Flow equation). Here, the general idea is to approximate an evolution process by a sequence of minimizers of iteratively convexified energy (regularization) functionals.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. This note introduces a polynomial which extends both the Alexander-Conway and Jones invariants of links. The new invariant was discovered independently and almost simultaneously by four groups; an outline of each approach is given after a statement of the common result. (Yet another approach has been found by J. Przytycki and P. Traczyk, Univ. Warsaw, 1985.) The main theorem asserts that there is a unique homogeneous 3-variable Laurent polynomial invariant of isotopy classes of tame oriented links which satisfies a Conway identity and which takes the values 1 on the unknot.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. Die Frage nach Flächen \(S\), auf welchen eine Schar dualer Kongruenzen \(K\), \(K'\) von der Beschaffenheit existiert, daß die eine von ihnen vollständig konjugiert, die andere vollständig harmonisch auf \(S\) ist im Sinne einer früheren Untersuchung des Verf. (Rendiconti Accad. d. L. Roma (6) 4 (1926), 81-85; F. d. M. 52, 716 (JFM 52.0716.*)), führt mit den an dieser Stelle entwickelten Hilfsmitteln auf drei wesentliche Typen: auf die Flächen von \textit{Tzitzéica-Wilczynski}, auf gewisse von Verf. näher charakterisierte Flächen, welche von endlich vielen Konstanten abhängen, und auf die Flächen konstanter metrischer Krümmung mit ihren Dual- und Projektivbiegungsverwandten, welche in diesem Zusammenhang also eine projektive Charakterisierung gestatten.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The author develops geometric techniques which reduce the study of isotopy of links in the total space \(N\) of an oriented circle fibration over a closed oriented surface \(F\) to the study of certain equivalence classes of (purely 2-dimensional) objects on \(F\), which he calls shlinks (shadow links). More precisely, for each abelian group \(A\supset\mathbb{Z}\), the author defines a shadow on \(F\) over \(A\) to be a finite family of immersed closed curves on \(F\) with only double transversal crossings, together with a labelling of the components of the complement by elements of \(A\) (``gleams''). A shlink is an equivalence class of shadows modulo suitably defined ``Reidemeister moves''. (It should be noted that crossings are treated as if labelled by \((-2)\in\mathbb{Z}\subset A\).) The sum over all labels of a shadow is called the total gleam, and it is an invariant of the shlink. The author shows how to associate to each (isotopy class of) a link \(K\) in \(N\) a shlink \(S(K)\) on \(F\). Then he proves that it is possible to almost reconstruct \(K\) from \(S(K)\). More precisely, the map \(K\mapsto S(K)\) establishes a bijective correspondence between the set of isotopy classes of links in \(N\), modulo a natural action of \(H_ 1(F)\), and the set of integral shlinks on \(F\) with total gleam equal to the negative of the Euler number of the bundle \(N\to F\). An interesting example is the Hopf fibration \(S^ 3\to S^ 2\), where in fact the transition from links to shlinks is faithful. The author goes on to discuss framed and colored shlinks. Then he sets up an IRF model for colored complex shlinks, which enables to define invariants of shlinks from the representation theory of \(U_ q(sl_ 2(\mathbb{C}))\). For the Hopf fibration example, existence of the Jones polynomial and the Jones type invariants of links in \(\mathbb{R}^ 3\) colored by irreducible finite dimensional representations of \(U_ q(sl_ 2)\) [\textit{N. Yu. Reshetikhin} and the author, Commun. Math. Phys. 127, 1-26 (1990; Zbl 0768.57003)] is proved. The important philosophical point of the work is that shlinks over \(A=\mathbb{Q},\mathbb{R},\mathbb{C},\dots\) should be treated as interesting geometric objects on its own. So, in the Hopf fibration example, the transition from integer shlinks to complex shlinks, can be understood as a ``completion'' of the set of isotopy classes of links in \(S^ 3\), which needs to be explored in the future.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. This paper investigates the production/inventory policy in an integrated vendor-buyer inventory system with defective goods in the buyer's arrival order lot. We assume that shortages are allowed and partially backlogged on the buyer's side, and that the lead time demand distribution is unknown but has given finite first and second moments. The objective is to optimize the buyer's order quantity, reorder point and the number of shipments from vendor to buyer simultaneously in the proposed model. Since the lead time demand distribution is unknown, we apply the minimax distribution-free procedure to solve this problem. An algorithm is developed to obtain the optimal production/inventory strategy. Furthermore, numerical example is provided to illustrate the solution procedure and sensitivity analysis of the optimal solution with respect to major parameters is carried out.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. In dieser Abhandlung, deren Resultate teilweise bereits in den C. R. 115 kurz angegeben sind (JFM 24.0506.02), giebt der Verfasser eine Ausdehnung der bekannten, für Riemann'sche Flächen und deren Functionen grundlegenden Definitionen und Sätze auf Mannigfaltigkeiten von beliebig vielen Dimensionen. Er definirt insbesondere im ersten Teil ihre analytische Darstellung durch Gleichungen, resp. Ungleichungen oder aber durch unabhängige Parameter, ihre Scheidung in einseitige und zweiseitige auf Grund des Verhaltens der bezüglichen Functionaldeterminante bei Beschreibung geschlossener Wege, die auf den Flächen existirenden Integralfunctionen, die beim Schnitt von Mannigfaltigkeiten auftretenden Vorzeichenbestimmungen, die der Charakteristik in der Ebene, resp. der Summe der Punktcharaktere entsprechen und durch Functionaldeterminanten festgelegt werden, u. s. w. u. s. w., das letztere übrigens nur unter Annahme von zweiseitigen Flächen. Im zweiten Teil wendet sich der Verfasser zur Verallgemeinerung der gruppentheoretischen Formirungen. Er definirt die zu jeder \(M\) gehörige Gruppe von Substitutionen, die ein Functionssystem bei Beschreibung geschlossener Wege auf der \(M\) erleidet, die ihnen entsprechenden Fundamentalbereiche, endlich das Abbild dieser Bereiche, welches durch Aufsteigen zu dem um eine Dimension höheren Raume und durch Zusammenfalten der entsprechenden Grenzgebiete entsteht. Hierfür, sowie für die zugehörigen Gruppen wird eine grössere Zahl von Beispielen ausführlicher behandelt. Er untersucht dann schliesslich eingehend die Frage, ob eine Mannigfaltigkeit höherer Dimension durch die Betti'schen Zusammenhangszahlen im Sinne der Analysis situs eindeutig definirt ist. Die Antwort lautet, dass dies nicht der Fall ist; der Verfasser zeigt an verschiedenen Beispielen, dass man mit gegebenen Betti'schen Zahlen noch unendlich viele Mannigfaltigkeiten bilden kann, die im Sinne der Analysis situs nicht in einander defininirbar sind. Als besondere Resultate der Abhandlung sind noch zu nennen: erstens der Satz, dass für eine geschlossene Mannigfaltigkeit die gleichweit von den ersten und letzten abstehenden Betti'schen Zahlen einander gleich sind; zweitens die Ausdehnung des Theorems von Euler auf Polyeder von beliebigen Dimensionen und beliebigem Zusammenhang, jedoch unter der Annahme, dass alle Grenzgebiete der verschiedenen Dimensionen einfach zusammenhängend sind. Ist \(\alpha_0\) die Zahl der Ecken, \(\alpha_1\) die der Kanten, ..., \(\alpha_p\) die der \(p\)-dimensionalen Grenzgebiete, so hat \[ N = \alpha_p - \alpha_{p-1} + \alpha_{p-2} -\cdots\mp\alpha_1\mp\alpha_0, \] wenn \(p\) gerade ist, den Wert \[ N = 3 - P_1 + P_2 -\cdots+ P_{p-1}, \] wo die \(P_i\) die Betti'schen Zahlen sind; für ungerades \(p\) dagegen ist \(N\) stets gleich Null, welches auch die \(P_i\) sind.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval \([-1,1]\) and on the ellipse in the complex plane with the focuses \(-1, 1\) and the semiaxes \(R\). We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions. We present in detail various properties of the classes of functions, satisfying Bernstein type inequalities and various approaches to establishing such inequalities.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. The authors show how some algebraic input (over a commutative ring \(K\) with 1) gives rise to (unoriented) topological quantum field theories (TQFT). A TQFT here is a functor from the category of surfaces (with homomorphisms being 3-dimensional bordisms) into the category of finite- dimensional \(K\)-modules. Such a TQFT in particular assigns to each closed unoriented (maybe non-orientable) 3-manifold \(M\) an element \(| M | \in K\). The basic idea for the construction is that of admissable colorings of triangulated surfaces and 3-manifolds. Fix some finite set \(I\) and a distinguished subset of unordered triples of elements of \(I\), called admissable. Then an admissable coloring of a triangulated surface (resp. 3-manifold) is a function from the corresponding set of edges into \(I\), such that the colors of the boundary of each 2-simplex determine an admissable triple. The algebraic input now allows to assign to each compact triangulated 3-manifold \(M\) (possibly with boundary) and admissable coloring \(\varphi\) an element \(| M |_ \varphi \in K\). Using this and assuming independence of triangulations, the TQFT is constructed in a purely formal way. The \(K\)-module associated to a surface is essentially the free module on admissable colorings of \(F\). The invariant of a closed 3-manifold \(M\) is the sum \(\sum| M |_ \varphi\) over all colorings \(\varphi\) of \(M\). In order to prove that \(| M | _ \varphi\) does not depend on the triangulation the authors define equivalent notions of coloring and \(|\;|_ \varphi\) for a category of simple 2-complexes (dual to triangulations), and then translate the Alexander moves (moves relating different triangulations of a dimensionally homogeneous polyhedron) into this dual framework. This is necessary to reduce to a finite set of moves. It should be mentioned that the authors prove a relative version of the Alexander theorem. The relation to simple spines of manifolds, and a resulting method of computation of \(| M|_ \varphi\) from Heegard diagrams is discussed. It is shown that for each \(r\)-th root of unity \(q\), \(r>2\), the quantum \(6j\)-symbols (associated with the quantized enveloping algebra \(U_ q(\text{sl}_ 2(\mathbb{C})))\) provide the necessary algebraic input. Thus for each such \(q\) the authors have constructed a TQFT. For these TQFTs explicit calculations of \(| M |\) for \(S^ 3\), \(\mathbb{R} P^ 3\), \(L (3,1)\) and \(S^ 2 \times S^ 1\) and topological interpretations (Betti numbers) of \(| M |\) for \(r=3\) and arbitrary 3-manifolds are given.

As the authors have stated, this paper ``is a highly subjective chronology describing how physicists have begun to use ideas from \(n\)-category theory in their work, often without making this explicit.'' They have begun with the discovery of special relativity and quantum mechanics, ending at the dawn of the twenty-first century. Since the developments in the twenty-first century are so thick that the authors have given up putting them in their proper perspective. The chronology begins with a citation from \textit{J. C. Maxwell} [Matter and motion. With notes and appendices by Joseph Larmor. Reprint of the 1876 original. Reprint of the 1876 original. Amherst, NY: Prometheus Books (2002; Zbl 1060.01011)]. The second is \textit{H. Poincaré}'s fundamental groups [J. de l'Éc. Pol. (2) I. 1--123 (1895; JFM 26.0541.07)] hinting at the unification of space and symmetry, which was later to become one of the main themes of \(n\)-category theory. The third is \textit{H. A. Lorentz} transformations [Leiden. E. J. Brill. 139 S. \(8^\circ\) (1895; JFM 26.1032.06); ibid. 138 S. \(8^\circ\) (1895; JFM 25.1632.01); Amst. Ak. Versl. 12, 986--1009 (1904; JFM 35.0837.03)]. The fourth is again \textit{H. Poincaré} [C. R. Acad. Sci., Paris 140, 1504--1508 (1905; JFM 36.0911.02)] as a keyman in the formation of special relativity. The fifth is \textit{A. Einstein} [Ann. der Phys. (4) 17, 891--921 (1905; JFM 36.0920.02)] surely as an inventor of special relativity. The sixth is \textit{H. Minkowski} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1908, 53--111 (1908; JFM 39.0909.02)] for his Minkowski spacetime. The succeeding two are \textit{W. Heisenberg} [Z. Phys. 33, 879--893 (1925; JFM 51.0728.07)] and \textit{M. Born} [Z. Phys. 37, 863--867 (1926; JFM 52.0973.03)]. It seems that the authors have made an error in attributing Born's discovery of the interpretation of \(\psi^{\ast}\psi\)\ in the Schrödinger equation as the probability density function, for which he won his Nobel prize, not to 1926, when Schrödinger found out his famous equation to be called the Schrödinger equation, but mysteriously to 1928. It is well known that Einstein was against this interpretation, saying that God does not play dice. The ninth and the tenth are \textit{J. von Neumann} [Mathematische Grundlagen der Quantentheorie. Berlin: Julius Springer (1932; Zbl 0005.09104)] and \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)], respectively. The eleventh is [\textit{S. Eilenberg} and \textit{S. MacLane}, Trans. Am. Math. Soc. 58, 231--294 (1945; Zbl 0061.09204)], inventing the notion of a category. The twelfth is Feynman's famous lecture in a small conference at Shelter Island in 1947, which surpassed the understanding of most physicists except Schwinger. The reviewer would like to mention [\textit{A. Wüthrich}, The genesis of Feynman diagrams. Dordrecht: Springer (2010; Zbl 1226.81008)] as a good reference here. The thirteenth is \textit{C. N. Yang} and \textit{R. Mills}' generalization of Maxwell's equations [``Conservation of isotopic spin and isotopic gauge invariance'', Phys. Rev. 96, No. 1, 191--195 (1954; \url{doi:10.1103/PhysRev.96.191})], which is now known as the Yang-Mills theory. The fourteenth is \textit{S. MacLane}'s introduction of monoidal categories [Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. The fifteenth is \textit{F. W. Lawvere}'s thesis on functorial semantics [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)], whose impact is apparent in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. York: Springer-Verlag (1973; Zbl 0285.55012); \textit{S. MacLane}, Bull. Am. Math. Soc. 71, 40--106 (1965; Zbl 0161.01601); \textit{J. P. May}, The geometry of iterated loop spaces. York: Springer-Verlag (1972; Zbl 0244.55009)] and so on, reaching ``the definitions of conformal field theory and topological quantum field theory propounded by Segal and Atiyah in the late 1980s. ``The sixteenth is \textit{J. Bénabou} et al.'s introduction of bicategories [Reports of the Midwest Category Seminar. York: Springer-Verlag (1967; Zbl 0165.33001)]. The seventeenth is \textit{R. Penrose} [in: Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221--244 (1971; Zbl 0216.43502)]. The eighteenth is [\textit{G. Ponzano} and \textit{T. Regge}, in: F. Bloch et al., Spectroscopic and group theoretical methods in physics. Amsterdam: North-Holland Publishing Company. 75--103 (1968; Zbl 0172.27401)], who ``applied Penrose's theory of spin networks before it was invented to relate tetrahedron-shaped spin networks to gravity in three-dimensional spacetime. The nineteenth is Grothendieck's ``Pursuing Stacks'', a very long letter to D. Quillen in 1983, in which he ``fantasized about \(n\)-categories for higher \(n\) -- even \(n=\infty\) -- and their relation to homotopy theory.'' The twentieth is string theory which experienced the outburst in the 1980s. The reviewer would like to mention [\textit{R. Blumenhagen} et al., Basic concepts of string theory. Berlin: Springer (2012; Zbl 1262.81001)] as a recommendable textbook on the subject, though the great influence of [\textit{M. B. Green} et al., Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology. Cambridge etc.: Cambridge University Press (1987; Zbl 0619.53002)] on the above textbook is apparent. Just as essentially one-dimensional Feynman diagrams in quantum field theory are replaced by two-dimensional diagrams depicting string worldsheets in string theory, the mathematics of categories should be replaced by that of bicategories. The twenty-first is [\textit{A. Joyal} and \textit{R. Street}, Adv. Math. 102, No. 1, 20--78 (1993; Zbl 0817.18007)]. The twenty-second is [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. ``The work of Jones led researchers toward a wealth of fascinating connections between von Neumann algebras, higher categories, and quantum field theory in two- and three-dimensional spacetime.'' The twenty-third is [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)] and [Zbl 0638.57003]. The twenty-fourth is [\textit{V. G. Drinfel'd}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798--820 (1987; Zbl 0667.16003)], which is ``the culmination of a long line of work on exactly solvable problems in low-dimensional physics''. The twenty-fifth is [\textit{G. B. Segal}, in: Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060)], which proposed axioms describing a conformal field theory. The twenty-sixth is [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)], whose goal was to formalize [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. Ironically, ``these invariants have led to a revolution in our understanding of four-dimensional topology'', while it has never successfully dealt with the Donaldson theory. The twenty-seventh is \textit{R. H. Dijkgraaf}' purely algebraic characterization of two-dimensional topological quantum field theories in terms of commutative Frobenius algebras in his [A geometric approach to two-dimensional conformal field theory. Utrecht, NL: University of Utrecht (PhD Thesis) (1989)]. The twenty-eighth is [\textit{S. Doplicher} and \textit{J. E. Roberts}, Invent. Math. 98, No. 1, 157--218 (1989; Zbl 0691.22002)], which shows ``one could start with a fairly general quantum field theory and compute its gauge group instead of putting the group in by hand''. The twenty-ninth is [\textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev}, Commun. Math. Phys. 127, No. 1, 1--26 (1990; Zbl 0768.57003)], where the author summarizes a bit of the theory of quantum groups in its modern form. The thirtieth is [\textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010); ibid. 17, 361--451 (1994; Zbl 0818.57014)], which gave an intrinsically three-dimensional description of the Jones polynomial by using a gauge field theory in three-dimensional spacetime called the Chern-Simons theory. The thirty-first is loop quantum gravity initiated by \textit{C. Rovelli} and \textit{L. Smolin} in [``Loop space representation of quantum general relativity'', Nucl. Phys. B 331, No. 1, 80--152 (1990; \url{doi:10.1016/0550-3213(90)90019-A})]. The thirty-second is [\textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); ibid. 69, No. 2, 455--485 (1993; Zbl 0774.17018); Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009)] and [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008); ibid. 4, No. 2, 365--421 (1991; Zbl 0738.17011); \textit{I. Grojnowski} and \textit{G. Lusztig}, Contemp. Math. 153, 11--19 (1993; Zbl 1009.17502)] related with canonical bases. Their appearance hints ``that quantum groups are just shadows of more interesting structures where the canonical basis elements become objects of a category, multiplication becomes the tensor product in this category, and addition becomes a direct sum in this category'', which might be called a categorified quantum group. The thirty-third is [\textit{M. M. Kapranov} and \textit{V. A. Voevodsky}, Proc. Symp. Pure Math. 56, 177--259 (1994; Zbl 0809.18006)], initiating \(2\)-vector spaces and what are now called braided monoidal bicategories. The authors of the paper argued that, just as any solution of the Yang-Baxter equation gives a braided monoidal category, any solution of the Zamolodchikov tetrahedron equation gives a braided monoidal bicategory. The thirty-fourth is [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)], constructing invariants of 3-manifolds from quantum groups, which were later seen to be part of a full-fledged three-dimensional topological quantum field theory known as the Witten-Reshetikhin-Turaev theory. The thirty-fifth is [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)], constructing another invariant of 3-manifolds from the modular category stemming from quantum SU(2). It is now known as a part of a full-fledged three-dimensional topological quantum field theory, and its relation with the Witten-Reshetikhin-Turaev theory ``subtle and interesting.'' The thirty-sixth is [\textit{M. Fukuma} et al., Commun. Math. Phys. 161, No. 1, 157--175 (1994; Zbl 0797.57012)], which ``found a way to construct two-dimensional topological quantum field theories from semisimple algebras''. The authors ``essentially created a recipe to turn any two-dimensional cobordism into a string diagram'' and, with a little extra work, it gives a topological quantum field theory called \(Z\). The thirty-seventh is [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], which can be seen as ``a categorical version of the Fukuma-Hosono-Kawai construction''. The thirty-eighth is [\textit{V. G. Turaev}, J. Differ. Geom. 36, No. 1, 35--74 (1992; Zbl 0773.57012)], followed by [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], which ``amounts to a four-dimensional analogue of the Traev-Viro-Barrett-Westbury construction''. The thirty-ninth is famous [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] arriving at a deeper understanding of quantum groups, which was based on ideas of Witten. The fortieth is [\textit{R. J. Lawrence}, in: Quantum topology. Based on an AMS special session on topological quantum field theory, held in Dayton, OH, USA on October 30-November 1, 1992 at the general meeting of the American Mathematical Society. Singapore: World Scientific. 191--208 (1993; Zbl 0839.57017)] followed by [\textit{R. J. Lawrence}, J. Pure Appl. Algebra 100, No. 1--3, 43--72 (1995; Zbl 0827.57011)]. ``The essential point is that we can build any \(n\)-dimensional spacetime out of a few standard building blocks, which can be glued together locally in a few standard ways''. The forty-first is [\textit{L. Crane} and \textit{I. B. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)] discussing ``algebraic structures that provide topological quantum field theories in various low dimensions''. The forty-second is [\textit{D. S. Freed}, Commun. Math. Phys. 159, No. 2, 343--398 (1994; Zbl 0790.58007)] exhibiting ``how higher-dimensional algebraic structures arise naturally from the Lagrangian formulation of topological quantum field theory''. The forty-third is [\textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)], ``which led to a burst of work relating string theory to higher categorical structures''. The forty-fourth is [\textit{R. Gordon} et al., Mem. Am. Math. Soc. 558, 81 p. (1995; Zbl 0836.18001)], which ``is a precise working out of the categorical Eckman-Hilton argument''. Now we have finally come to [\textit{J. C. Baez} and \textit{J. Dolan}, J. Math. Phys. 36, No. 11, 6073--6105 (1995; Zbl 0863.18004)] as the forty-fifth. Its key part is the periodic table of \(n\)-categories. The last (the forty-sixth) is [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. The authors have stated that they are scientists rather than historians of science, so they were trying to make a specific scientific point rather than accurately describe every twist and turn in a complex sequence of events. But the reviewer should say regrettably that the authors have not made due effort to be exact and adequate in references and citations even as scientists. Nevertheless, the paper marvelously succeeded in following the threads of various ideas and intertwining them. The story that the authors narrate is highly interesting and even thrilling. Besides, the authors have tried every effort to make the paper more accessible by including a general introduction to \(n\)-categories. Applications of \(n\)-categories started only around the 1980s, particularly in string theory and spin foam models of loop quantum gravity. It should be stressed again and again that these physical theories are spectulative at present and are not ready for experimental tests at all. The history of physics is full of theories resulting in fiasco (e.g., Kaluza-Klein theory). Therefore the authors have modestly and wisely chosen to speak of a prehistory in place of a history. It is suggested that ``mathematical modelling is a way of life'' for everyone. The \(M\)'s of mathematics -- methods, models and modelling, and the \(R\)'s of mathematics -- reasoning, reading, 'riting and 'rithmetic are described. Their role in promoting the ``way of life'' is outlined and implications for the 16--19 curriculum are discussed. (Orig.)