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noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces representation theory of finite-dimensional algebras; tame hereditary algebras; tame bimodules; noncommutative curves of genus zero; noncommutative function fields of genus zero D. Kussin, Parameter curves for the regular representations of tame bimodules, J. Algebra, 320 (2008), no. 6, 2567--2582.Zbl 1197.16017 MR 2437515 Representations of associative Artinian rings, Noncommutative algebraic geometry, Special algebraic curves and curves of low genus, Rings arising from noncommutative algebraic geometry, Representation type (finite, tame, wild, etc.) of associative algebras Parameter curves for the regular representations of tame bimodules.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite fields; symmetric tensor rank; algebraic function field; tower of function fields; modular curve; Shimura curve Computational aspects of algebraic curves, Effectivity, complexity and computational aspects of algebraic geometry, Number-theoretic algorithms; complexity Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces tensor products of cyclic algebras; division algebras of prime index; division algebras over function fields; cubic divisors; central division algebras; ramification divisors; Brauer groups; exponents Michel Van den Bergh, Division algebras on \?² of odd index, ramified along a smooth elliptic curve are cyclic, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) Sémin. Congr., vol. 2, Soc. Math. France, Paris, 1997, pp. 43 -- 53 (English, with English and French summaries). Finite-dimensional division rings, Arithmetic theory of algebraic function fields, Quaternion and other division algebras: arithmetic, zeta functions, Brauer groups of schemes Division algebras on \(\mathbb{P}^2\) of odd index, ramified along a smooth elliptic curve are cyclic

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces birational classification of real rational surfaces; classification of function fields; ruled surface Silhol, R., Classification birationnelle des surfaces rationnelles réelles, 308-324, (1990), Berlin Special surfaces, Topology of real algebraic varieties, Rational and birational maps, Families, moduli, classification: algebraic theory, Arithmetic theory of algebraic function fields Birational classification of real rational surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces tensor product of quaternion algebras; central simple algebras; orthogonal involution; Brauer-Severi variety; involution variety; function fields; generic isotropic splitting field; Brauer groups; Quillen \(K\)-theory D. Tao, ''A variety associated to an algebra with involution'',J. Algebra,168, 479--520 (1994). Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Rings with involution; Lie, Jordan and other nonassociative structures, Brauer groups of schemes, Computations of higher \(K\)-theory of rings, Homogeneous spaces and generalizations A variety associated to an algebra with involution

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Brauer groups; indecomposable division algebras; noncrossed products; ramification; function fields of smooth curves; non-crossed product central division algebras; exponents; indices; periods; tensor products of central algebras E. Brussel, K. McKinnie, and E. Tengan, Indecomposable and noncrossed product division algebras over function fields of smooth \?-adic curves, Adv. Math. 226 (2011), no. 5, 4316 -- 4337. Finite-dimensional division rings, Skew fields, division rings, Algebraic functions and function fields in algebraic geometry, Brauer groups (algebraic aspects) Indecomposable and noncrossed product division algebras over function fields of smooth \(p\)-adic curves.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces division algebras; cyclic algebras; ramifications; étale cohomology; function fields of surfaces; affine schemes; Brauer groups; central algebras; fields of fractions; cyclic Galois extensions Colliot-Thélène, J.-L.: Conjectures de type local-global sur image des groupes de Chow dans la cohomologie étale. In: Algebraic K-theory (Seattle, WA, 1997), Proceedings of Symposia in Pure Mathematics, vol. 67, pp. 1-12. Amer. Math. Soc., Providence (1999) Finite-dimensional division rings, Étale and other Grothendieck topologies and (co)homologies, Brauer groups of schemes, Brauer groups (algebraic aspects) Division algebras over surfaces.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces perfect fields; simple two-sided central vector spaces; non-commutative symmetric algebras; division rings of fractions; non-commutative surfaces Hart, J.; Nyman, A.: Duals of simple two-sided vector spaces, Comm. algebra 40, 2405-2419 (2012) Bimodules in associative algebras, Vector spaces, linear dependence, rank, lineability, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry Duals of simple two-sided vector spaces.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic groups; adjoint groups; R-equivalence; nondyadic local fields; function fields of curves; algebras with involution; Hermitian forms; Rost invariant R. Preeti and A. Soman, Adjoint groups over \Bbb Q_{\?}(\?) and R-equivalence, J. Pure Appl. Algebra 219 (2015), no. 9, 4254 -- 4264. Linear algebraic groups over local fields and their integers, Quadratic forms over general fields, Bilinear and Hermitian forms, Classical groups, Galois cohomology of linear algebraic groups, Rational points, Other nonalgebraically closed ground fields in algebraic geometry, Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures Adjoint groups over \(\mathbb Q_p(X)\) and R-equivalence.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces abelian Galois extensions; relative Brauer groups; cyclic extensions; indecomposable division algebras of prime exponent; central simple algebras; Brauer class; rational function fields Finite-dimensional division rings, Equations in general fields, Valued fields, Brauer groups of schemes, Separable extensions, Galois theory Dec groups for arbitrarily high exponents

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite groups; finite simple groups; applications of simple groups; Brauer groups; Riemann surfaces; polynomials; function fields Guralnick, Robert, Applications of the classification of finite simple groups.Proceedings of the International Congress of Mathematicians---Seoul 2014. Vol. II, 163-177, (2014), Kyung Moon Sa, Seoul Finite simple groups and their classification, Primitive groups, Coverings of curves, fundamental group, Algebraic field extensions Applications of the classification of finite simple groups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces flat localizations of Abelian categories; structure presheaves of modules; quantized algebras; noncommutative schemes in categories; left spectrum; maximal left ideals; completely prime left ideals; categories of rings; Levitzki radical; quasi-affine schemes; projective spectra; quantized rings; quantum planes; algebra of \(q\)-differential operators; Weyl algebras; quantum envelopes; coordinate rings; generalized Weyl algebras; skew polynomial rings; Serre subcategories; Grothendieck categories; hyperbolic rings; skew PBW monads; monoidal category; Kac-Moody and Virasoro Lie algebras; semigroup-graded monads; Gabriel-Krull dimension Rosenberg, A.L.: Algebraic Geometry Representations of Quantized Algebras. Kluwer Academic Publishers, Dordrecht, Boston London (1995) Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Torsion theories; radicals on module categories (associative algebraic aspects), Rings of differential operators (associative algebraic aspects), Local categories and functors, Abelian categories, Grothendieck categories, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, ``Super'' (or ``skew'') structure, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects) Noncommutative algebraic geometry and representations of quantized algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite dimensional representation; symmetric algebra; stable isomorphism; invariant fields; reductive linear groups; division algebras; function field; Brauer-Severi variety David J. Saltman, Invariant fields of linear groups and division algebras, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 279 -- 297. Division rings and semisimple Artin rings, Other matrix groups over rings, Vector and tensor algebra, theory of invariants, Representation theory for linear algebraic groups, Brauer groups of schemes, Galois cohomology, Rational and unirational varieties Invariant fields of linear groups and division algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces two dimensional global fields; algebraic function field in one; variable over algebraic number field; Galois cohomology group; \(H^ 3\); Hasse principles; local-global principles; reduced norms; division algebras; quadratic forms; sum of squares K.~Kato, {A {H}asse principle for two dimensional global fields. With an appendix by {J}.-{L} {C}olliot-{T}hélène.}, J. Reine Angew. Math. {366} (1986), 142--180. DOI 10.1515/crll.1986.366.142; zbl 0576.12012; MR0833016 Galois cohomology, Brauer groups of schemes, Quadratic forms over global rings and fields, Galois cohomology, Quaternion and other division algebras: arithmetic, zeta functions, Waring's problem and variants, Arithmetic theory of algebraic function fields A Hasse principle for two dimensional global fields. Appendix by Jean-Louis Colliot-Thélène

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces global deformations of Lie algebras; Lie algebras of meromorphic vector fields on marked Riemann surfaces; cuspidal cubic; nodal cubic; current Lie algebra; Krichever-Novikov Lie algebra Homological methods in Lie (super)algebras, Cohomology of Lie (super)algebras, Infinite-dimensional Lie (super)algebras, Lie algebras of vector fields and related (super) algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Virasoro and related algebras, Families, moduli of curves (algebraic), Elliptic curves Deformations of the Witt, Virasoro, and current algebra

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic surface in projective space; function fields of surfaces; subfields of function fields of algebraic surfaces; dominant rational maps; plane curves Lee, Y; Pirola, G, On subfields of the function field of a general surface in \({\mathbb{P}}^3\), Int. Math. Res. Not., 24, 13245-13259, (2015) Surfaces of general type, Plane and space curves, Real and complex fields On subfields of the function field of a general surface in \(\mathbb{P}^{3}\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central division algebras; cyclic algebras; ramification; curve points; nodal points; Brauer groups; curves over local fields; \(p\)-adic curves; field extensions; algebraic function fields; curves over rings of integers of \(p\)-adic fields D. J. Saltman, ''Cyclic algebras over \(p\)-adic curves,'' J. Algebra, vol. 314, iss. 2, pp. 817-843, 2007. Finite-dimensional division rings, Curves over finite and local fields, Arithmetic ground fields for curves, Brauer groups of schemes, Skew fields, division rings, Algebras and orders, and their zeta functions, Algebraic functions and function fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local ground fields in algebraic geometry, Special surfaces Cyclic algebras over \(p\)-adic curves.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces graded algebras; global dimension; homogeneous coordinate rings of projective surfaces; Artin-Schelter regular algebras; skew polynomial rings; elliptic algebras D. R. Stephenson, ''Artin-Shelter regular algebras of global dimension three,'' J. Algebra, 183, No. 1, 55--73 (1996). Graded rings and modules (associative rings and algebras), Homological dimension in associative algebras, Noetherian rings and modules (associative rings and algebras), Ordinary and skew polynomial rings and semigroup rings, Elliptic curves Artin-Schelter regular algebras of global dimension three

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite subgroups of rotation group; groups; linear algebra; infinite dimensional spaces; systems of linear differential equations; symmetry; free groups; generators; relations; Todd-Coxeter algorithm; bilinear forms; spectral theorems; linear groups; group representations; rings; algebraic geometry; factorization; modules; function fields and their relations to Riemann surfaces; Galois theory Artin, M.: Algebra. Prentice-Hall, Englewood Cliffs (1991) Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory Algebra

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces elliptic surfaces; elliptic curves over function fields; generators of Mordell-Weil group; Kodaira-Néron model; number of minimal sections; specialization homomorphisms Arithmetic varieties and schemes; Arakelov theory; heights, Rational points, Elliptic curves, Elliptic curves over global fields, Finite ground fields in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations Mordell-Weil lattices and Galois representation. II

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Cremona group; noncommutative algebras; deformation quantization of rational surfaces Usnich, Alexandr. \(Action of the Cremona group on a noncommutative ring\). Adv. Math. 228 (2011), no. 4, 1863-1893. Birational automorphisms, Cremona group and generalizations, Noncommutative algebraic geometry Action of the Cremona group on a noncommutative ring

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces skew-symmetric tensors; algebra of polynomial functions; maximal unipotent subgroups; algebras of invariants; explicit construction F. D. Grosshans, The symbolic method and representation theory, Adv. Math., to appear. Representation theory for linear algebraic groups, Actions of groups on commutative rings; invariant theory, Vector and tensor algebra, theory of invariants, Classical groups (algebro-geometric aspects) The symbolic method and representation theory

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces groups; linear algebra; infinite dimensional spaces; systems of linear differential equations; symmetry; finite subgroups of rotation group; free groups; generators; relations; Todd-Coxeter algorithm; bilinear forms; spectral theorems; linear groups; group representations; rings; algebraic geometry; factorization; modules; function fields and their relations to Riemann surfaces; Galois theory Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory Algebra

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces indecomposable division algebras; noncrossed product division algebras; patching over fields; smooth projective curves; completions of function fields; Brauer groups Chen, F.: Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings, (2010) Finite-dimensional division rings, Brauer groups (algebraic aspects), Brauer groups of schemes, Algebraic functions and function fields in algebraic geometry Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces moduli space of marked Riemann surfaces; singular Riemann surfaces; Lie algebras of meromorphic vector fields; elliptic curves; complex tori; algebraic geometric degeneration; Riemann sphere M. Schlichenmaier, ''Degenerations of Generalized Krichever-Novikov Algebras on Tori,'' J. Math. Phys. 34, 3809--3824 (1993). Virasoro and related algebras, Riemann surfaces; Weierstrass points; gap sequences Degenerations of generalized Krichever-Novikov algebras on tori

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central simple algebras; irreducible lattices; rings of invariants; function fields; normal varieties; coordinate rings; reduced traces; Cayley-Hamilton algebras; étale local classes; smooth orders Lieven Le Bruyn, ''Non-smooth algebra with smooth representation variety (asked in MathOverflow)'', Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Finite-dimensional division rings, Algebraic functions and function fields in algebraic geometry, Actions of groups and semigroups; invariant theory (associative rings and algebras) Local structure of Schelter-Procesi smooth orders

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces field of definition; Belyi's theorem; minimal surfaces; ruled surfaces; number fields Other nonalgebraically closed ground fields in algebraic geometry, Global ground fields in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.) Fields of definition and Belyi type theorems for curves and surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces alternative algebra; quadratic algebra; composition algebras; algebraic curves of genus zero; locally ringed spaces; Cayley-Dickson doubling process; Zorn's vector matrices; octonion algebras; Zorn algebras; function fields of genus zero; polynomial rings Petersson, H.: Composition algebras over algebraic curves of genus 0. Trans. Am. Math. Soc. 337, 473--491 (1993) Composition algebras, Curves in algebraic geometry, Quadratic algebras (but not quadratic Jordan algebras), Alternative rings Composition algebras over algebraic curves of genus zero

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finitely generated modules; trace ideals; finite groups; skew group rings; actions; rings of invariants; symmetric algebras; rational representations; reductive algebraic groups M. P. Holland, \(K\)-theory of endomorphism rings and of rings of invariants , J. Algebra 191 (1997), no. 2, 668-685. Grothendieck groups, \(K\)-theory, etc., Automorphisms and endomorphisms, Endomorphism rings; matrix rings, \(K_0\) of other rings, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups \(K\)-theory of endomorphism rings and of rings of invariants

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic \(F\)-representations; retract rational extensions; stably isomorphic extensions; lifting property; Azumaya algebras; fields of invariants; rational function fields; generic division algebras; central simple algebras D. J. Saltman, J.-P. Tignol, Generic algebras with involution of degree 8m, J. Algebra 258 (2002), no. 2, 535--542. Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Geometric invariant theory Generic algebras with involution of degree \(8m\).

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces bilinear complexity; congruence function fields; descent of function fields; tensor rank; finite fields; Artin--Schreier extensions Ballet, Stéphane; Le Brigand, Dominique; Rolland, Robert, On an application of the definition field descent of a tower of function fields.Arithmetics, geometry, and coding theory (AGCT 2005), Sémin. Congr. 21, 187-203, (2010), Soc. Math. France, Paris Number-theoretic algorithms; complexity, Curves over finite and local fields, Arithmetic theory of algebraic function fields, Arithmetic ground fields for curves On an application of the definition field descent of a tower of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central simple algebras; involution of orthogonal type; \(\sigma\)- symmetric elements; reduced norm; similitudes; quaternion algebras; discriminant extensions; Clifford algebras; algebras with involution; Clifford bimodules; Clifford groups; homogeneous varieties; semisimple linear algebraic groups; Brauer groups Merkurjev, A.; Tignol, J., The multipliers of similitudes and the Brauer group of homogeneous varieties, Journal für die Reine und Angewandte Mathematik, 461, 13-47, (1995) Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures, Brauer groups of schemes, Clifford algebras, spinors, Linear algebraic groups over arbitrary fields, Group actions on varieties or schemes (quotients) The multipliers of similitudes and the Brauer group of homogeneous varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces simple Lie algebras; Lie algebras of algebraic; symplectic and Hamiltonian vector fields; smooth affine curves; Danielewski surfaces; locally nilpotent derivations Lie algebras of vector fields and related (super) algebras, Classification of affine varieties Bracket width of simple Lie algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Riemann surfaces; Abelian integrals; algebraic number theory; function fields; valuations; function fields of curves; Abel-Jacobi theorem Cohn, P. M.: Algebraic numbers and algebraic functions, Chapman \& Hall math. Ser. (1991) Algebraic number theory: global fields, Arithmetic theory of algebraic function fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Algebraic functions and function fields in algebraic geometry Algebraic numbers and algebraic functions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ruled surfaces; structure function-pitch function-angle function of ruled surface; Minkowski space Rational and ruled surfaces, Minkowski geometries in nonlinear incidence geometry Structure and characterization of ruled surfaces in Minkowski 3-space

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces del Pezzo surfaces; fibrations; function fields of curves; rational points; intermediate Jacobians Hassett, B; Tschinkel, Y, Quartic del Pezzo surfaces over function fields of curves, Cent. Eur. J. Math., 12, 395-420, (2014) Arithmetic ground fields (finite, local, global) and families or fibrations, Families, moduli, classification: algebraic theory, Structure of families (Picard-Lefschetz, monodromy, etc.), Jacobians, Prym varieties, Rational points, Fano varieties Quartic del Pezzo surfaces over function fields of curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central division algebras; anisotropic orthogonal involutions; Springer-Satz for orthogonal involutions; quadratic forms; motives of quadrics; function fields; Brauer-Severi varieties; Witt index; Chow motives А. С. Меркурьев, А. А. Суслин, \textit{K-когомологии многообpaзий Севери-Брауэра и гомоморфизм норменного вычета}, Изв. АН СССР, cep. мат \textbf{46} (1982), no. 5, 1011-1046. Engl. transl.: A. Merkurjev, A. Suslin, \textit{K-cohomology of Severi\(-\)Brauer varieties and the norm residue homomorphism}, Math. of the USSR-Izvestiya \textbf{21} (1983), 2, 307-340. Finite-dimensional division rings, Algebraic theory of quadratic forms; Witt groups and rings, Rings with involution; Lie, Jordan and other nonassociative structures, \(K\)-theory of quadratic and Hermitian forms, (Equivariant) Chow groups and rings; motives, Rational and birational maps On anisotropy of orthogonal involutions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces SDP-Hopf algebra; Lie algebra; smooth formal group (law); co(ntra)variant bialgebra; sequence of divided powers; primitive element; pure primitive; curve; symmetric function; quasisymmetric function; noncommutative symmetric function; composition; Lyndon composition Symmetric functions and generalizations, Formal groups, \(p\)-divisible groups, Hopf algebras and their applications Various \(S\)-adic symmetric functions and smooth formal groups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces artinian algebras; thick points; noncommutative deformations; deformation functor; versal deformation; moduli suite; extensions; simple modules; phase space functor; Hochschild cohomology; Massey products; representations of associative algebras; Toy model; blow-ups; desingularizations, Hilbert schemes; Chern classes, Dirac derivation, de Rham complex; Jacobian conjecture; dynamical structure; swarms; metrics, gravitation; quantum gravitation; energy; Clifford algebras; Chern-Simons classes; Yang-Mills theory; heat equation; thermodynamics; Kepler laws; heat equation; Navier-Stokes equation; Schrödinger equation; Einstein field equation; entropy; cosmology; cosmological time; density of mass; inflation; cyclical cosmology; conformally trivial cosmological model; universe, observers; photons; red-shift; entanglement; consciousness; super symmetry bosonic fields; fermionic fields; gluons; quarks; charge; black energy; black mass Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Plane and space curves, Simple and semisimple modules, primitive rings and ideals in associative algebras, Representations of orders, lattices, algebras over commutative rings, Representation theory of associative rings and algebras, Rings arising from noncommutative algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory, Noncommutative geometry in quantum theory, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Quantum field theory; related classical field theories, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory, Relativistic cosmology Mathematical models in science

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central division algebras; indecomposable division algebras; Schur indices; Brauer groups; Brauer-Severi varieties; excellent field extensions; multidimensional Laurent series fields; nondegenerate projective conics; projective quadrics; \(u\)-invariants; tensor products of quaternion algebras Finite-dimensional division rings, Field extensions, Homological methods (field theory), Non-Archimedean valued fields, Brauer groups of schemes, Special algebraic curves and curves of low genus, Algebraic theory of quadratic forms; Witt groups and rings, Brauer groups (algebraic aspects) On some elements of the Brauer group of a conic.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central simple algebras; Brauer groups; adjoint semisimple linear algebraic groups; Borel varieties; twisted flag varieties; projective varieties; change of Schur index; function fields; groups of inner type; parabolic subgroups; index reduction formula; Brauer-Severi varieties A. S. Merkurjev, I. A. Panin, A. R. Wadsworth, \textit{Index reduction formulas for twisted flag varieties}. I, \(K\)-Theory \textbf{10} (1996), no. 6, 517-596. Finite-dimensional division rings, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, \(K\)-theory of schemes Index reduction formulas for twisted flag varieties. I

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces compact Riemann surfaces; algebraic function fields of one variable Riemann surfaces; Weierstrass points; gap sequences, Algebraic functions and function fields in algebraic geometry, Compact Riemann surfaces and uniformization Compact Riemann surfaces and algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces multiplicative structure; division algebras over number fields; reduced norm; group of rational points; anisotropic algebraic group; maximal cyclic subfield; SL(1,D); Hasse norm principle; central simple skew field Quaternion and other division algebras: arithmetic, zeta functions, Class field theory, Linear algebraic groups over global fields and their integers, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Rational points, Other algebraic groups (geometric aspects) About the multiplicative structure of division algebras over number fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces triviality of algebras over rational function fields; rationality of conic bundle; local global principle I. I. Voronovich, A local-global principle for algebras over fields of rational functions, Dokl. Akad. Nauk BSSR 31 (1987), no. 10, 877 -- 880, 956 (Russian, with English summary). Quaternion and other division algebras: arithmetic, zeta functions, Brauer groups of schemes, Rational and unirational varieties, Algebraic functions and function fields in algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) A local-global principle for algebras over rational function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces lattices; invariant fields; Picard groups; division algebras; tensor squares; symmetric squares; exterior squares Lemire, Nicole; Lorenz, Martin, On certain lattices associated with generic division algebras, J. Group Theory, 1433-5883, 3, 4, 385-405, (2000) Finite-dimensional division rings, Rationality questions in algebraic geometry, Representations of orders, lattices, algebras over commutative rings, Linear algebraic groups over arbitrary fields, Geometric invariant theory, Representations of finite symmetric groups On certain lattices associated with generic division algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces linear algebraic group; R-equivalence; function fields of surfaces; weak approximation; Hasse principle Colliot-Thélène, J.-L.; Gille, P.; Parimala, R., Arithmetic of linear algebraic groups over two-dimensional geometric fields, Duke Math. J., 121, 285-341, (2004) Forms of degree higher than two, Modular and Shimura varieties, Linear algebraic groups over adèles and other rings and schemes, Linear algebraic groups over arbitrary fields, Global ground fields in algebraic geometry Arithmetic of linear algebraic groups over 2-dimensional geometric fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central simple algebras; Brauer groups; semisimple linear algebraic groups; Borel varieties; twisted flag varieties; projective varieties; change of Schur index; function fields; groups of inner type; parabolic subgroups; index reduction formula; Brauer-Severi varieties Merkurjev, A.; Panin, A.; Wadsworth, A., \textit{index reduction formulas for twisted flag varieties II}, J. K-Theory, 14, 101-196, (1998) Finite-dimensional division rings, Group actions on varieties or schemes (quotients), Quadratic spaces; Clifford algebras, Representation theory for linear algebraic groups, \(K\)-theory of schemes Index reduction formulas for twisted flag varieties. II

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function field of a curve; ultrametric valuation; function fields of surfaces; absolute values of a field; product formula; infinite extensions Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Valued fields Arithmetic on infinite extensions of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces semisimple linear algebraic groups; motivic invariants; indices of Tits algebras; Grothendieck filtrations; algebras with orthogonal involution; quadratic forms; function fields of Severi-Brauer varieties Quéguiner-Mathieu, A.; Semenov, N.; Zainoulline, K., J. Pure Appl. Algebra, 216, 2614-2628, (2012) Linear algebraic groups over arbitrary fields, (Equivariant) Chow groups and rings; motives, Rings with involution; Lie, Jordan and other nonassociative structures, Algebraic theory of quadratic forms; Witt groups and rings The \(J\)-invariant, Tits algebras and triality.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces non commutative projective geometry; morphisms to projective bundles; functor of points; symmetric algebras; bimodules Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry Morphisms to noncommutative projective lines

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces affine Lie algebras; abelian varieties; modular invariant; partition function; rational conformal field theory; Jacobian of a Fermat curve; triangulated surfaces; Riemann surface M. Bauer, A. Coste, C. Itzykson and P. Ruelle, ''Comments on the links between SU(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards,'' J. Geom. Phys. 22 (1997), 134--189. Jacobians, Prym varieties, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Complex multiplication and abelian varieties Comments on the links between \(su(3)\) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces rationally simply connected varieties; projective homogeneous spaces; rational points; function fields of algebraic surfaces Fibrations, degenerations in algebraic geometry, Homogeneous spaces and generalizations, Rationally connected varieties, Stacks and moduli problems Homogeneous space fibrations over surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces monads; higher Chow complex; operads; unital \(k\)-algebras; little \(n\)-cubes operad; tensor category; braid algebras; mixed Tate motives; symmetric monoidal category; operad of spaces; iterated loop spaces; higher Chow groups; Adams operations; derived category; integral mixed Tate modules; derived categories of modules; DGA; triangulated category; Tannakian category; Hopf algebra; co-Lie algebra; Beilinson-Soulé conjecture; operadic tensor product; cellular approximation theorem Kriz, I.; May, J. P., Operads, algebras, modules and motives, Astérisque, 233, (1995), iv+145 pp Research exposition (monographs, survey articles) pertaining to category theory, Applied homological algebra and category theory in algebraic topology, Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Homological algebra in category theory, derived categories and functors, Higher algebraic \(K\)-theory, \(K\)-theory in geometry, Generalizations (algebraic spaces, stacks) Operads, algebras, modules and motives

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function field; tower of function fields; tensor rank; algorithm; finite field Pieltant, Julia; Randriam, Hugues, New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields, Math. Comp., 0025-5718, 84, 294, 2023-2045, (2015) Algebraic functions and function fields in algebraic geometry, Number-theoretic algorithms; complexity, Finite fields (field-theoretic aspects) New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces theta functions; bounded symmetric domains; imaginary quadratic number fields; rings of integers; lattices; modular forms K. Matsumoto: Algebraic relations among some theta functions on the bounded symmetric domain of type \(I_r,r\) , Kyushu J. Math. 60 (2006), 63--77. Other groups and their modular and automorphic forms (several variables), Theta series; Weil representation; theta correspondences, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Theta functions and abelian varieties, Algebraic numbers; rings of algebraic integers Algebraic relations among some theta functions on the bounded symmetric domain of type \(I_{r,r}\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hasse-Weil-Serre bound; zeta function of curves over finite fields; rational points K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, Institut de Mathématiques de Luminy, preprint, 1999, pp. 99--29. Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic ground fields for curves Improved upper bounds for the number of rational points on algebraic curves over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces hyperbolic fibre space; higher dimensional analogue of Mordell's conjecture for curves; hyperbolic manifolds; algebraic families of hyperbolic varieties; Mordell's conjecture over function fields Noguchi, J.Hyperbolic fiber spaces and Mordell's conjecture over function fields, Publ. Research Institute Math. Sciences Kyoto University21, No. 1 (1985), 27--46. Hyperbolic and Kobayashi hyperbolic manifolds, Holomorphic bundles and generalizations, Families, moduli, classification: algebraic theory Hyperbolic fibre spaces and Mordell's conjecture over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces non commutative algebraic geometry; surface; blow up; graded algebras of Gelfand-Kirillov dimension three; Abelian categories; Rees algebra; pseudo-compact rings; completion functors; derived categories; Del Pezzo surfaces; quantum version of projective three space M. Van~den Bergh, \emph{Blowing up of non-commutative smooth surfaces}, Mem. Amer. Math. Soc. \textbf{154} (2001), no.~734, x+140. \MR{1846352 (2002k:16057)} Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Blowing up of non-commutative smooth surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces \(K3\) surfaces; Chow groups; Hilbert schemes of points; Lie algebras \(K3\) surfaces and Enriques surfaces, (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes) A Lie algebra action on the Chow ring of the Hilbert scheme of points of a \(K3\) surface

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces classification of Brauer groups; rational function fields over global fields; Ulm invariants B. Fein, M.M. Schacher and J. Sonn, Brauer groups of rational function fields, Bull. Amer. Math. Soc. 1, 766-768. Arithmetic theory of algebraic function fields, Galois cohomology, Transcendental field extensions, Brauer groups of schemes Brauer groups of rational function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; valuation; value group; rank; direct sum of n infinite cyclic groups MacLane, S. - Schilling, O.F.G.\(\,\): Zero-dimensional branches of rank 1 on algebraic varieties, Annals of Math. 40 (1939), 507-520 Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Valued fields Zero-dimensional branches of rank one on algebraic varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Enriques surfaces; quaternion algebras; moduli schemes of sheaves \(K3\) surfaces and Enriques surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Rank one sheaves over quaternion algebras on Enriques surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces 2-adic valuations of ratio of products of factorials; parity of degrees of determinantal varieties; subspaces of real skew symmetric matrices; subspaces of real rectangular matrices; parity of number of plane partitions; parity of number of symplectic tableaux Beauville, A.: Surfaces algébriques complexes, Astérisque 54, Soc. Math. de France (1978) Binomial coefficients; factorials; \(q\)-identities, Combinatorial aspects of representation theory, Congruences; primitive roots; residue systems, Determinantal varieties, Hermitian, skew-Hermitian, and related matrices 2-adic valuations of certain ratios of products of factorials and applications

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces pole placement; feedback control; orthogonal Grassmannian; geometry of Grassmannian manifolds; Lagrangian Grassmannian; skew-symmetric matrix Pole and zero placement problems, Grassmannians, Schubert varieties, flag manifolds, Geometric methods Complex static skew-symmetric output feedback control

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hilbert function; points in \(\mathbb{P}^3\); irreducible surface; number of generators of the ideal of distinct points; resolution of points; surfaces of low degree Guardo E., Parisi O.,Maximum number of generators of an ideal of points on an irreducible surface of lows degree, Le Matematiche,50 (1995), 137--162. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, complete intersections and determinantal ideals, Low codimension problems in algebraic geometry, Special surfaces Maximum number of generators of an ideal of points on an irreducible surface of low degree

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; Brauer group; theorem of Davenport-Halberstam Serre, Jean-Pierre, Spécialisation des éléments de \(\operatorname{Br}_2(\mathbf{Q}(T_1, \ldots, T_n))\), C. R. Acad. Sci. Paris, Sér. I, 311, 7, 397-402, (1990) Brauer groups of schemes, Galois cohomology Spécialisation des éléments de \(Br_ 2({\mathbb{Q}}(T_ 1,\cdot \cdot \cdot,T_ n))\). (Specialization of elements of \(Br_ 2({\mathbb{Q}}(T_ 1,\cdot \cdot \cdot,T_ n)))\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Abelian varieties over finite fields; Jacobian; zeta function; abelian surfaces DOI: 10.1016/j.crma.2012.10.001 Finite ground fields in algebraic geometry, Abelian varieties of dimension \(> 1\), Varieties over finite and local fields On the number of points on abelian and Jacobian varieties over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces gonality; deformations of space curves; Hilbert scheme; ruled surfaces; Zariski tangent space; Brill-Noether number E. Mezzetti and G. Sacchiero, Gonality and Hilbert schemes of smooth curves, in Algebraic Curves and Projective Geometry, (Trento, 1988), 183--194, Lecture Notes in Math. 1389, Springer, Berlin, 1989. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Gonality and Hilbert schemes of smooth curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Picard numbers; rank of the Mordell-Weil group; elliptic curves over function fields; automorphisms Peter F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific J. Math. 128 (1987), no. 1, 157 -- 189. Picard groups, Special surfaces, Group actions on varieties or schemes (quotients) The Picard numbers of elliptic surfaces with many symmetries

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces normal function; Hermitian symmetric domain; Mumford-Tate group; variation of Hodge structure; algebraic cycle Variation of Hodge structures (algebro-geometric aspects), Algebraic cycles, Homogeneous spaces and generalizations, Lie algebras of linear algebraic groups, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Period matrices, variation of Hodge structure; degenerations Normal functions over locally symmetric varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Cox rings; algebraic varieties; homogeneous spaces; graded algebras and rings; line bundles; toric varieties; geometric invariant theory; actions of groups; algebraic surfaces; Mori Dream Spaces; Zariski decompositions; Manin's conjecture; Hasse principle; Brauer-Manin obstructions; del Pezzo surfaces; \(K3\) surfaces; Enriques surfaces; GKZ decompositions; GALE transformations; flag varieties; combinatorial methods in algebraic geometry Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, Jürgen; Laface, Antonio, Cox rings, Cambridge Studies in Advanced Mathematics 144, viii+530 pp., (2015), Cambridge University Press, Cambridge Research exposition (monographs, survey articles) pertaining to algebraic geometry, Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Cox rings

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces automorphisms of Riemann surfaces; symmetric Riemann surfaces; Fuchsian and NEC groups; ovals Riemann surfaces, Automorphisms of curves, Fuchsian groups and their generalizations (group-theoretic aspects) The groups generated by maximal sets of symmetries of Riemann surfaces and extremal quantities of their ovals

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; constructions of linear codes; algebraic curves; algebraic-geometric codes; Goppa codes Ferruh Özbudak and Henning Stichtenoth, Constructing codes from algebraic curves, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2502 -- 2505. Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory), Arithmetic theory of algebraic function fields, Applications to coding theory and cryptography of arithmetic geometry Constructing codes from algebraic curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces rational points; hypersurface; counting function; multiple exponential sum; singular locus; Deligne's bounds for exponential sums; number of points; hypersurfaces over finite fields Heath-Brown, DR, The density of rational points on nonsingular hypersurfaces, Proc. Indian Acad. Sci. Math. Sci., 104, 13-29, (1994) Arithmetic algebraic geometry (Diophantine geometry), Rational points, Estimates on exponential sums The density of rational points on non-singular hypersurfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; biquadratic curves; biquadratic covers; number of points over finite fields; arithmetic statistics Curves over finite and local fields, Coverings of curves, fundamental group, Relations with random matrices Statistics for biquadratic covers of the projective line over finite fields. With an appendix by Alina Bucur

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite groups; automorphism groups of function fields; hyperelliptic function-field R. Brandt, Über die Automorphismengruppen von algebraischen Funktionenkörpern, PhD thesis, Universität Essen, 1988. Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On the groups of automorphisms of algebraic function fields.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces resolutions; minimal contraction of a ruled surface; index of non-rational numerical Del Pezzo surfaces; Moishezon surface Fujisawa T.: On non-rational numerical del Pezzo surfaces. Osaka J. Math. 32, 613--636 (1995) Rational and ruled surfaces, Compact complex surfaces On non-rational numerical Del Pezzo surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces automorphism groups of algebraic function fields; realization of group as Galois group; Galois theory Separable extensions, Galois theory, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Representations of groups as automorphism groups of algebraic systems Zur Realisierbarkeit endlicher Gruppen als Automorphismengruppen algebraischer Funktionenkörper

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Gauss conjecture; modular curves; Drinfeld modular curves; class field tower; congruence function fields; ring of \(S\)-integers; ideal class number; class number Lachaud, G.; Vladut, S.: Gauss problem for function fields, J. number theory 85, No. 2, 109-129 (2000) Arithmetic theory of algebraic function fields, Cyclotomic function fields (class groups, Bernoulli objects, etc.), Class field theory, Finite ground fields in algebraic geometry, Jacobians, Prym varieties, Arithmetic aspects of modular and Shimura varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Curves over finite and local fields Gauss problem for function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces valued function fields; genus change; algebraic function field; reduction of constants; rigid analytic geometry; non-discrete valuation Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Non-Archimedean valued fields, Arithmetic ground fields for surfaces or higher-dimensional varieties Genre des corps de fonctions values après Deuring, Lamprecht et Mathieu. (Genus of valued function fields after Deuring, Lamprecht and Mathieu)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces towers of function fields; Drinfeld modules; curves with many points Algebraic functions and function fields in algebraic geometry, Families, moduli of curves (algebraic), Arithmetic theory of algebraic function fields, Computational aspects of algebraic curves Good towers of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; domain of regularity; Hilbert's irreducibility theorem Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Über die Kennzeichnung algebraischer Funktionenkörper durch ihren Regularitätsbereich

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric spaces; prehomogeneous vector space of commutative parabolic type; zeta function Bopp, N.; Rubenthaler, H.: Fonction zêta associée à la série principale sphérique de certains espaces symétriques. Ann. sci. École norm. Sup. (4) 26, No. 6, 701-745 (1993) Homogeneous spaces and generalizations, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Zeta function associated to the spherical principal series of some symmetric spaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields of one variable over finite fields; Gauss sum; non- polynomial class {\#}1 rings Thakur D. : Gauss sums for function fields , J. Number Theory 37 (1991) 242-252. Arithmetic theory of algebraic function fields, Other character sums and Gauss sums, Drinfel'd modules; higher-dimensional motives, etc., Finite ground fields in algebraic geometry Gauss sums for function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces extensions of function field; generic Galois extension; Kummer theory; Leopoldt's conjecture; cyclotomic fields; geometric class field theory C. Greither, Cyclic Galois extensions of commutative rings. Lecture Notes in Mathematics, vol. 1534. Springer, Berlin-Heidelberg-New York, 1992. Zbl0788.13003 MR1222646 Galois theory and commutative ring extensions, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to number theory, Extension theory of commutative rings, Cyclotomic extensions, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Ramification and extension theory, Ramification problems in algebraic geometry, Coverings in algebraic geometry Cyclic Galois extensions of commutative rings

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Mots; words; trees; permutations; toric variety; Weyl chambers; semigroups; Lie algebra; Burnside problem for semigroups; symmetric group; skew tableaux; hypermaps; combinatorial theory; representations; continued fractions; differential algebra; probability measures; grammars of zigzags; complexity; finite automaton M. Lothaire , Mots . Hermès Paris 1990 . MR 1252659 | Zbl 0862.05001 Proceedings, conferences, collections, etc. pertaining to combinatorics, Proceedings, conferences, collections, etc. pertaining to computer science, Proceedings, conferences, collections, etc. pertaining to group theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Collections of articles of miscellaneous specific interest, Festschriften Words. Miscellany offered to M.-P. Schützenberger

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces unirationality for conic bundles; rational point; rational conic bundle surfaces; finite dimensional central simple algebras; large arithmetic fields Yanchevskiĭ, V. I., Astérisque, 209, 311-320, (1992), Soc. Math. France, Paris Rational and unirational varieties, Rational points, Global ground fields in algebraic geometry, Varieties over finite and local fields \(K\)-unirationality of conic bundles over large arithmetic fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hasse principle; function fields of \(p\)-adic curves Parimala, R.: A Hasse principle for quadratic forms over function fields. Bull. amer. Math. soc. (N.S.) 51, No. 3, 447-461 (2014) Quadratic forms over general fields, Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields A Hasse principle for quadratic forms over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Jacobian Kummer surfaces; Hessian model; Weber hexad; Hutchinson-Weber involution; degeneration; Comessatti surface; outer automorphisms of the symmetric group \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties, Complex multiplication and abelian varieties, Symmetric groups Hutchinson-Weber involutions degenerate exactly when the Jacobian is Comessatti

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; holomorphic semisimple differentials; p- extensions of \({\mathbb{Z}}_ pfields\) of CM-type; p-class group G. Villa and M. Madan,Structure of semisimple differentials and p-class groups in \(\mathbb{Z}\) p -extensions. Manuscripta Mathematica57 (1987), 315--350. Cyclotomic extensions, Arithmetic theory of algebraic function fields, Iwasawa theory, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Algebraic functions and function fields in algebraic geometry Structure of semisimple differentials and p-class groups in \({\mathbb{Z}}_ p\)-extensions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; plane curves of genus one; exceptional points Nagell, T.: [3] ''Les points exceptionnels sur les cubiques planes du premier genre'', II, ibid. Nova Acta Reg. Soc. Sci. Upsaliensis, Ser. IV, 14, 1946, No. 3. Algebraic functions and function fields in algebraic geometry, Special algebraic curves and curves of low genus Les points exceptionnels sur les cubiques planes du premier genre

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces number of rational points; Deligne-Lusztig curves; function fields; large groups of automorphisms; Goppa codes HP Johan~P. Hansen and Jens~Peter Pedersen, \emph Automorphism groups of Ree type, Deligne-Lusztig curves and function fields, J. Reine Angew. Math. \textbf 440 (1993), 99--109. Algebraic functions and function fields in algebraic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Arithmetic ground fields for curves, Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields Automorphism groups of Ree type, Deligne-Lusztig curves and function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces division algebras; tensor products; Schur indices; ramification; Picard groups; Brauer groups; products of curves; domains Louis Rowen and David J. Saltman, Tensor products of division algebras and fields, J. Algebra 394 (2013), 296 -- 309. Finite-dimensional division rings, Brauer groups of schemes, Picard groups, Brauer groups (algebraic aspects), Galois cohomology Tensor products of division algebras and fields.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces parabolic curve; asymptotic fields of lines; real algebraic surfaces; quadratic differential forms Affine differential geometry, Surfaces in Euclidean and related spaces, Real algebraic sets, Enumerative problems (combinatorial problems) in algebraic geometry, Implicit functional-differential equations, Nonlinear differential equations in abstract spaces On the geometric structure of certain real algebraic surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Kodaira's theory; Meyer function; algebraic surfaces of general type; Lefschetz fibrations Ashikaga, T.; Endo, H.: Various aspects of degenerate families of Riemann surfaces, Sūgaku expo. 19, No. 2, 171-196 (2006) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic) Various aspects of degenerate families of Riemann surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces global function fields; Artin-Schreier extensions; genus; rational places; towers; limit of towers; asymptotically good towers Arithmetic theory of algebraic function fields, Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry A problem of Beelen, Garcia and Stichtenoth on an Artin-Schreier tower in characteristic two

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Chowla's conjecture; \(L\)-functions; zeta functions of curves; Carlitz extensions; cyclotomic function fields; abelian varieties over finite fields Zeta and \(L\)-functions in characteristic \(p\), Zeta functions and \(L\)-functions, Cyclotomy, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Vanishing of Dirichlet \(L\)-functions at the central point over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces modular Springer theory; Schur algebras; Schur functors; Schur-Weyl duality; perverse sheaves; nilpotent cones; affine Grassmannians; categories of polynomial representations; general linear groups; representations of symmetric groups Mautner, C., A geometric Schur functor, Selecta Math. (N.S.), 20, 4, 961-977, (2014) Schur and \(q\)-Schur algebras, Representation theory for linear algebraic groups, Coadjoint orbits; nilpotent varieties, Representations of finite symmetric groups A geometric Schur functor.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces twistor spaces; slice regular functions; functions of hypercomplex variables; rational and ruled surfaces Altavilla, A., Twistor interpretation of slice regular functions, J. Geom. Phys., 123, 184-208, (2018) Twistor methods in differential geometry, Functions of hypercomplex variables and generalized variables, Global differential geometry of Hermitian and Kählerian manifolds, Rational and ruled surfaces Twistor interpretation of slice regular functions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Lie algebra of vector fields; polynomial Lie algebras Бухштабер, В. М.; Лейкин, Д. В., Функц. анализ и его прил., 36, 4, 18-34, (2002) Lie algebras of vector fields and related (super) algebras, Applications of Lie algebras and superalgebras to integrable systems, Singularities in algebraic geometry Polynomial Lie algebras