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321,916 |
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? The first question is to what extent are the notions different. I believe that a bounded measurable function can have a non-measurable "area under graph" (it should be doable by transfinite induction), but I am not completely sure, so treat it as a part of my question. (EDIT: I was very wrong. The two notions coincide and the argument is very straightforward, see Nik Weaver's answer and one of the comments). What are the advantages of the Lebesgue integration over area-under-graph integration? I believe that behaviour under limits may be indeed worse. Is it indeed the main reason? Or maybe we could develop integration with this alternative approach? Note that if a non-negative function has a measurable area under graph, then the area under the graph is the same as the Lebesgue integral by Fubini's theorem , so the two integrals shouldn't behave very differently. EDIT: I see that my question might be poorly worded. By "area under the graph", I mean the measure of the set of points $(x,y) \in E \times \mathbb{R}$ where $E$ is a space with measure and $y \leq f(x)$ . I assume that $f$ is non-negative, but this is also assumed in the standard definition of the Lebesuge integral. We extend this to arbitrary function by looking at the positive and the negative part separately. The motivation for my question concerns mostly teaching. It seems that the struggle to define measurable functions, understand their behaviour, etc. might be really alleviated if directly after defining measure, we define integral without introducing any additional notions.
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Actually, in the following book the Lebesgue integral is defined the way you suggested: Pugh, C. C. Real mathematical analysis .
Second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015. First we define the planar Lebesgue measure $m_2$ . Then we define the Lebesgue integral as follows: Definition. The undergraph of $f:\mathbb{R}\to[0,\infty)$ is $$ \mathcal{U}f=\{(x,y)\in\mathbb{R}\times [0,\infty):0\leq y<f(x)\}. $$ The
function $f$ is Lebesgue measurable if $\mathcal{U}f$ is Lebesgue measurable
with respect to the planar Lebesgue measure and then we define $$
\int_{\mathbb{R}} f=m_2(\mathcal{U}f). $$ I find this approach quite nice if you want to have a quick introduction to the Lebesgue integration. For example: You get the monotone convergence theorem for free: it is a
straightforward consequence of the fact that the measure of the union
of an increasing sequence of sets is the limit of measures. As pointed out by Nik Weaver, the equality $\int(f+g)=\int f+\int g$ is not obvious, but it can be proved quickly with the following trick: $$
T_f:(x,y)\mapsto (x,f(x)+y)
$$ maps the set $\mathcal{U}g$ to a set disjoint from $\mathcal{U}f$ , $$
\mathcal{U}(f+g)=\mathcal{U}f \sqcup T_f(\mathcal{Ug})
$$ and then $$ \int_{\mathbb{R}} f+g= \int_{\mathbb{R}} f +\int_{\mathbb{R}} g $$ follows immediately once you prove that the sets $\mathcal{U}(g)$ and $T_f(\mathcal{U}g)$ have the same measure. Pugh proves it on one page.
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322,020 |
It is well-known that the simply typed lambda calculus is strongly normalizing (for instance, Wikipedia) . Hence, it is not strong enough to be Turing-complete, as also mentioned on the Wikipedia page for Turing-completeness . Its strength is usually compared to propositional logic (I think intuitionistic), and one way to show this is the Curry-Howard Correspondence. However, it also seems to be well known that the simply typed lambda calculus is equivalent to "simple type theory," which is equivalent to higher order logic and hence has no sound, complete, effective proof system. For example, see the article " Seven Virtues of Simple Type Theory ", which cites Godel's theorem and explicitly addresses the "virtue" that STT can create categorical theories (such as second-order PA). How can these two things possibly both be true? What is the correct way of understanding this? EDIT: people have asked for some references on the equivalence between the terms "Simply Typed Lambda Calculus" and "Simple Type Theory." When I said that, I didn't mean they were two different systems that admit some technical "equivalence," but rather that I have generally seen the two terms used interchangeably to mean the same thing, which is the thing defined in Alonzo Church's 1940 paper . To be clear, this paper describes a simply typed lambda calculus with two base types - that of propositions and that of "individuals" (not the same as a primitive "integer" type, but more general and without any particular description of the inhabitants of that type). He also defines as primitives negation, logical OR, universal quantification, and a definite description operation, from which he further derives existential quantifiers, an implication relation, a propositional bidirectional implication, an equality relation on "individuals," an encoding of numerals with a "successor relation," and so on. Church also gives an inference system as a list of additional axioms. His axioms 1-4 are sufficient to derive the law of excluded middle, adding his axioms 5-6 are sufficient for the "logical functional calculus" (which I believe is his term for first-order logic, given that these axioms define how quantifiers work), axioms 7-9 describe the universe of individuals and yield that there are infinitely many, 10-11 give axioms of extension and choice. Church describes which axioms are required to prove different theories; axioms 1-4 are sufficient for "propositional calculus," 1-6 are sufficient for "logical functional calculus" (FOL?), 1-9 are sufficient for "elementary number theory," 1-11 are sufficient for "classical real analysis." Church then goes onto derive the Peano axioms from the above, including the Peano induction axiom. I am not sure how strong the induction axiom is. There are a few other papers describing ways to simplify Church's system: for instance, you can derive quantifiers from lambda plus definite description (Quine 1956, Henkin 1963). A good reference for these is Stanford's Encyclopedia page on Church's Type Theory . Here are a few examples in which the terms "Simply Typed Lambda Calculus" and "Simple Type Theory" are used to describe the same system from Church's paper: Referring to Church's System as "Simply Typed Lambda Calculus" Wikipedia's page on the Simply Typed Lambda Calculus states in the first paragraph "The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties." In general, Wikipedia is fairly uniform in defining the "simply typed lambda calculus" as the typed lambda calculus without polymorphic types, dependent types, etc, and explicitly citing Church's version. Thierry Coquand's course notes on Type Theory says: "Church formulated then an elegant formulation of higher-order logic, using simply typed λ-calculus 5 , which can be seen as a simplification of the type system used in Principia Mathematica, but also is in some sense a return to Frege." In general, an arXiv search for "Simply Typed Lambda Calculus" "Higher Order Logic" yields plenty of results. For instance, see the paper " An overview of type theories " by Nino Guallart, which says "Simply typed lambda calculus. Simply typed lambda calculus was also originally developed by Church (1940,1941). It is a higher order logic system based on lambda calculus and it uses the same syntax." Referring to Church's System as Simple Type Theory The article Seven Virtues of Simple Type Theory refers to Church's system instead as "Simple Type Theory," and says "In 1940 A. Church presented an elegant formulation of simple type theory, known as Church's type theory..." They claim that the abstract "simple type theory" that Church's version is "a formulation of" is equivalent to Russell's "ramified theory of types plus the Axiom of Reducibility." The same article writes "Simple type theory, also known as higher-order logic, is a natural extension of first-order logic. It is based on the same principles as first-order logic but differs from first-order logic in two principal ways. First, terms can be higher-order, i.e., they can denote higher-order values such as sets, relations, and functions. Predicates and functions can be applied to higher-order terms, and quantification can be applied to higher-order variables in formulas." Seven Virtues gives an explicit formulation of "a version" of "Simple Type Theory" which is claimed to be "a version of" Church's theory and equivalent to it. Their derivation seems to be equivalent to the one on Stanford's page , which shows that some of Church's primitives (such as universal quantification) are redundant and can be derived from lambda plus equality. The "Seven Virtues" paper proves as a theorem that any nth-order logic can be embedded in their STT, which they prove in Theorem 2. The Stanford Encyclopedia of Philosophy has an article on " Church's Type Theory ," for which they make clear that they consider Church's theory "a formulation of" type theory, and also state "Type theories are also called higher-order logics, since they allow quantification not only over individual variables (as in first-order logic), but also over function, predicate, and even higher order variables." Stanford also has a subsection on "adding types" to their page on the lambda calculus, in "Lambda Calculus - Adding Types" , citing Church's theory. They also have a page on "Simple Type Theory and the Lambda Calculus" which goes into detail about how Russell's type structure from Principia can be derived using Church's typed lambda calculus. All such examples above referring to "Simple Type Theory" or "Church's Type Theory" do not incorporate any notion of polymorphic types, dependent types, etc. In general it is also not difficult to find references citing Church's paper and using the term "Simple Type Theory" for it. Here is an arXiv paper called Formalising Mathematics In Simple Type
Theory that says "Higher-order logic is based on the work of Church 10 , which can be seen as a simplified version of the type theory of Whitehead and Russell." So that was my point. I have seen the terms "STLC" and "STT" used interchangeably to describe the same system, which is Church's typed system, or various equivalent formulations of it. The terminology seems messy and I am not sure exactly in what sense Church's system is or isn't stronger than FOL.
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The simply-typed $\lambda$ -calculus is not stronger than second-order logic. The simply-typed $\lambda$ -calculus has: product types $A \times B$ , with corresponding term formers (pairing and projections) function types $A \to B$ , with corresponding term formers (abstraction and application) equations governing the term formers and substitution The simply-typed $\lambda$ -calculus does not postulate the existence of any types. Sometimes we postulate the unit type $1$ , and often we postulate the existence of a collection of basic types, but without any assumptions about them being inhabited. This is akin to using a collection of propositional symbols in the propositional calculus, where we make no claims as to their truth value. Simple type theory is simply-typed $\lambda$ -calculus and additionally at least: the type of truth values $o$ , with the corresponding term formers (constants $\bot$ and $\top$ , connectives, quantifiers at every type) the type of natural numbers $\iota$ , with the corresponding term formers (zero, successor, primitive recursion into arbitrary types) equations governing the term formers and substitution There are several variations: we may postulate excluded middle for truth values we may include a definite description operator we may include the axiom of choice we may vary the extensionality principles We quickly obtain a formal system that expresses Heyting (or Peano) arithmetic and more, which suffices for incompleteness phenomena to arise. What I think is confusing you is the fact that there are two ways to relate logic to type theory: The Curry-Howard correspondence relates the propositional calculus to the simply-typed $\lambda$ -calculus by an interpretation of propositional formulas as types . Higher-order logic embeds into simple type theory by an interpretation of logical formulas as terms of the type $o$ of truth values. There is a difference of levels, which makes all the difference. To illustrate, consider the propositional formula $$p \land q \Rightarrow (r \Rightarrow p \land r).$$ In the simply typed $\lambda$ -calculus it is interpreted as the type $$P \times Q \to (R \to P \times R).$$ To prove the formula amounts to giving a term of the type.
In contrast, in simple type theory it is interpreted as the term $$p \land q \Rightarrow (r \Rightarrow p \land r) : o$$ (with parameters $p, q, r$ of type $o$ ).
Now proving the formula amounts to proving the equation $(p \land q \Rightarrow (r \Rightarrow p \land r)) =_o \top$ in the simple type theory. A higher-order formula, such as $(\forall r : \mathsf{Prop} . r \Rightarrow p) \Rightarrow p$ cannot be encoded in the simply-typed $\lambda$ -calculus, whereas in the simple type theory it is again just a term of type $o$ (just replace the sort of propositions $\mathsf{Prop}$ with the type $o$ ). Also note that the pure simply-typed $\lambda$ -calculus does not postulate the natural numbers. If we add the natural numbers to the simply-typed $\lambda$ -calculus we get a fragment of simple type theory known as Gödel's System T (or a version of it, depending on minutiae of how equality is treated), which suffers from – or enjoys, depending on your point of view – the incompleteness phenomena already.
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322,302 |
Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields. Question What are the conjectures in your field proved or disproved (counterexample found) in recent years, which are noteworthy, but not so famous outside your field? Answering the question you are welcome to give some comment for outsiders of your field which would help to appreciate the result. Asking the question I keep in mind by "recent years" something like a dozen years before now, by a "conjecture" something which was known as an open problem for something like at least dozen years before it was proved and I would say the result for which the Fields medal was awarded like a proof of fundamental lemma would not fit "not so famous", but on the other hand these might not be considered as strict criteria, and let us "assume a good will" of the answerer.
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Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: https://arxiv.org/abs/1812.10454 . This was probably considered the biggest open problem in the combinatorics of simplicial complexes. See Gil Kalai's blog post: https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ .
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322,551 |
Common folklore dictates that the Surreals were discovered by John Conway as a lark while studying game theory in the early 1970's, and popularized by Donald Knuth in his 1974 novella. Wikipedia disagrees; the page claims that Norman Alling gave a construction in 1962 which modified the notion of Hahn series fields to realize a field structure on Hausdorff's $\eta_\alpha$ -sets, and that this construction yields the Surreals when considered over all ordinals -- the article pretty explicitly characterizes Conway's construction as a popularized rediscovery more than 10 years later (!). The given reference links to a paper from 1962 (received by editors in 1960) which seems to fit the bill, however this paper gives no indication that Alling considered his construction extended to all ordinals nor does it contain mention of proper classes that I can find. Is there any evidence indicating that Alling considered a proper class sized version of his construction almost a decade before Conway did? Alling published a book in 1987 which constructs the Surreals in exactly this manner*, but this is (of course) more than a decade after Conway's construction was popularized. Was this construction known to Alling earlier and only written up in book form when he realized there was a more widespread interest, or were proper-class considerations in this arena genuinely not in play before Conway? *This is incorrect, thanks to Philip Ehrlich for pointing out my mistake. Alling's 1987 book doesn't construct the surreals as modified formal power series in the fashion of his 1962 paper; the methods involved are more subtle. On January 3, 1983 and December 20, 1982, Norman Alling and Philip Ehrlich respectively submitted papers which constructed isomorphic copies of the surreals using modified versions of the constructions in Allings 1962 paper, and these submissions eventually lead to the publication of two papers to this effect. The 1987 book mentioned above contains expansions of these works in sections 4.02 and 4.03, respectively. (see Philip's excellent answer below for a better description of events)
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Norman Alling's Conway's field of surreal numbers (1985) gives full credit to Conway: Conway introduced the Field No of numbers, which Knuth has called
the surreal numbers. No is a proper class and a real-closed field,
with a very high level of density, which can be described by extending
Hausdorff's $\eta_\xi$ condition. In this paper the author applies a
century of research on ordered sets, groups, and fields to the study
of No . References to Alling's own 1962 paper appear only in passing, on page 373 and 377–378. The 1987 book referred to in the OP follows up on this 1985 exposition, which makes it clear that Alling in no way claims independence from Conway.
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322,965 |
$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a cubic polynomial with integer coefficients. Indeed, we can take $k=1$ and $$ p(n) =n^3+(n-1)^3=(2n-1)(n^2-n+1)=1,9,35,91,\dots \qquad $$ (this one generalizes in the obvious way to the odd zeta values $\zeta(5),\zeta(7),...$ ) or, as shown by Apéry, $k=6$ and $$ p(n) = (2n-1)(17n^2-17n+5)= 5,117,535,1463,\dots
. $$ Numerically, I have found that $k=\dfrac87$ and $p(n) = (2n-1)(3n^2-3n+1)$ also works. (Is that known? Maybe Ramanujan obtained that as some by-product?) The question: Are there other values of $k$ where such a polynomial exists? Must all those polynomials have a zero at $\dfrac12$ for some deeper reason?
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See NOTE below. This MO inquiry is over 3 yrs old now. By the date the question about the $\zeta(3)$ CF with $k=8/7$ was made (Feb, 2019), it can be answered in the negative nowadays, since it was ' (re)-discovered ' (and tagged as a new conjectured CF for Apéry's constant) by a team from Technion - Institute of Technology (Israel) using a highly specialized CAS named The Ramanujan Machine . See here . These findings for $\zeta(3)$ and several other constants were published in Arxiv (Table 5. pg. 16) in May, 2020 and also in Science Journal Nature (Feb, 2021). See Ref. below. So, I think it deserves to be called Wolfgang's $\zeta(3)$ CF. It provides about 1.5 decimal digits per iteration. Are there other values of k where such a polynomial exists?. Answer is yes.
In addition to known $k=1,\,8/7,\,6$ , ( $k=6$ CF is equivalent to Apéry's recursion employed to prove $\zeta(3)$ 's irrationality), $k=5/2$ is currently also known (June, 2019) and $k=12/7$ is conjectured (2020). This youtube video shows at 26:10 $\zeta(3)$ Continued Fractions for $k=8/7$ (Wolfgang's) and $k=12/7$ . Must all those polynomials have a zero at 1/2 for some deeper reason? $k=5/2$ CF does not have a zero at 1/2, but $k=1,6,8/7,12/7$ do (all have companion numerator sequence polynomials $q(n)=-n^6$ ). There are some conjectures in this paper to look for candidates to prove (or improve) the irrationality (measure) of some constants based on the type of factors and roots that $q(n)$ must have. NOTE . As Wolfgang has pointed out in his answer, there was a "Séminaire de Théorie des Nombres" held in Bordeaux University on March 21th, 1980. In the Exposé N°23 by Christian Batut and Michael Olivier "Sur l'accéléracion de la convergence de certaines fractions continues" (in French), the $\zeta(3)$ CF with $k=8/7$ is found on page 23-20 3.2.4 . In this presentation, Apéry's equivalent $\zeta(3)$ CF with $k=6$ is also found (23-19 3.2.2), together with some CFs for Catalan's and other constants. Other $\zeta(3)$ CFs like $k=5/2$ or $k=12/7$ are not shown. To preserve the spirit of the original response, I will leave this answer at this point. Ref: Raayoni, G., Gottlieb, S., Manor, Y. et al. Generating conjectures on fundamental constants with the Ramanujan Machine. Nature 590, 67–73 (2021). https://doi.org/10.1038/s41586-021-03229-4
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322,968 |
Let $\alpha$ be a root of a polynomial $a_nx^n + \ldots + a_1x$ with integral coefficients. I would like to determine $\varepsilon > 0$ depending on $a_1, \ldots, a_n$ so that $|\alpha| < \varepsilon$ implies $\alpha = 0$ . Is it possible to give a "formula" for such an $\varepsilon$ without refering to the complete list of roots?
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See NOTE below. This MO inquiry is over 3 yrs old now. By the date the question about the $\zeta(3)$ CF with $k=8/7$ was made (Feb, 2019), it can be answered in the negative nowadays, since it was ' (re)-discovered ' (and tagged as a new conjectured CF for Apéry's constant) by a team from Technion - Institute of Technology (Israel) using a highly specialized CAS named The Ramanujan Machine . See here . These findings for $\zeta(3)$ and several other constants were published in Arxiv (Table 5. pg. 16) in May, 2020 and also in Science Journal Nature (Feb, 2021). See Ref. below. So, I think it deserves to be called Wolfgang's $\zeta(3)$ CF. It provides about 1.5 decimal digits per iteration. Are there other values of k where such a polynomial exists?. Answer is yes.
In addition to known $k=1,\,8/7,\,6$ , ( $k=6$ CF is equivalent to Apéry's recursion employed to prove $\zeta(3)$ 's irrationality), $k=5/2$ is currently also known (June, 2019) and $k=12/7$ is conjectured (2020). This youtube video shows at 26:10 $\zeta(3)$ Continued Fractions for $k=8/7$ (Wolfgang's) and $k=12/7$ . Must all those polynomials have a zero at 1/2 for some deeper reason? $k=5/2$ CF does not have a zero at 1/2, but $k=1,6,8/7,12/7$ do (all have companion numerator sequence polynomials $q(n)=-n^6$ ). There are some conjectures in this paper to look for candidates to prove (or improve) the irrationality (measure) of some constants based on the type of factors and roots that $q(n)$ must have. NOTE . As Wolfgang has pointed out in his answer, there was a "Séminaire de Théorie des Nombres" held in Bordeaux University on March 21th, 1980. In the Exposé N°23 by Christian Batut and Michael Olivier "Sur l'accéléracion de la convergence de certaines fractions continues" (in French), the $\zeta(3)$ CF with $k=8/7$ is found on page 23-20 3.2.4 . In this presentation, Apéry's equivalent $\zeta(3)$ CF with $k=6$ is also found (23-19 3.2.2), together with some CFs for Catalan's and other constants. Other $\zeta(3)$ CFs like $k=5/2$ or $k=12/7$ are not shown. To preserve the spirit of the original response, I will leave this answer at this point. Ref: Raayoni, G., Gottlieb, S., Manor, Y. et al. Generating conjectures on fundamental constants with the Ramanujan Machine. Nature 590, 67–73 (2021). https://doi.org/10.1038/s41586-021-03229-4
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322,987 |
I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually. By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc. I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?
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What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level. But still you may try these books: Michio Kuga, Galois' dream, David Mumford, Caroline Series, David Wright, Indra's Pearls, Hermann Weyl, Symmetry. Marcel Berger, Geometry revealed, D. Hilbert and Cohn-Vossen, Geometry and imagination, T. W. Körner, Fourier Analysis, T. W. Körner, The pleasures of counting. A. A. Kirillov, What are numbers? V. Arnold, Huygens and Barrow, Newton and Hooke. Mark Levi, Classical mechanics with Calculus of variations
and optimal control. Shlomo Sternberg, Group theory and physics, Shlomo Sternberg, Celestial mechanics. All these books are written in a leisurely informal style, with a lot of
side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss: Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics.
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323,136 |
Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do binary coproducts always exist in the category of noetherian commutative rings? End of edit. I asked the question on Mathematics Stackexchange but got no answer. Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one. Let $A$ be in $\mathsf{CRing}$ . Question 1. Is there a functor from a small category to $\mathsf{Noeth}$ whose limit in $\mathsf{CRing}$ is $A$ ? Let $f:A\to B$ be a morphism in $\mathsf{CRing}$ such that the map $$
\circ f:\text{Hom}_{\mathsf{CRing}}(B,C)\to\text{Hom}_{\mathsf{CRing}}(A,C)
$$ sending $g$ to $g\circ f$ is bijective for all $C$ in $\mathsf{Noeth}$ . Question 2. Does this imply that $f$ is an isomorphism? Yes to Question 1 would imply yes to Question 2. Question 3. Does the inclusion functor $\iota:\mathsf{Noeth}\to\mathsf{CRing}$ commute with colimits? That is, if $A\in\mathsf{Noeth}$ is the colimit of a functor $\alpha$ from a small category to $\mathsf{Noeth}$ , is $A$ naturally isomorphic to the colimit of $\iota\circ\alpha$ ? Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do not exist in $\mathsf{Noeth}$ : see this answer of Martin Brandenburg. Here are two particular cases of the above questions: Question 4. Is $\mathbb Z[x_1,x_2,\dots]$ a limit of noetherian rings? (The $x_i$ are indeterminates.) Question 5. Do binary coproducts exist in $\mathsf{Noeth}$ ? One may try to attack the first question as follows: Let $A$ be in $\mathsf{CRing}$ and $I$ the set of those ideals $\mathfrak a$ of $A$ such that $A/\mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/\mathfrak a$ with $\mathfrak a\in I$ , and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective. [Edit. By a comment of Laurent Moret-Bailly the morphism in question is not always bijective.]
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The answer is no to all questions except 4. Negative answers to 1,2 and 3 :
It is easy to construct a ring $A$ with an element $a$ satisfying: (i) $a≠0$ , (ii) $a$ is nilpotent, (iii) for each $n≥1$ there is $y_n\in A$ such that $a=y_n^n$ . For every morphism $\varphi:A\to C$ , the image of $a$ inherits properties (ii) and (iii), hence it is zero if $C$ is noetherian (observe that the radical of $C$ is finitely generated, hence nilpotent). In other words, any morphism from $A$ to a noetherian ring $C$ factors through $B:=A/(a)$ , and of course the same holds if $C$ is a limit of noetherian rings. This proves that $A$ is not such a limit ( Question 1 ) and the natural morphism $A\to B$ provides a negative answer to Question 2 . For Question 3 , consider the following special case: let $k$ be a nonzero noetherian ring and take $$A:=k[X_1,X_2,\dots]/(X_1, X_{mn}^m-X_n)_{m,n≥1}.$$ If $x_n$ denotes the class of $X_n$ , we easily check that $x_n^n=0$ for all $n$ , $x_n≠0$ if $n≥2$ , and each $x_n$ is an $m$ -th power for all $m$ . If $\varphi:A\to C$ is a morphism with $C$ noetherian, the above argument (with $a=$ any $x_n$ ) shows that $\varphi$ factors through $A/(x_n)_{n≥1}\cong k$ . Now if $\Lambda$ is the ordered set of finitely generated $k$ -subalgebras of $A$ , then of course $A$ is the colimit of $\Lambda$ in CRing, but the above shows that there is a colimit in Noeth, which is $k$ . Positive answer to 4 : $\mathbb{Z}[x_1,x_2,\dots]$ is a Krull domain (even a UFD), hence an intersection of discrete valuation rings inside its fraction field. Negative answer to 5 : Given a ring $R$ , let us call a noetherian $R$ -algebra $S$ a noetherian hull of $R$ if it is an initial object of the category of noetherian $R$ -algebras. Proposition. If $R$ has a noetherian hull $S$ , the natural map $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ is surjective. In particular, $\mathrm{Spec}(R)$ is a noetherian space. Proof: for each $p\in \mathrm{Spec}(R)$ its residue field $\kappa(p)$ (i.e. $\mathrm{Frac}(R/p)$ ) is a noetherian $R$ -algebra, hence $R\to\kappa(p)$ factors through $S$ . QED (Remark: it is easy to see that $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ is in fact bijective, with trivial residue field extensions). Now, if $A$ and $B$ are two noetherian rings, a coproduct of $A$ and $B$ in Noeth is the same thing as a noetherian hull of $A\otimes B$ : this is clear from the universal properties. Thus, to answer Question 5 negatively, it suffices to find two noetherian rings $A$ , $B$ such that $\mathrm{Spec}(A\otimes B)$ is not a noetherian space. There are plenty of examples, for instance $A=B=\overline{\mathbb{Q}}$ . Note: the same example was given by François Brunault in his answer, with a different argument. Since I have edited my answer several times, I am not sure who came first!
|
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|
323,764 |
Let $S_n$ be the group of $n$ -permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$ . QUESTION. Assume $n>2$ . Does this cancellation property hold true? $$\sum_{\sigma\in S_n}(-1)^{\ell(\sigma)}\sum_{i=1}^ni(i-\sigma(i))=0.$$
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Let $n$ be some integer greater than 2. Since the number of even and odd permutations in $S_n$ is the same we have $\sum_{\sigma\in S_{n}}(-1)^{\ell(\sigma)}=0$ therefore the contribution of $\sum_{\sigma\in S_{n}}(-1)^{\ell(\sigma)}\left(\sum_{i=1}^n i^2\right)$ is zero. It remains to show that $$\sum_{\sigma\in S_{n}}(-1)^{\ell(\sigma)}\sum_{i=1}^n i\sigma(i)=0.$$ Notice that if we write $P(x)=\det\left(x^{ij}\right)_{i,j=1}^n$ then this sum is simply $P'(1)$ . However the order of vanishing of $P$ at $1$ is $\binom{n}{2}$ (notice that the matrix is pretty much a Vandermonde matrix) and this is greater than $1$ since $n>2$ , therefore $P'(1)=0$ .
|
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|
324,257 |
The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points? Sorry if this is a silly question. Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme: Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics. That wrong statement was correctly applied to some other field, or somehow used in calculations etc. The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering. This led to some real-world harm or danger. It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project. Sorry if I made it even sillier. Feel free to close the question.
|
Quoting from Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes , Scientific American Vol. 208 , No. 2 (February 1963), pp. 148-158: Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry . To give an example, it might be thought that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
|
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|
324,281 |
In the nice paper "On the rank function of a differential poset" (2011) by Richard Stanley and Fabrizio Zanello a number of interesting questions was asked about such posets. I would like to know which of them are already answered, what are new questions appeared in the field and so on, what are now considered to be the most important and so on. Maybe, some new surveys appeared?
|
Quoting from Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes , Scientific American Vol. 208 , No. 2 (February 1963), pp. 148-158: Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry . To give an example, it might be thought that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
|
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|
324,797 |
A recently asked question (linked here ) deals with the remarkable identity $$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$ where $\mathrm{sinc}(x)=\sin(x)/x$ . It is easy to construct functions $f$ other than $\mathrm{sinc}(x)$ such that $\sum_{n\in\mathbb Z} f(n+x)$ is constant for all real $x$ : define $f$ outside of $[0,1)$ to ensure convergence and then let $f(x)=C-\sum_{n\in\mathbb Z\setminus\{0\}}f(n+x)$ for $x\in[0,1)$ . I wonder, however, whether there are analytic functions other than $\mathrm{sinc}(x)$ with this property? The set of such functions is a vector space over the complex numbers; is it finite-dimensional? If so, what is its dimension?
|
If the Fourier transform $F(k)$ of $f(x)$ vanishes outside of the interval $(-1,1)$ then, by virtue of Poisson summation, $$\sum_{n=-\infty}^\infty f(x+n)=\sum_{n=-\infty}^\infty F(n)e^{2\pi inx}=F(0)$$ independent of $x$ . An example is $F(k)=k^2-a^2$ for $|k|<a$ and $F(k)=0$ for $|k|>a$ , with $0<a<1$ . Then $$f(x)=\frac{2}{\pi x^3}(a x \cos a x-\sin a x)\;\;\text{and}\;\;\sum_n f(x+n)=-a^2.$$
|
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|
324,867 |
Given a set $X$ , let $\beta X$ denote the set of ultrafilters. The following theorems are known: $X$ canonically embeds into $\beta X$ (by taking principal ultrafilters); If $X$ is finite, then there are no non-principal ultrafilters, so $\beta X = X$ . If $X$ is infinite, then (assuming choice) we have $|\beta X| = 2^{2^{|X|}}$ . These are reminiscent of similar claims that can be made about vector spaces and double duals: $V$ canonically embeds into $V^{\star \star}$ ; If $V$ is finite-dimensional, then we have $V = V^{\star \star}$ ; If $V$ is infinite-dimensional, then (assuming choice) we have $\dim(V^{\star \star}) = 2^{2^{\dim(V)}}$ . This suggests that the operation of taking the collection of ultrafilters on a set can be viewed as a double iterate of some 'duality' of sets. Can this be made precise: that is to say, is there some notion of a 'dual' of a set $X$ , $\delta X$ , such that the following are true? The double dual $\delta \delta X$ is (canonically isomorphic to) the set $\beta X$ of ultrafilters on $X$ ; If $X$ is finite, then $|\delta X| = |X|$ (but not canonically so); If $X$ is infinite, then (assuming choice) $|\delta X| = 2^{|X|}$ . Apart from the tempting analogy between $\beta X$ and $V^{\star \star}$ , further evidence for this conjecture is that $\beta$ can be given the structure of a monad (the 'ultrafilter monad'), and monads can be obtained from a pair of adjunctions.
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This is a quite standard idea in functional analysis. Let $X$ be any set and let $c_0(X)$ be the space of all functions from $X$ to $\mathbb{C}$ which go to zero at infinity. Then the algebra homomorphisms from $c_0(X)$ to $\mathbb{C}$ are precisely the point evaluations at elements of $X$ , i.e., the spectrum of $c_0(X)$ is naturally identified with $X$ . Going to the second dual we get $l^\infty(X)$ , the space of all bounded functions from $X$ to $\mathbb{C}$ , whose spectrum is naturally identified with $\beta X$ . [deleted an additional comment which wasn't accurate]
|
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|
325,105 |
Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in which the problem was solved. I think that most mathematicians as a whole, even upon solving major open problems, are an extremely humble lot. But as an outsider I appreciate the understated manner in which some results are dropped. The very recent example that inspired this question: Andrew Booker's recent solution to $a^3+b^3+c^3=33$ with $(a,b,c)\in\mathbb{Z}^3$ as $$(a,b,c)=(8866128975287528,-8778405442862239,-2736111468807040)$$ was publicized on Tim Browning's homepage . However the homepage had merely a single, austere line, and did not even indicate that this is/was a semi-famous open problem. Nor was there any indication that the cubes actually sum to $33$ , apparently leaving it as an exercise for the reader. Other examples that come to mind include: In 1976 after Appel and Hakken had proved the Four Color Theorem, Appel wrote on the University of Illinois' math department blackboard "Modulo careful checking, it appears that four colors suffice." The statement "Four Colors Suffice" was used as the stamp for the University of Illinois at least around 1976. In 1697 Newton famously offered an "anonymous solution" to the Royal Society to the Brachistochrone problem that took him a mere evening/sleepless night to resolve. I think the story is noteworthy also because Johanne Bernoulli is said "recognized the lion by his paw." As close to a literal "mic-drop" as I can think of, after noting in his 1993 lectures that Fermat's Last Theorem was a mere corollary of the work presented, Andrew Wiles famously ended his lecture by stating "I think I'll stop here." What are other noteworthy examples of such announcements in math that are, in some sense, memorable for being understated ? Say to an outsider in the field? Watson and Crick's famous ending of their DNA paper, "It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material," has a bit of the same understated feel...
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The best known lower bound for the minimal length of superpermutations was originally posted anonymously to 4chan . The story is told at Mystery Math Whiz and Novelist Advance Permutation Problem , and a publication with a cleaned-up version of the proof is at A lower bound on the length of the shortest superpattern, with " Anonymous 4chan Poster " as the first author. The original 4chan source is archived here .
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|
325,186 |
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be $$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$ where $N_m$ is the number of points over $\mathbb{F}_{p^m}$ . I was wondering what is the motivation for this definition. The sum in the exponent is vaguely logarithmic. So maybe that explains the exponential? What sort of information is the zeta function meant to encode and how does it do it? Also, how does this end up being a rational function?
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The definition using exponential of such an ad hoc looking series is admittedly not too illuminating. You mention that the series looks vaguely logarithmic, and that's true because of denominator $m$ . But then we can ask, why include $m$ in the denominator? A "better" definition of a zeta function of a curve (more generally a variety) over $\mathbb F_p$ involves an Euler product. The product will be over all points $P$ of $V$ which are defined over the algebraic closure $\overline{\mathbb F_p}$ (this isn't exactly true, see below). Any such point has a minimal field of definition, namely the field $\mathbb F_{p^n}$ generated by the coordinates of this point. We shall define the norm of this point $P$ as $|P|=p^n$ . Then we can define $$\zeta_{V,p}(s)=\prod_P(1-|P|^{-s})^{-1}.$$ (again, this is not quite right) Why would this definition be equivalent to yours? It's easiest to see by taking the logarithms. Then for a point $P$ , the logarithm of the corresponding factor of the product will contribute $$-\log(1-|P|^{-s})=\sum_{k=1}^\infty\frac{1}{k}|P|^{-ks}=\sum_{k=1}^\infty\frac{1}{k}p^{-nks}=\sum_{k=1}^\infty\frac{n}{nk}(p^{nk})^{-s}.$$ In the last step I have multiplied the numerator and the denominator by $n$ , because the point $P$ contributes precisely to numbers $N_{nk}$ , since $P$ is defined over all the fields $\mathbb F_{p^{nk}}$ . But we see a problem - this way, we have counted each point $n$ times because of $n$ in the numerator. The resolution is rather tricky - instead of taking a product over points, we take a product over Galois orbits of the points - if we have a point $P$ minimally defined over $\mathbb F_{p^n}$ , then there are exactly $n$ points (conjugates of $P$ ) which we can reach from $P$ by considering the automorphisms of $\mathbb F_{p^n}$ . If we were to write $Q$ for this set of conjugates, and we define $|Q|=p^n$ , then repeating the calculation above we see that we always count $Q$ $n$ times - which is just right, since it consists of $n$ points! Thus we arrive at the following (this time correct) definition of the zeta function: $$\zeta_{V,p}(s)=\prod_Q(1-|Q|^{-s})^{-1},$$ the product this time over Galois orbits. Apart from being (in my opinion) much better motivated, it has other advantages. For instance, from the product formula it is clear that the series has integer coefficients. Further, it highlights the similarity with the Riemann zeta function, which has a very similar Euler product. Both of those are generalized to the case of certain arithmetic schemes, but that might be a story for a different time. As for your last question, regarding rationality, this is a rather nontrivial result, even, as far as I know, for curves. If you are interested in details, then I recommend taking a look at Koblitz's book " $p$ -adic numbers, $p$ -adic analysis and zeta functions". There he proves, using moderately elementary $p$ -adic analysis, rationality of zeta functions of arbitrary varieties. As KConrad says in the comment, the proof of rationality actually is much simpler for curves than for general varieties. I imagine it is by far more illuminating than Dwork's proof as presented in Koblitz.
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|
325,192 |
I was stunned when I first saw the article Counterexample to Euler's conjecture on sums of like powers by L. J. Lander and T. R. Parkin:. How was it possible in 1966 to go through the sheer astronomical space of possibilities, on a CDC 6600 computer? 1) Did Lander and Parkin reveal their strategy? 2) How would you, using all the knowledge that was accessable until 1965, go to search for counter-examples, if you have access to a computer with 3 MegaFLOPS ? (As a comparison, todays home computers can have beyond 100 GigaFLOPS, using GPUs even TeraFLOPs)
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The search strategy is laid out in a paper devoted to finding all "small" non-negative solutions of $$x_1^5+\cdots x_n^5=y^5\text{ with }n \leq 6$$ L. Lander & T. Parkin, " A counterexample to Euler's sum of powers
conjecture ," Math. Comp., v. 21, 1967, pp. 101-103. The one result was striking enough grab the title and get separate mention in the Bulletin. The search was carried out for $n=6$ and $y \leq 100$ turning up $10$ primitive solutions of which two had $n=5$ $n=5$ and $y \leq 250$ turning up $3$ more including the unexpected one for $n=4$ $n=4$ and $y \leq 750$ turning up nothing else new. That last search covers more than $5000$ times as many cases as going to $133^5.$ The length of the paper hints that the method is fairly simple. Space must have been limited because a table of fifth powers was maintained but not a table of sums $a^5+b^5$ for a meet in the middle attack.
|
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325,383 |
Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists when $C$ is left proper, and it seems that it also always exists as a left semi-model structure, but I don't known if there is any concrete example where it is known to not be a Quillen model structure. PS: I technically already asked this question a year ago but it was mixed with other related questions and this part was not answered, so I thought it was best to ask it again as a separate question.
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A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the structure of a model category. Here, we know that a counterexample must fail to be left proper, so start with a diagram $\require{AMScd}$ $$
\begin{CD}
a @>\sim>> b\\
@VVV @VVV\\
c @>>> d
\end{CD}
$$ in which $a \to b$ is a weak equivalence, $a \to c$ is a cofibration, but $c \to d$ is not a weak equivalence. Then $a \to c$ also cannot be a weak equivalence (otherwise $b \to d$ would be one too). Since $a \to c$ and $c \to d$ are not weak equivalences, they must be both cofibrations and fibrations and therefore the same is true of $a \to d$ . Then $a \to d$ cannot be a weak equivalence (or it would be an isomorphism), so $b \to d$ is also not a weak equivalence, and therefore is a fibration too. In summary, all the maps are fibrations and $a \to c$ , $b \to d$ , $c \to d$ are cofibrations while $a \to b$ is a weak equivalence. One can check that this does in fact yield a model category structure (probably the easiest way is to verify that the (acyclic) cofibrations/fibrations are closed under composition and pushout/pullback, and that the factorization axioms hold). Now, let's try to form the left Bousfield localization at the map $a \to c$ , which is already a cofibration between cofibrant objects. All objects are fibrant in the original structure, and the local objects are the ones which have the same maps from $a$ and from $c$ , which are the objects $c$ and $d$ . The map $c \to d$ was not a weak equivalence originally, so it has to still not be one in the localization. However, making $a \to c$ a weak equivalence also makes $b \to d$ a weak equivalence because it is the pushout of the acyclic cofibration $a \to c$ , which contradicts two-out-of-three.
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|
325,397 |
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
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According to a well known result of Martinet, every compact orientable $3$ -dimensional manifold has a contact structure [2], see also [1] for various proofs. On the other hand we have Theorem. For $n\geq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure. For $n=2$ , $SU(3)/SO(3)$ has no contact structure and for $n>2$ , $SU(3)/SO(3)\times\mathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [3]. [1] H. Geiges, An introduction to contact topology , Cambridge studies in advanced mathematics 109. [2] J. Martinet, Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971. [3] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.
|
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|
325,566 |
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$ , is true. However, more evidence is discovered, which might lead to a consensus that $\neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification. Some examples that come to mind: Because many decision problems, such as graph non-isomorphism , have nice probabilistic protocols, i.e. they are in $\mathsf{AM}$ , but are not known to have certificates in $\mathsf{NP}$ , a reasonable conjecture was that $\mathsf{NP}\subset\mathsf{AM}$ . However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $\mathsf{NP}=\mathsf{AM}$ , etc. I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)\in \mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions. Numerical evidence suggests that for all $x$ , $y$ , we have $\pi(x+y)\leq \pi(x)+\pi(y)$ . This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood. Number theory may also be littered with other such examples. I'm interested if it has ever happened whether the process has ever repeated itself. That is: Have there ever been situations wherein it is reasonable to suppose $P$ , then, after some heuristic analysis, it is reasonable to supposed $\neg P$ , then, after further consideration, it is reasonable to suppose $P$ ? I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.
|
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/\mathbb{Q}$ , both the size of the torsion subgroup of $E(\mathbb Q)$ and its rank were bounded independently of $E$ . The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$? ), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/\mathbb{Q}$ for which the rank of $E(\mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
|
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|
325,963 |
I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, but is there some particular reason that numerical verifications give credence to the truth of the Riemann hypothesis or some way that the computations assist in proving the hypothesis (as we know, historically hypotheses and conjectures have had numerical verification to the point where it seemed that they must be true but the conjectures then turned out to be false, especially hypotheses related to prime numbers and things like that). Is there something special about this hypothesis which makes this kind of argument more powerful than normal? Would one be able to use these arguments somewhere in the case for a proof of the hypothesis or would they never be used in the proof at all (and yes, until it is proven we cannot know that, sure).
|
People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the nontrivial zeros (after "unfolding" them to have average spacing 1, assuming they are on a vertical line to begin with). Odlyzko found striking numerical support for such conjectures by making calculations with zeros very high up the critical line: hundreds of millions of zeros around the $10^{20}$ -th zero. See the Katz--Sarnak article here and look at the picture on the second and fourth pages. These vertical distribution conjectures do not look convincing by working with low-lying zeros. If you're not interested in considering large-scale statistics of the zero locations, there is a small refinement of RH worth keeping in mind since the calculations supporting RH are based on it: the (nontrivial) zeros of $\zeta(s)$ are expected to be simple zeros. This has always turned out to be the case in numerical work, and the methods used to confirm all zeros in a region lie exactly on -- not just nearby -- the critical line would not work in their current form if a multiple zero were found. The existence of a multiple zero on the critical line would of course not violate RH, but if anyone did detect one because a zero-counting process doesn't work out (say, suggesting there's a double zero somewhere high up the critical line), I don't know if there is an algorithm waiting in the wings that could be used to prove a double zero exists if a computer suggests a possible location. I think it is more realistic to expect a computer to detect a multiple zero than to detect a counterexample to RH. Of course I really don't expect a computer to detect either such phenomena, but if I had to choose between them... From Wikipedia's table on its RH page, the latest exhaustive numerical checks on RH (all zeros up to some height) go up to around the $10^{13}$ -th zero. There are other conjectures that have been tested numerically far beyond $10^{13}$ data points, e.g., the $3x+1$ problem has been checked for all positive integers up to $80 \cdot 2^{60} \approx 10^{19}$ , Goldbach's conjecture has been checked for the first $2 \cdot 10^{18}$ even numbers greater than $2$ , and the number of twin prime pairs found so far is over $8\cdot 10^{14}$ . With such examples in mind, I would not agree that the numerical testing of RH is out of line with how far people are willing to let their computers run to test other open problems.
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|
326,602 |
I am wondering if there is some example of a mathematician or physicist who published other papers at the same time as their PhD work and independently of it which actually eclipsed the content of the PhD thesis. The only semi-example I can think of immediately is Einstein, whose other publications in 1905 (especially on special relativity and the photoelectric effect) eclipsed his PhD thesis which was published in the same year. Although it contained important insights, it was somewhat forgotten to the point where he felt that he had to remind people about it. Although this is a soft question, I didn't ask in Academia as I didn't want examples outside of mathematics and physics.
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Anatoly Karatsuba discovered the Karatsuba algorithm in 1960, and reported it to Kolmogorov who published it under his (Karatsuba's) name without his knowledge. It seems fair to say that this first example of a "divide and conquer" algorithm eclipsed Karatsuba's 1966 thesis on "The method of trigonometric sums and intermediate value theorems". For a physics example (from my own university) I note George Uhlenbeck , who with Goudsmit introduced the electron spin in a 1925 publication, while his 1927 Ph.D. thesis on quantum statistics was much less influential. ( Here is the story how two Ph.D. students discovered the electron spin, which was missed by a giant like Pauli.)
|
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327,177 |
It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples: Faber's Theorem on polynomial interpolation: Interpreted as saying that polynomial interpolants are useless, but they are quite useful if the function is Lipschitz-continuous. Squire's Theorem on hydrodynamic instability: Applies in the limit $t \to \infty$ but (nondimensional) $t$ is rarely more than $100$ . Trefethen, Lloyd N. " Inverse Yogiisms ." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28. Google books link . In my own experience, I have witnessed the several negative-result theorems
proved in Marvin Minsky and Seymour A. Papert. Perceptrons: An Introduction to Computational Geometry , 1969. MIT Press . impede progress in neural-net research for more than a decade. 1 Q . What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields? 1 Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]" RG link .
|
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
|
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|
330,146 |
I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “internal perspective,” I mean a constant expression of worry from set theorists and logicians about the relevance of their work to the broader community / “real world”, with these worries sometimes leading to career-defining decisions on the direction of research. For me, this situation is unwanted. I studied set theory because I thought it was interesting, not because I wanted to be a soldier in some kind of movement. Furthermore, I don’t see why an area needs defending when it produces a lot of deep theorems. That part is hard enough. To what degree does there exist, in the various areas of mathematics, a widespread feeling of pressure to defend the relevance of the whole subject? Are there some areas in which it is enough to pursue the research that is considered interesting, useful, or important by experts in the field? Of course there will always be a demand to explain “broader impacts” to funding agencies, but I am talking about situations where the pressure comes from one’s own colleagues or even one’s own internalized sense of what is proper research.
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Overall, people in academia in general and mathematicians in particular are very lucky in being free to study (and being able to make a good living) according to the standards of their discipline, without feeling pressure to defend the relevance of their whole subject. In fact even within our disciplines we have a lot of freedom to pursue our individual visions and tastes. (To appreciate how lucky we are compare the situation with musicians, writers, artists, film directors, actors, ...) Relations with other areas of mathematics or outside mathematics are nice but they are one (and not necessarily a major one) among variety of criteria to appreciate mathematical progress. I think we do have some duty to try to explain what we are doing outside our community and even outside the mathematical community. (But also this task is easier in some areas and harder in others.) Another thing that I found useful in similar contexts is the " sure thing principle ". Given an unwanted situation that has no implication on your action why worry about it at all too much. For example, suppose a paper you wrote and regard as a good paper is rejected. If the rejection was unjust then the conclusion is: "Improve your paper", and if the rejection was just then the conclusion is "Improve your paper".
|
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|
330,977 |
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical: $$ \sqrt{1+2\sqrt{1+3\sqrt{1+\ldots}}}.
\qquad(*)
$$ Assuming the above is well-defined, there is a slick trick showing that it evaluates to $3$ . But such careless assumptions can lead to trouble, as in the example of the expression: $$ -5 + 2(-6 + 2(-7 + 2(-8 + \ldots))).
\qquad(**)
$$ Applying the identity $n = -(n + 2) + 2(n + 1)$ repeatedly for $n=3,4,5,\ldots$ , we get \begin{align}
3 &= -5 + 2(4) \\
&= -5 + 2(-6 + 2(5))\\
&= -5 + 2(-6 + 2(-7 + 2(6))\\
&= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\\
&=\ldots,
\end{align} which would falsely suggest that $(**)$ evaluates to $3$ . What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence? Edit: clarifying in light of some of the discussion in the comments. I can see how $(**)$ can also invite examples of false conclusions from an assumption of existence. That was not the intent of the question; the sole point of $(**)$ was to show that the solution technique to $(*)$ provided at the link can in general yield false conclusions if existence is assumed without additional proof. The spirit of the question is to exhibit cases where the limit exists, but its value given the existence is much easier to establish than the existence itsef.
|
Brownian motion is an example of this phenomenon in probability. I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$ . (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on. But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923). (From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$ ; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.) (The historical notes are from Pitman and Yor, Guide to Brownian Motion , which see for more references.)
|
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|
330,980 |
Let $p(n)$ be the partition function. Are $n=1,2,3$ the only cases for which $np(n)$ is a perfect square?
|
Brownian motion is an example of this phenomenon in probability. I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$ . (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on. But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923). (From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$ ; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.) (The historical notes are from Pitman and Yor, Guide to Brownian Motion , which see for more references.)
|
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|
331,030 |
I was surprised to see that On the construction of balanced incomplete block designs by Raj Chandra Bose was published (in 1939) in a journal named Annals of Eugenics (see here ) (published between 1925 and 1954). I double checked and it seems that the journal was indeed focused on eugenics and the article does not relate to eugenics or genetics in any way. So why would a mathematician decide to publish in such a journal and why was the article accepted? It appears that there are also other mathematics articles published in the same journal.
|
Eugenics and agriculture were two areas of application that motivated much of early 20th century statistics. The paper was a continuation of research on the use of Latin squares in experimental design for those areas. It says so explicitly on the second page: It was, however, only about [1925] that the importance of
combinatorial problems, for the proper designing of biological
experiments, began to be understood, mainly through the work of Prof.
R. A. Fisher and his associates. At the time, Fisher was head of the Department of Eugenics at University College London.
|
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|
331,248 |
Why are Thompson's groups called $F$ , $T$ and $V$ ? I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?
|
The " $F$ " actually stands for "free homotopy idempotent", since $F$ is the universal group encoding a free homotopy idempotent (the endomorphism sending each standard generator $x_i$ to $x_{i+1}$ is idempotent up to conjugation). The universality was proved by Freyd–Heller (who called it " $F$ " for this reason), and independently Dydak. The group was subsequently sometimes called the "Freyd–Heller group", giving the notation $F$ a double meaning. Then people realized Thompson had worked with the group before Freyd–Heller did, and started calling it "Thompson's group $F$ ". The " $T$ " just stands for "Thompson". The " $V$ " is a little mysterious; for a while it was denoted " $G$ " but I think people just realized "G" was bad notation for a specific group, and somehow " $V$ " emerged as a good option since it wasn't being used, or something (actually, according to Cannon–Floyd–Parry , Thompson called it $\hat{V}$ in his unpublished handwritten notes, so maybe that's part of the reason). (P.S. This is Matt Zaremsky, I don't really use MathOverflow but I thought I should answer this question!)
|
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|
332,074 |
Let $W_F$ denote the Weil group of a finite extension of $\mathbb{Q}_p$ . Let $I$ denote the inertia subgroup and $I^{>0}$ the (pro- $p$ ) subgroup of wild inertia. (I hope I've got my notation right...apologies if I haven't.) We have $W_F/ I = \mathbb{Z}$ (canonically), and $I/I^{>0} \cong \prod_{\ell \ne p} \mathbb{Z}_\ell$ (non-canonically). Consider a Weil-Deligne representation $(V, \rho, N)$ , where $V$ is a finite-dimensional complex vector space, $\rho : W_K \to GL(V)$ is a rep of $W_K$ , and $N$ is a nilpotent operator satisfying a well-known condition. Naively, I would have thought that a Weil-Deligne representation $(V,\rho,N)$ is tamely ramified if $\rho$ vanishes on $I^{>0}$ , no conditions on $N$ However, when I read literature on the Deligne-Langlands conjecture (e.g. Kazhdan-Lusztig, Chriss-Ginzburg) it seems that tamely ramified actually means $\rho$ vanishes on $I$ , no conditions on $N$ [Under this assumption an $F$ -semi-simple, tamely ramified, Weil-Deligne representation becomes a (conjugacy class of a) pair $(s,N)$ consisting of a semi-simple element $s$ , a nilpotent element $N$ , such that $sNs^{-1} = qN$ , which is what one expects from affine Hecke algebras, and what we are told the Deligne-Langlands conjecture predicts that the answer should be. [EDIT (following Ben Zvi's comments below): Actually we also need an irreducible representation of the component group of the centraliser of this pair. This extra piece of data is absent for $GL_n$ as this group is always connected.] Passing back through Grothedieck's equivalence, the above seems to give that a tamely-ramified and continuous representation of $W_K$ in a $\mathbb{Q_\ell}$ -vector space, should have each $\mathbb{Z}_{\ell'}$ factor (for $\ell' \ne \ell$ ) acting trivially, which seems a little strange. (I.e. the finite monodromy which could come from the other factors should magically be trivial.) Finally, an admittedly lazy question. I can read in several places that, under the local Langlands correspondence, tamely ramified should correspond to representations admitting an Iwahori fixed vector . Is this explained somewhere? Why should tamely ramified (whatever that actually means, c.f. the above question) translate into having an Iwahori fixed vector? I'm sure this is easy for the pros! Thanks in advance.
|
$\def\R{\mathbf{R}}$ $\def\Z{\mathbf{Z}}$ $\def\Q{\mathbf{Q}}$ $\def\Qbar{\overline{\Q}}$ $\def\F{\mathbf{F}}$ $\def\GL{\mathrm{GL}}$ $\def\Gal{\mathrm{Gal}}$ Here are some thoughts on your question from a number theorist. First of all, I think the only sensible
answer to your question is that your naive guess is the right one, namely, that a tamely ramified Weil-Deligne
representation is one for which the wild inertia group $I_{>0}$ acts trivially. But let me take the opportunity
to give a more expansive answer which touches on a few other concepts. This discussion will all be at the cartoon
level, but that may still be useful to some. Let's fix ideas and always talk about the
case of $G = \GL_n(\Q_p)$ . Let $\rho$ be a Weil-Deligne representation and let $\pi$ be the corresponding
representation associated to $\rho$ by local Langlands. The following three things are then equivalent: The Weil-Deligne representation is trivial on the entire inertia group. $\pi$ has an Iwahori fixed vector. The corresponding $l$ -adic representations for $l \ne p$ are unipotent on inertia. With that out of the way: I would certainly only ever use "tamely-ramified" to refer to the weaker condition that the image of wild inertia is trivial, exactly as you hypothesized. A really basic example to keep in mind is the case of characters. We have $\GL_1(\Q_p) = \Q^{\times}_p$ .
Let $\chi$ be a (continuous) character of $\Q^{\times}_p$ . Let's specialize the conditions above in this case. We get The Weil-Deligne representation is trivial on the entire inertia group. $\pi$ has a fixed vector under $\Z^{\times}_p \subset \Q^{\times}_p$ . The corresponding $l$ -adic representations for $l \ne p$ are trivial (equivalently in dimension one unipotent) on inertia. The nice thing about the case $n = 1$ is that the local Langlands correspondence is very transparent, namely, it is given by local class field theory. To recall (briefly) what class field theory says (in one formulation), it gives a canonical map $$\Q^{\times}_p \hookrightarrow \Gal(\Qbar_p/\Q_p)^{\mathrm{ab}}$$ with dense image. Indeed, the image is precisely the image of the Weil group, and the image of $\Z^{\times}_p$ is the image of the inertial
part of the Weil group. It follows that the wild inertia group maps to the pro- $p$ part of $\Z^{\times}_p$ which is $1 + p \Z_p$ .
A (Galois) character of $\Gal(\Qbar_p/\Q_p)$ canonically gives (by restriction) a character of $\Q^{\times}_p$ .
Certainly one would want to allow a "tamely-ramified" Galois character to be tamely ramified, so (in dimension one) we get the following equivalences: The Weil-Deligne representation is trivial on $1 + p \Z_p \subset \Q^{\times}_p$ , $\pi$ has a fixed vector under $1 + p \Z_p$ . The corresponding $l$ -adic representations are tamely ramified on inertia. I honestly only found very few papers in the literature in which "tamely ramified" was implied to have the meaning (1), (2), or (3). I think they were just in error.
Here are a few speculations on how they could get confused. First of all, the maximal tamely ramified extension of a local field has a particularly simple Galois group.
Namely, it is given by (the profinite completion of) the group $$\Gamma: \langle \tau, \sigma | \sigma \tau \sigma^{-1} = \tau^q \rangle,$$ where $\sigma$ is Frobenius, $\tau$ is a (pro-finite) generator of tame inertia, and $q$ is the order of the residue field (So $q = p$ for $\Q_p$ ).
In characteristics $l \ne p$ , unipotent implies pro- $l$ implies tame, and thus unipotent implies tamely ramified.
Let us now compare condition 3 above with a new condition 4. The corresponding $l$ -adic representations for $l \ne p$ are unipotent on inertia, that is, they are representations of $\Gal(\Qbar_p/\Q_p)$ which factor through $\Gamma$ and which therefore give rise to matrices $T$ and $F$ satisfying $FTF^{-1} = T^q$ , and where $T$ is unipotent. The corresponding $l$ -adic representations for $l \ne p$ are tamely ramified, that is, they are representations of $\Gal(\Qbar_p/\Q_p)$ which factor through $\Gamma$ and which give rise to
matrices $T$ and $F$ satisfying $FTF^{-1} = T^q$ . One reason these look quite similar is that the condition that $T$ is conjugate to $T^q$ almost looks as though it should imply that $T$ is unipotent, or equivalently that the (generalized) eigenvalues of $T$ are all $1$ . But this is not quite true, it doesn't force the eigenvalues to satisfy $\lambda = 1$ but only $\lambda^{q-1}= 1$ and so
only $q-1$ th roots of unity. ( Correction: as Will Sawin points out, this should be $\lambda^{q^k - 1} = 1$ for some $k \le n$ . Indeed, in the supercuspidal case you can get primitive $(q^n-1)$ th roots of unity.) So 3 and 4 are quite similar, but 4 gives a more relaxed condition
and is the "correct" definition of tamely ramified. So now, let us return to your next question (slightly modified): Why should [...] translate into having an Iwahori fixed vector? What do tamely ramified representations correspond do? OK, so answering "why" to questions in Langlands is always tricky. (What is Langlands? A vast generalization of class field theory. What is class field theory? A vast generalization of quadratic reciprocity. What is quadratic reciprocity? NOBODY KNOWS.) But let me give a little attempt to answer your question by a somewhat circuitous route. I want to start by talking about conductors. (I'm not actually going to say very much interesting about conductors, but I'm going to start out this way.) If you are a graduate student in number theory, sooner or later you are going to read
about the modularity of elliptic curves. And to understand the statement, there is some mysterious invariant $N$ associated to an elliptic curve $E/\Q$ .
At first glance, it does something reasonable to measure the bad reduction of $E$ : if $p |N$ , then $E$ (or a suitably good model) has bad reduction at $p$ .
And there's some sense that the power of $p$ dividing $N$ measures "how bad" the reduction really is in some controllable way for $p > 3$ . And then for $p = 3$ or $p = 2$ everything goes crazy, but it's OK because some guy called Tate came up with an algorithm and $\texttt{gp}$ can compute it for you. But if you try to think about it, it makes bugger all sense. Then you have to believe the same $N$ turns up as the level of the modular form, and this is even more confusing.
In order to even begin to understand it, it's really key to once again step back and think about the case of $\GL(1)$ .
In this case, we have a character $\chi$ which we can think of as a character of $\mathbf{Q}^{\times}_p$ .
And what class field theory tells us is that the ramification behavior of $\chi$ is all wrapped up in the restriction of $\chi$ to $\Z^{\times}_p$ .
The "automorphic" and "Galois" sides are almost literally the same here, but let me still try to distinguish them. We can now define the conductor as follows: The automorphic conductor of $\chi$ is the smallest $p^n$ such that $\chi$ has an invariant under the subgroup $1 + p^n \Z_p$ . The Galois conductor of $\chi$ is the smallest $p^n$ such that $\chi$ is trivial on $1 + p^n \Z_p$ . Of course these are clearly the same. But even though I haven't really said it here, what is important in class field theory is that the filtration of $\Z^{\times}_p$ by $1+p^n \Z_p$ is intimately related to the inertia filtrations on the Galois groups. The special case that $1+p \Z_p$ corresponds to $I_{>0}$ is clear,
but the others are more subtle. So here the "equality" of conductors coming from "local Langlands for $\GL(1)$ " is expressing something rather
deep in class field theory. To give at least one concrete example, there is Hasse's Führerdiskriminantenproduktformel which
expresses the discriminant of the corresponding
cyclic fixed field in terms of the conductors of the non-trivial powers of $\chi$ . The study of relations of this flavour were actually key to Artin's formulation
of his various reciprocity laws and recognizing the correct way to define $L$ -functions. I think Noah Snyder wrote an interesting undergraduate thesis
about this once. Anyway, I'm drifting slightly from my main topic. To get back on point, the key thing about this example which generalizes is that the following
two things are intimately related: The restriction of the Weil--Deligne representation $\rho$ to inertia. The restriction of $\pi$ as a $\GL_n(\Q_p)$ representation to $\GL_n(\Z_p)$ . There is (some sort of) converse to this, in that when $\pi$ is spherical (unramified), then it is determined
by the Hecke operators at $p$ , and the classical Hecke operators at $p$ come from the appropriate actions of diagonal matrices with powers
of $p$ , which are (in a non-technical sense) ``orthogonal'' to the copy of $\GL_n(\Z_p)$ . (Warning, this is a very vague remark.)
Of course, this connection is much deeper when $n > 1$ than when $n = 1$ . For example, when $n > 1$ , then $\pi$ is infinite dimensional, and so its restriction to $\GL_n(\Z_p)$ decomposes into infinitely many different representations.
(each with finite multiplicity by admissibility). Here, I think, is one reasonable answer to your question: Claim: $\rho$ is tamely ramified if and only if $\pi$ has an invariant vector under the $p$ -congruence subgroup $K(p)$ of $\GL_n(\Z_p)$ . The broader takeaway is that the image of inertia in $\rho$ is determined by the restriction of $\pi$ to $K = \GL_n(\Z_p)$ . Of course I can't really prove this because it requires (at least) the local Langlands correspondence itself. But let's talk about some examples,
even for the case when $n = 2$ because you see most of the phenomena in this case. In fact, for many people, the first thing to do is
just to understand the statement of local Langlands for $n = 2$ and $p > 2$ by just understanding the objects of both sides.
Kevin Buzzard wrote a great note about this
here: http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf and then he even gave a 20 lecture course as well: https://www.youtube.com/watch?v=Rv59aRUMfio In order to talk about it, I guess one has to say a little bit about what the representations $\pi$ really are.
For $n = 2$ , there are two types of basic construction: (parabolic) induction from the Borel $B$ of $\GL_2(\Q_p)$ ,
and (compact) induction from representations of the maximal compact $K = \GL_2(\Z_p)$ . (I won't even try to keep track of normalizations which I always find confusing). Let's also denote by $I$ the Iwahori subgroup of $K$ and by $P$ the pro- $p$ Iwahori (the inverse image in $K$ of the unipotent subgroup the Borel of $\GL_2(\F_p)$ ), and by $K(p)$ the full congruence subgroup of $K$ . Tamely Ramified Principal Series: Let $\chi_1$ , $\chi_2$ be a pair of tamely ramified admissible characters of $\Q^{\times}_p$ ,
which one can think of as a character of the torus $T$ and thus the Borel $B$ , and let $\pi$ be
the corresponding principal series representation, which is roughly the induction from $B$ to $G$ of the corresponding character of $B$ .
Suitably defined, $\pi$ has invariants under the pro- $p$ -Iwahori subgroup $P$ of $K$ . Note that $P$ is normalized by $I$ , and thus $\pi^{P}$ has an action of $I/P$ .
But $I/P$ is nothing but $(\F^{\times}_p)^2$ , and this acts on $\pi^{P}$ .
The assumption that $\chi_i$ are tame means that their restriction to $\Z^{\times}_p$ give characters on $\F^{\times}_p$ ,
and hence the pair naturally give a character of $I/P$ .
If $\pi$ is irreducible (as it will be most of the time) then $\pi^{P}$ is $1$ -dimensional and $I/P$ acts exactly by this character.
So $\pi$ never has invariants under the Iwahori unless the $\chi_i$ are both unramified. In this case, one is either just talking about a unramified principal series, or a (possibly unramified twist of) the Steinberg representation, which have invariants under $K$ and $I$ respectively. (tamely ramified) Supercuspidals: If we want something with $K(p)$ -invariants, we can take a representation $\tau$ of $K/K(p) = \GL_2(\F_p)$ and take the compact induction from $K$ to $G$ . If you take $\tau$ to be induced from a Borel then you again get a $\pi$ which has invariants under
the pro-p Iwahori. But if you take $\tau$ to be one of the "exotic" representations of dimension $p - 1$ , then $\tau$ turns out to have no $P$ -invariant vectors.
The compact inductions in these cases now turn out to be irreducible. Let's compare $\pi$ on the automorphic side with the Weil-Deligne
side. On the automorphic side, I always think of $\tau$ as "almost" constructed as follows: take a character $\psi$ of the non-split Cartan (which is isomorphic to $\F^{\times}_{p^2}$ ), and take the induction to $\GL_2(\F_p)$ .
Of course, this is not quite correct, because inducing something from the non-split Cartan is not irreducible, but it at least
sees $\tau$ and other similar representations. At any rate, this automorphic side is related to the induction of a character $\psi$ of $\F^{\times}_{p^2}$ . But now the Weil-Deligne side is very similar --- the representation is exactly induced from a tamely ramified
character ( $\psi$ ) of the quadratic unramified extension of $\Q_p$ . But, by class field theory, on inertia $\psi$ is tamely ramified and hence the same as a character of
the units in the residue field $k^{\times} = \F^{\times}_{p^2}$ . These examples actually exhaust all the $\pi$ which are tamely ramified. Even for $\mathrm{GL}_n(\Q_p)$ , the situation is similar. There are representations built up from smaller Levi subgroups, and there are the supercuspidals. But at least the tame supercuspidals are "easy" in that we know on the Weil-Deligne side they should be irreducible (and tamely ramified) and hence induced from an unramified cyclic degree $n$ extension of $\mathbf{Q}_p$ . (The supercuspidals were constructed by Howe - the construction requires at least one understands the irreducibles of $\mathrm{GL}_n(\mathbf{F}_p)$ .) And on the automorphic side they can be constructed from compact inductions from "known" representations of $\mathrm{GL}_n(\mathbf{F}_p)$ . Back to conductors: You might guess that the conductor of $\pi$ is given by the minimal $p^n$ such that $\pi$ has an invariant vector under $\Gamma(p^n)$ .
But in fact this is not quite correct --- the automorphic conductor is the minimal $p^n$ such that $\pi$ has an invariant vector under
the group $\Gamma_1(p^n)$ . In the case of the tamely ramified principal series representation the conductor is thus $p^2$ if $\chi_i$ are both ramified and $p$ if exactly one of $\chi$ is ramified. In particular, $\pi$ will have a twist of conductor $p$ .
Similarly, taking ramified principal series which may now be wildly ramified, their explicit construction shows that the conductor
in the automorphic sense is the product (or sum if you take the exponent) of the conductors of $\chi_1$ and $\chi_2$ .
And similarly, the conductor on the "Galois" side is the same.
On the other hand,
the tamely ramified supercuspidals have conductor $p^2$ , and all their twists have conductor $p^2$ . In particular, they have no
invariant vectors under the pro- $p$ Iwahori subgroup $P$ (this can be checked more directly). On the Galois side,
the representation is tamely ramified but irreducible so also has conductor $p^2$ .
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332,210 |
I presume this is a GAGA-style result, but I cannot find a reference.
|
Also like affine varieties, we have: Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold. For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables , Theorem 5.3.9. For the converse, an argument is contained on pp 109-110 of Hörmander, immediately after the definition of Stein manifold.
|
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|
332,281 |
I had a discussion with my advisor about what am I interested as my future research direction and I said it is special functions and q-series. He laughed and said that the topic is essentially dead and the people who study it are dinosaurs. I'm really confused by this statement and don't know what to think. Is this area really dead and not worth pursuing a research in it?
|
I cannot answer the question of whether the field of $q$ -series is dead. I would make this a comment, but I lack sufficient reputation. I am a Banach space theorist, and Banach space theory is not fully dead. However, it is sufficiently unfashionable as to make it terribly difficult for me to find a job (even with a very strong research record). In fact, I have been completely unable to find employment, and I am now forced to leave academia because of it. Granted, geographical factors, personal connections, and luck may all be different for you. However, I made the mistake of choosing my research area only because of its intellectual interest to me, without any practical consideration of the job market, and I would caution you against doing the same. If your advisor says the area is dead, I would listen. I would also say that, if you like that area, you can study it as much as you want after you find a permanent position.
|
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|
332,305 |
I have wondered for a while if there are any interesting rational solutions to $a^b = b^a$ . I have tried but cannot find any solutions other than $a=2$ and $b=4$ , or vice versa. Thank you in advance.
|
There is an infinite number of rational solutions $$a=\left(\frac{n+1}{n}\right)^n,\;\;b=\left(\frac{n+1}{n}\right)^{n+1},\;\;n\in\mathbb{Z},\;\;0\neq n\neq -1.$$ For a proof that these are all the rational solutions of $a^b=b^a$ with $a\neq b$ , see Marta Sved's article (1990). As she describes, this question has a long history, it was first answered by Euler in 1748 and has been generalized in various ways. I show a screen shot from Euler's proof that there is an infinite number of rational solutions (Euler uses the word "innumerabilia" -- uncountable, obviously not in the technical sense of the word).
|
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332,404 |
I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was told that's for engineers and physicist mostly, not for mathematicians. I also looked for Fisher and Flanigan, but they don't seem to have as many topics as Brown. I wonder which book is best for the subject or if one of the two previously mentioned will do to master most of the topics of complex variables as a mathematician. Thanks.
|
There are many good books, but the choice depends on your background and on your needs and on your taste. For what purpose do you study complex variables? Do you like geometry or formulas? If your aim is to use complex variables (for example in engineering and physics problems) Whittaker and Watson is an excellent choice. It is somewhat outdated, but contains most of the things useful in applications. By far more than modern texts. And I have to warn you that this is a difficult reading, but it has an enormous number of exercises. The standard textbook for mathematicians (US graduate students) is Ahlfors. An excellent choice for the very beginning (mathematician) is Cartan (translated into English from the French, no exercises). One very good recent one is by Don Marshall. The last 3 are oriented on pure mathematicians, while Whittaker Watson is universal, can be read by engineers and mathematicians with equal profit.
Remmert has an excellent book 2 volumes, both translated into English: Theory of complex functions and 2. Classical topics in complex function theory. What makes these books special is the great attention to history of the subject. Ahlfors and WW are very different in contents, which reflects the change of fashion in mathematics. Ahlfors is more geometric, while WW is full of formulas. Another classical book with more formulas then geometry is Titchmarsh. On the very minimum level an old but excellent little book by Knopp can be recommended. He has 2 separate small books of exercises. By difficulty I can order these books as follows: Knopp < Cartan < Marshall = Remmert << Ahlfors=Titchmarsh << WW. If you read foreign languages, I can also recommend Hurwitz-Courant which does not exist in English. It is of the same epoch as WW but written from a completely different point of view. It begins on a very basic level, but ends with more advanced material then all other texts that I mention (the things which are covered nowadays under the title Riemann surfaces, and not included in CV textbooks anymore). For this reason it does not fit into the ordering I wrote above. But the first part can be considered as a superb
minimal introduction to the subject, written by one of the greatest masters
of it (Hurwitz). There are very good, corrected and amended editions: the German (by Rohrl) and in Russian (by Evgrafov). It has no exercises. There is a completely different approach to the self study of complex variables. Take a good problem book and solve problems. (Keep a companion text besides and look into it when necessary). By "good problem book" on the subject I mean the above mentioned Knopp, and also Volkovyski (Translated from the Russian by Dover), and Polya Szego.
The difficulty ranking is Knopp < Volkovyski <<< Polya-Szego. Volkovyskii is especially recommended: first it is very large, and second, every chapter has a short background. So you can really use it without a textbook. In any case, solving problem is a very important part of self-study. You cannot claim that you understood something, until you solve a couple of problems.
|
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|
332,852 |
I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field. I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!
|
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
|
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|
332,906 |
Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my question (if you don't have an answer, then probably there is no), and that there is some (little) chance of getting some good answers to the question - and not asking this question here or banning it right away would reduce the chance of getting a good answer to 0. The question is: If one tries to prove something, are there some tricks for getting a creative idea? Ok, I now what you think, yes, there is no algorithm to finding a creative idea, otherwise the idea wouldn't be creative . However, there are some general "tricks": if for minutes one stares at ones sheet of papers with no new ideas, just moving in the same thought cycles, it certainly helps to go and talk to a colleague, because somehow talking awakes the creative ability of the brain (and, additionally, together with a colleague one can mutually pick up an idea of the other and think it a bit further). Also, forgetting the problem for a moment and go and attend talks (even if they are about another topic) or even just rest helps. Do you have any other general "tricks" for getting creative ideas for solving mathematical problems? To make the question a bit more concrete, do you know of any tricks for finding or looking for a good lemma (or several lemmas)? I have the feeling that often the most creativity in proving a theorem lies in finding the right lemma (not even the proof of it, but just the statement). I noticed that whenever I see a proof about which I afterwards say "wow, that's genius, I don't even rudimentally see how one could have come up with it", the crucial point was a lemma (or several lemmas). This also seems to me to be one difference between doing research and doing like homework problems: in homework assignments the proofs usually require only one or two main ideas, and if it requires a lemma, this lemma often is stated in the task as a subtask - while in research, one doesn't even know how much one has to "go down", how many levels of lemmas one has to show.
|
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
|
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|
333,157 |
I am pretty distant from anything analytic, including analytic number theory but I decided to read the Wikipedia page on the Riemann hypothesis ( current revision ) and there is some pretty interesting stuff there: Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, (Ireland & Rosen 1990, p. 359) say The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!! What is surprising is that both a statement and its negation are useful for proving the same theorem. Do similar situations arise with other major, notorious conjectures in mathematics? I only care about algebraic geometry and algebraic number theory for the most part but I guess it will make little sense to have such questions devoted to each area of mathematics so post whatever you've got. To give an initial direction: are there any interesting statements one can prove assuming both some hard conjecture about motives (e.g. motivic $t$ -structure, the standard conjectures, Hodge/Tate conjectures) and its negation?
|
I think this is much less surprising today. It's not uncommon to argue that a statement about some structures is true because one can decompose structures into random-like and structured parts ('structure and randomness'), and in either case get to the desired conclusion. Usually one is trying to prove that a certain statement is true for a whole class of structures by this method; there is no one special structure for which we really want to prove it. Take Roth's theorem as an example: we want to prove any positive-density set of integers in $N$ (for some large $N$ ) contains a three-term AP. We can do this if the set looks random (small Fourier coefficients) easily enough, but if the set does not look random, then there is a large Fourier coefficient, so the set looks structured, namely it has a noticeably larger density on some long AP than on $[N]$ . But then we can pass to the AP and iterate this argument. It just happens that when we try to prove things about the primes, we really only want to know about the one structure, contained in a class of structures with prime-like properties (which might not have more than one member, depending on the properties used in the proof). But the proof method is still to show that the whole class of structures satisfies the desired conclusion. In this case (G)RH is a statement that the primes behave in some random-like way, in a fairly strong quantitative sense. And so its negation is the assertion of some structure beyond the obvious - which of course can be useful. Of course, if you want to find other examples of theorems which you can prove by appealing to some major open problem being either true or false, you should probably have some reason why either outcome helps. Quite a few major open problems (or major theorems) can be read as some kind of quasirandomness statement, and for that there is a reason why the conjecture failing might be useful, so in principle there should be a decent list of possible candidates.
|
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|
334,726 |
If people think this is the wrong forum for this question, I'll cheerfully take it elsewhere. But: How did Solomon Lefschetz do mathematics with no hands? Presumably there was an amanuensis to whom he dictated his papers, and then dictated his revisions. Does history record that person's identity? Was it someone trained in mathematics? Someone we might have heard of independently? Or maybe a series of graduate students? And what about the research phase, when most of us make a lot of idle notes, come back, revisit, cross things out, etc. Was Lefschetz also dictating all these idle thoughts to someone --- or did he learn to hold them in his mind until he was ready to write a paper? Or something else? Or did he have some sort of prosthetics that allowed him to write? I expect there are people alive --- and perhaps active on this site --- who were at Princeton during the time Lefschetz was active, or who have had direct contact with such people. Maybe one of them can answer? Edited to add: If anybody is tempted to answer that "he used a Lefschetz Pencil, of course!", I've just pre-empted you.
|
I just found this in Halmos's autobiography: "The natural hands were replaced by a neat-looking pair of wooden hands, covered by gloves. As prosthetic devices they were awkward, but they were good enough to hold a pen or a piece of chalk." I guess this question can be closed or deleted now
|
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|
334,944 |
In his paper, "Minds, Machines and Gödel", J.R. Lucas writes the following: Gödel's theorem [First Incompleteness Theorem, that is—my comment] must apply to cybernetic machines, because it is of the essence of being a machine, that it should be a concrete instantiation of a formal system. It follows that given any machine which is consistent and capable of doing simple arithmetic, there is a formula which it is incapable of producing as being true—i.e., the formula is unprovable-in-the-system—but which we can see to be true. It follows that no machine can be a complete or adequate model of the mind, that minds are essentially different from machines. Contrariwise, the following papers, Wilfrid Sieg and Clinton Field, " Automated search for Gödel's Proofs ", Annals of Pure and Applied Logic 133 (2005) 319-338 ( MSN ) Lawrence C. Paulson, " A Mechanized Proof of Gödel's Incompleteness Theorems using Nominal Isabelle " ( published suggest that computers can not only show that the Gödel sentence is not provable from ZF − Infinity , but can also show that it is true, provided ZF − Infinity is consistent. Why this is important is because Lucas, in the paragraph I quoted, makes the mistake that we as humans 'see' that the Gödel sentence is true. In point of fact, we actually infer the truth of the Gödel sentence much as a theorem-prover might infer its truth (if in fact the theorem-prover (via its respective metatheory) can infer the truth of the Gödel sentence, assuming ZF − Infinity is consistent). So that is the question before us: Can computers that run theorem-proving software infer that that the Gödel sentence is true (note that Sieg and Field, as well as Paulson, use ZF − Infinity rather than PA as the object-theory for their theorem-proving software).
|
Yes, computers can infer that the Gödel sentence is true. This is performed in a meta-theory which is stronger than the object theory, as it has to be. For example, Russell O'Connor formalized Gödel's incompleteness theorems in Coq . As he points out in Section 7.1, Coq can prove that the natural numbers form a model of Peano arithmetic $PA$ . I cannot find in his formalization an explicit statement that Gödel's sentence is true (which is not to say it isn't there), but I am quite confident that it would take little effort to formalize such a statement. [This paragraphs has been made obsolete as the question was edited to address the issue.] Also, let me point out that you might be confusing meta-theory with object-theory. Paulson uses the meta theory called "Nominal Isabelle" to prove Gödel's incompleteness theorem, but the way you phrased your question sounds as if you think Paulson's mechanised proof is carried out in $ZF$ without infinity. Lastly, I would just like to say that I never understood how one could hold the position that ugly bags of mostly water are superior to machines in their ability to understand and create mathematics. A machine is not subject to uncontrollable chemical processes, fatigue, emotions, and temptations to sacrifice just a little bit of truth for a great deal of fame.
|
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335,039 |
Q . Which high-degree derivatives play an essential role
in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the 3rd derivative (jolt or jerk) certainly play several roles in applications.
And the torsion of a curve in $\mathbb{R}^3$ can be expressed
using 3rd derivatives. Beyond this, I'm out of my depth of experience. I know of the biharmonic equation $\nabla ^4 \phi=0$ .
There is a literature on the solvability of quintics ,
but it seems this work is neither aimed at applications nor essential to
further theoretic developments.
(I am happy to have my ignorance corrected here.) Q . What are examples of applications that depend on 4th-derivatives
(snap/jounce) or higher?
Are there substantive theorems that require existence of $\partial^4$ or higher as
assumptions, but do not require (or are not known to require) smoothness—derivatives of all orders?
|
Given two sets $A$ and $B$ in $\mathbb{R}^n$ , the Minkowski sum written $A+B$ is the set $\{a+b:a\in A,b\in B\}$ . If $A$ and $B$ are convex subsets of $\mathbb{R}^2$ with real-analytic boundaries then the boundary of $A+B$ is only guaranteed to be ' $6\frac{2}{3}$ times differentiable,' by which I mean $6$ times differentiable with $6$ th derivative Hölder continuous with exponent $\frac{2}{3}$ . This is known to be sharp.
|
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|
335,347 |
This is quite a philosophical, soft question which can be moved if necessary. So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an enjoyable and rewarding experience. However, I have come to realise how much further I still have to go to reach a point where I could even think about publishing original contributions in the literature given how intensively everything has already been studied and the discoveries already made. For example, I have just finished a 600 page textbook on graduate level mathematics. Although it took me a while to understand everything in it, I learned from this and enjoyed doing the exercises, but realised by the end that I still basically know nothing and that it is really intended as a springboard to slightly more advanced texts. I picked up another book which starts to delve more into one of the specific aspects in the book and again, it is 500 pages long. Do I have to read another 500 page book to get a sense of something more specific which I can contribute? At this rate, it will be years and years before I am ever able to publish anything. Later: I am reading this a few years later and realise the question could be hard to answer, as depends on many things (there are some problems where one could contribute decisively without knowing any math at all). However, I will leave the question as I think it's something that many students ask themselves and there is some useful generic advice in the answers.
|
It is something of a myth that everything has already been studied and that you have to master thousands of pages of prior work before you can contribute something new. To be sure, there are some subfields of mathematics that are highly technical, and you're unlikely to be able to contribute something new to them unless you've studied a lot of background material. However, there are also areas of mathematics that don't require that much background knowledge. For example, Aubrey de Grey recently made spectacular progress on a longstanding open problem in combinatorics, and almost no background knowledge was needed for that problem. Even in supposedly highly technical areas of mathematics, people sometimes come up with breakthroughs that employ very little advanced machinery. As others have mentioned, more crucial than "knowing everything" are (1) finding a good problem to work on, and (2) having problem-solving ability. If you have both of these, then you can typically learn what you need as you go along. When you're at an early stage in your career, finding a good problem generally requires an advisor, unless you have the rare ability to smell out good problems yourself just by reading the literature and listening to talks. Problem-solving ability is probably innate to some extent, but a lot of it comes down to experience and persistence. Of course you will be a more powerful problem solver if you have a lot of tools in your toolbox, but generally speaking, you get better at solving problems by spending your time directly attempting to solve problems, and only reading the 500-page books when it becomes clear that they are needed to solve the problem you have in mind.
|
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|
336,078 |
Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that the corresponding L-function $$L_{\{a_n\}}(s):=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$ $L_{\{a_n\}}(s)$ has analytic continuation to a meromorphic function on the whole complex plane $L_{\{a_n\}}(n)=0$ for all negative integers $n$ not all of $a_n$ are zero? Added : It was suggested in the answers that I should have used the term "Dirichlet series of integer sequences" instead of "L-function" as it lacks Euler product. I apologize for the confusion :)
|
Are L-functions uniquely determined by their values at negative integers? No. The rescaled Riemann zeta function $$
\zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} =
\sum_{n=1}^\infty \frac{a_n}{n^s},
$$ corresponding to the coefficient sequence $$
a_n = \begin{cases}
1 & \textrm{if $n$ is a square}, \\
0 & \textrm{otherwise},
\end{cases}
$$ is an example of an $L$ -function that has a meromorphic continuation to all of $\mathbb{C}$ and vanishes at the negative integers. Note that the formulation of your question seems to mix up the notion of an $L$ -function with the more general notion of a Dirichlet series. There are also some interesting Dirichlet series that are not $L$ -functions but still satisfy the properties you are asking about (meromorphic continuation and zeros at the negative integers). One such function is the so-called Witten zeta function of the group $SU(3)$ , as I proved in " On the number of $n$ -dimensional representations of $\operatorname{SU}(3)$ , the Bernoulli numbers, and the Witten zeta function " (see theorem 1.3 on page 5). The coefficient sequence for that function is \begin{align}
a_n &= \#\{ j,k\ge 1 : n = jk(j+k) \}
\\ &=
\textrm{the number of inequivalent irreducible}
\\ & \quad \textrm{ representations of $SU(3)$ of dimension $n/2$.}
\end{align}
|
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|
337,023 |
I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. Two examples come to mind: Prime Number Theorem . This was proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896. Sum of three fourth powers equals a fourth power. In 1986, Noam Elkies proved that there are infinitely many integer solutions to $a^4 + b^4 + c^4 = d^4$ . His smallest example was $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$ . Don Zagier reported that he found a solution independently just weeks later. Can you give other instances?
|
One of the most entertaining seminar talks I ever attended was by Michael Atiyah on the moment map on a Lie group. As the talk progressed, the people in the front row became more and more agitated, until Raoul Bott finally spoke up, said something like, "Michael, these guys just proved the same theorem last week!", and pointed to Victor Guillemin and Shlomo Sternberg.
|
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|
337,419 |
Recently, I formulated the following conjecture which seems novel. Conjecture . For any positive odd integer $n$ , we have the identity $$\sum_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}2.\tag{1}$$ Using Galois theory, I see that the sum is a rational number. The identity $(1)$ has some equivalent versions, for example, $$\sum_{0\le j<k\le n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac n4\left(n-(-1)^{(n-1)/2}\right)\tag{2}$$ and $$\sum_{1\le j<k\le n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=-\frac{n-(-1)^{(n-1)/2}}4\left(n+(-1)^{(n-1)/2}2\right).\tag{3}$$ It is easy to check $(1)$ - $(3)$ numerically. Question. How to prove the conjecture? Your comments are welcome!
|
Let $T_n$ be the $n$ -th Chebyshev polynomial, so that $$T_n(x)-1=\prod_{j=1}^n(x-\cos 2\pi j/n).$$ Taking a logarithmic derivative we have $$\sum_{j=0}^{n-1}\frac{1}{x-\cos 2\pi j/n}=\frac{T_n'(x)}{T_n(x)-1}.$$ For $x=-\cos 2\pi k/n$ , we easily find $T_n(x)=-1$ . We therefore have $$\sum_{j=0}^{n-1}\frac{1}{\cos 2\pi k/n+\cos 2\pi j/n}=\frac{T_n'(-\cos 2\pi k/n)}{2}.$$ Now the derivative of $T_n$ is equal to $nU_{n-1}$ , where $U_{n-1}$ is the Chebyshev polynomial of the second kind, satisfying $U_{n-1}(\cos t)=\frac{\sin nt}{\sin t}$ . For $k\neq 0$ , at $t=2\pi k/n+\pi$ , so that $\cos t=-\cos 2\pi k/n$ , we get $U_{n-1}(-\cos 2\pi k/n)=\frac{\sin(2\pi k+n\pi)}{\sin(2\pi k/n+\pi)}=0$ . For $k=0$ , we have to take a limit at $t\to\pi$ and we find $U_{n-1}(1)=n$ . Therefore we get $$\sum_{k=0}^{n-1}\sum_{j=0}^{n-1}\frac{1}{\cos 2\pi k/n+\cos 2\pi j/n}=\frac{T_n'(1)}{2}=\frac{n^2}{2}.$$
|
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337,558 |
Warning: I am only an amateur in the foundations of mathematics. My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about decidability) is that "plane geometry is decidable". The 2019 International Maths Olympiad happened recently, and there were two plane geometry questions in it (problems 2 and 6). Their solutions look really intimidating! However even as a student I felt that one should be able to solve these questions, in theory, by just "writing down coordinates of everything and doing the algebra". Tarski's work, which I will freely confess that I do not understand fully, might even vindicate my view. The question: Is there an algorithm for solving these kinds of questions, or have I misunderstood? If so, is this algorithm actually feasible to run in practice nowadays (on a computer say) for IMO-level problems? In other words -- are there computer programs which will take as input a planar geometry question of "olympiad level" (for example problems 2 and 6 in this year's IMO) and actually output a solution? Currently I am not too bothered about whether the solution is human-readable -- it could just be a formal proof in some kind of type theory or something, but the output would be some object that some expert could coherently argue was a solution of some sort. The reason I'm asking is that I was talking to some computer scientists about various goals in the long-term project of getting computers to do mathematics "better than humans", and having a computer program which could solve IMO problems by itself was a suggested milestone.
|
Arguably, the so-called " area method " of Chou, Gao and Zhang represents the state of the art in the field of machine proofs of Olympiad-style geometry problems. Their book Machine Proofs in Geometry features over 400 theorems proved by their computer program. Many of the proofs are human-readable, or nearly so. The area method is less powerful than Tarski–Seidenberg quantifier elimination in the sense that not every statement provable by the latter is provable by the area method, but the area method has the advantage of staying closer to the "synthetic" nature of (the vast majority of) Olympiad problems. EDIT (February 2022):
OpenAI has announced some success with solving (some) formal math olympiad problems . They did not restrict themselves to geometry problems.
|
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|
337,942 |
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces. If it is true that: In a Topological Space, if there exists a loop that cannot be contracted to a point then there exists a simple loop that cannot also be contracted to a point. then we can replace a loop by a simple loop in the definition of simply connected. If this theorem is not true for all spaces, then perhaps it is true for Hausdorff spaces or metric spaces or a subset of $\mathbb{R}^n$ ? I have thought about the simplest non-trivial case which I believe would be a subset of $\mathbb{R}^2$ . In this case I have a quite elementary way to approach this which is to see that you can contract a loop by shrinking its simple loops. Take any loop, a continuous map, $f$ , from $[0,1]$ . Go round the loop from 0 until you find a self intersection at $x \in (0,1]$ say, with the previous loop arc, $f([0,x])$ at a point $f(y)$ where $0<y<x$ . Then $L=f([y,x])$ is a simple loop. Contract $L$ to a point and then apply the same process to $(x,1]$ , iterating until we reach $f(1)$ . At each stage we contract a simple loop. Eventually after a countably infinite number of contractions we have contracted the entire loop. We can construct a single homotopy out of these homotopies by making them maps on $[1/2^i,1/2^{i+1}]$ consecutively which allows one to fit them all into the unit interval. So if you can't contract a given non-simple loop to a point but can contract any simple loop we have a contradiction which I think proves my claim. I'm not sure whether this same argument applied to more general spaces or whether it is in fact correct at all. I realise that non-simple loops can be phenomenally complex with highly non-smooth, fractal structure but I can't see an obvious reason why you can't do what I propose above. Update: Just added another question related to this about classifying the spaces where this might hold - In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?
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Here is an example of topological space $X$ , embeddable as compact subspace of $\mathbf{R}^3$ , that is not simply connected, but in which every simple loop is homotopic to a constant loop. Namely, start from the Hawaiian earring $H$ , with its singular point $w$ . Let $C$ be the cone over $H$ , namely $C=H\times [0,1]/H\times\{0\}$ . Let $w$ be the image of $(w,1)$ in $C$ . Finally, $X$ is the bouquet of two copies of $(C,w)$ ; this is a path-connected, locally path-connected, compact space, embeddable into $\mathbf{R}^3$ . It is classical that $X$ is not simply connected: this is an example of failure of a too naive version of van Kampen's theorem. However every simple loop in $X$ is homotopic to a constant loop. Indeed, since the joining point $w\in X$ separates $X\smallsetminus\{w\}$ into two components, such a loop cannot pass through $w$ and hence is included in one of these two components, hence one of the two copies of the cone $C$ , in which it can clearly be homotoped to the sharp point of the cone.
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|
338,607 |
To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error? , Examples of common false beliefs in mathematics , When have we lost a body of mathematics because errors were found? , etc... My background: I am a senior undergraduate student in mathematics. Recently, I got a nice chance in a REU program, and started to read some journal articles. My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result, and ad infinitum. On the other hand, some graduate students and professors in my university, who stand in quite intimate relations to me, say that, they do not check every details of proofs when they read mathematical monographs and research articles. They simply do not have enough time to read all the details and fill in the lines. (Clearly, I also do not read all the proofs in detail, if it seems to be so difficult or not much relevant to what I am interested in.) Finally, I've been heard of some stories on fatal mathematical errors. To be honest, I do not understand what the errors precisely are. What I've been heard about are some "urban legends".
(I intentionally didn't write down the details of these urban legends, since if I write down everything I've heard, maybe someone working in the mentioned field may feel insulted...) For the above reasons, recently I am afraid of the situation where a field in mathematics collapse down because of a single, fatal, but very subtle error in the foundations of that field. In mathematics, everything seems to be so much intertwined, and it seems that no one actually checks every single detail in every mathematical articles. But the mathematics community seems to be very sound.
Maybe at least one of the followings are true: Actually, a typical mathematical result does not depend that much on other results. So whenever if possible, a mathematician can check the details of every results which is of interest to him/her. Strictly speaking, rigor is actually not that important. Even if a mathematical result turns out to be false, there is still something true in the statement. Therefore, only minor changes will be needed, and all the results depending on the turned-out-to-be-false result remains sound. Here are my questions. Why the whole mathematics remains so sound, even though humans are imperfect and quite often produce errors? Are my explanations above correct? If a theorem turns out to be wrong, then mathematicians will try to correct (if possible) all the results depending on that theorem. How hard is this job? Isn't it very tedious and frustrating? I want to hear some personal stories. As an undergraduate student, I want to know if anybody who is much wiser, older, or experienced, had the same fear as mine. (Again, I want to hear some personal stories.) As an undergraduate student who will get into a graduate school in the near future, I want to get some advice. Should I stop worrying and believe the authors of the books and articles I read? When should I check all the details, and when should I just accept the theorem as given? Thanks to everyone for reading my question.
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In addition to the answers that have already been given, I think another reason that mathematics doesn't collapse is that the fundamental content of mathematics is ideas and understanding , not only proofs. If mathematics were done by computers that mindlessly searched for theorems and proof but sometimes made mistakes in their proofs, then I expect that it would collapse. But usually when a human mathematician proves a theorem, they do it by achieving some new understanding or idea, and usually that idea is "correct" even if the first proof given involving it is not. One recent and well-publicized story is that told by the late Vladimir Voevodsky in his note The Origins and Motivations of Univalent Foundations . Here's a bit of one story that he tells about his own experience: my paper "Cohomological Theory of Presheaves with Transfers," ... was written... in 1992-93. [Only] In 1999-2000... did I discover that the proof of a key lemma in my paper contained a mistake and that the lemma, as stated, could not be salvaged. Fortunately, I was able to prove a weaker and more complicated lemma, which turned out to be sufficient for all applications.... This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake.... A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail. I don't know any of the details of the mathematics in this story, but the fact that he was able to prove a "weaker and more complicated lemma which turned out to be sufficient for all applications" matches my own experience. For instance, while working on a recent project I discovered no fewer than nine mistaken theorem statements (not just mistakes in proofs of correct theorems) in published or almost-published literature, including several by well-known experts (and two by myself). However, in all nine cases it was simple to strengthen the hypothesis or weaken the conclusion in such a way as to make the theorem true, in a way that sufficed for all the applications I know of. I would argue that this is because the mistaken statements were based on correct ideas , and the mistakes were simply in making those ideas precise. Or to put it differently, mathematicians get our intuitions from "well-behaved" objects: sometimes that intuition can be wrong for "pathological" objects we didn't know about, but in such cases we simply alter the definitions to exclude the pathological ones from consideration. On the other hand, people do sometimes get mistaken ideas. For instance, here's another quote from Voevodsky's article: In October 1998, Carlos Simpson ... claimed to provide an argument that implied that the main result of the "∞-groupoids" paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!). I can see two factors that contributed to this outrageous situation: Simpson claimed to have constructed a counterexample, but he was not able to show where the mistake was in our paper. Because of this, it was not clear whether we made a mistake somewhere in our paper or he made a mistake somewhere in his counterexample. Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson’s paper appeared, both Kapranov and I had strong reputations. Simpson’s paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it. In this case I do know something about the mathematics involved, and my own opinion is somewhat different from Voevodsky's. In the 2000's I was a graduate student working on higher category theory, and my impression was that in the community of higher category theory it was taken for granted that Simpson's counterexample was correct and the Kapranov-Voevodsky paper was wrong, because the claimed KV result contradicted well-known ideas in the field. The point here is that a community of people developing ideas together is likely to have arrived at correct intuitions, and these intuitions can flag "suspicious" results and lead to increased scrutiny of them. That is, when looking for mistaken ideas (as opposed to technical slips), it makes sense to give differing amounts of scrutiny to different claims based on whether they accord with the intuitions and expectations of experience. So what do you do as a student? In addition to the other good advice that's been given, I think one of your primary goals should be to train your own intuition. That way you will be better-able to evaluate whether a given result, or something like it, is probably true, before you decide whether to read and check the proof in detail. Of course, there is also the position that Voevodsky was led to: And I now do my mathematics with a proof assistant. I have a lot of wishes in terms of getting this proof assistant to work better, but at least I don’t have to go home and worry about having made a mistake in my work. I have a lot of respect for that position; I do plenty of formalization in proof assistants myself, and am very supportive of it. But I don't think that mathematics would be in danger of collapse without formalization, and I feel free to also do plenty of mathematics that would be prohibitively time-consuming to formalize in present-day proof assistants.
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|
341,306 |
The questions Q1 . What are simple ways to think mathematically about the physical meanings of the Planck constant? Q2. How does the Planck constant appear in mathematics of quantum mechanics? In particular, quantization is an important notion in mathematical physics and there are various forms of quantization for classical Hamiltonian systems. What is the role of the Planck constant in mathematical quantization? Q3. How does the Planck constant relate to the uncertainty principle and to mathematical formulations of the uncertainty principle? Q4. What is the mathematical and physical meaning of letting the Planck constant tend to zero? (Or to infinity, if this ever happens.) Motivation: One purpose of this question is for me to try to get better early intuition towards a seminar we are running in the fall. Another purpose is that the Planck constant plays almost no role (and, in fact, is hardly mentioned) in the literature on quantum computation and quantum information, and I am curious about it. Related MO questions: Does quantum mechanics ever really quantize classical mechanics? ;
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Let's give it a try. Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. But if I understand correctly, you want to gain intuition ;) The first observation is that Planck's constant has units, it is not a numerical constant but carries physical dimension. In some sense, this makes all the difference: in your favorite unit system, $\hbar$ has the numerical value $1$ . Period. Nothing more to say... Now what is the impact? If you think of e.g. Fourier transform, the phase in your integral is $e^{ikx}$ and for a mathematician, everything is fine with this. Now for a physicist (depending on the application), $x$ has a unit of length. So there is no way to exponentiate a length per se, you can only plug in dimension-less quantities in a transcendental function. This means that the $k$ has a physical dimension of $1/\text{length}$ . The physical interpretation is that $k$ is an inverse wave length. Now in quantum physics, there comes the time that you want your wave functions in the momentum representation. So you have to replace $k$ by the physical momentum $p$ which has a different unit, namely that of momentum. This requires dividing the product $px$ in your phase by a quantity of dimension momentum times length which is action. So the observation in physics is that there is a universal constant providing a scale for doing exactly that, $\hbar$ . The familiar uncertainty relation you know from Fourier transform becomes thereby scaled by $\hbar$ as well. Another, perhaps more important observation is that in quantization in the Schrödinger approach you consider wave functions on configuration space, depending on the position $x$ of dimension length. Now the relevant operators are the momentum operators encoded by a derivative. But derivatives have dimension $1/\text{length}$ instead of momentum. Thus you have to rescale the derivative by a physical constant of dimension action to get a quantity of dimension momentum. Again, $\hbar$ is the one doing the job. From what I said (and much more can be said) it should be clear that $\hbar$ does not tend to zero at all (it's $1$ , right?). So what should these statements $\hbar \to 0$ then really mean from a physical point of view? This is in fact a quite subtle and, I guess, ultimately not well understood point. The observation in daily life is that quantum effects do not play any role, the world behaves classically. So classical physics is at least a perfect approximation in many situations. Quantum physics only enters the picture if we perform very precise measurements etc. Now the idea is that the more fundamental theory (quantum) has a certain less fundamental theory (classical) as limit. The interpretation of this limit is subtle. But in many situations, the limit relates to parameters of the system which carry dimensions, typically the dimension of action. Now it is the ratio of this system-dependent parameter (say a combination of masses, lengths etc) and $\hbar$ which tells us whether the classical theory is a good approximation or not. The main point is that we need parameters $\alpha$ of the same dimension as $\hbar$ to have a dimension-less ratio $\hbar/\alpha$ . Only such a dimension-less parameter can be considered to be "small" or "large". The classical limit is thus better understood as $\hbar/\alpha \to 0$ , meaning that we look at many different systems with different values of $\alpha$ (whatever its concrete physical interpretation may be) and get the classical limit as limiting scenario if these parameters assume values much larger than $\hbar$ . We can compare them since they have both the same physical dimension. Now why don't we see these arguments in math? The (perhaps pretty obscure) observation is that in the explicit situations we can handle, the mathematical limit $\hbar/\alpha \to 0$ can also be understood as $\hbar \to 0$ while $\alpha$ is fixed. Mathematically this is not a big deal, but physically an absurd interpretation: $\hbar$ is a fundamental constant of nature and we have no $\hbar$ -wheel where we can adjust its value. Hmm, lot of blabla, but I hope that this clarifies the physical side of the story a bit. Mathematically, one has several ways to incorporate "dimensional" constants. One nice way is to look at graded algebras where the grading refers to the power of the dimension you are looking at. Then the grading helps to keep track of the correct dimensions. In particular in quantization theory this turned out to be a very useful tool.
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341,845 |
For which $a,b,c$ does $axy+byz+czx$ represent all integers? In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$ , and I hope someone will know. I also conjectured that $axy+byz+czx+dx+ey+fz$ represents all integers when $\gcd(a,b,c,d,e,f)=1$ and each variable appears non-trivially, though I'm less optomistic about finding prior results on that. Here are some results: If $\gcd(a,b)=1$ then $axy+byz+czx$ represents all integers. [Proof: Find $r,s$ with $ar+bs=1$ , then take $x = r$ , $y = n - crs$ , $z = s$ .] $6xy+10yz+15zx$ , the first case not covered above, represents all integers up to 1000. Similarly $77xy+91yz+143zx$ represents all integers up to 100. [by exhaustive search] If $\gcd(a,b,c)=1$ then $axy+byz+czx$ represents all integers mod $p^r$ . [proved in the above link] The literature on this is hard to search because these are not positive definite forms, and many apparently relevant papers only consider the positive definite case. For old results, the most relevant parts of Dickson's History of the Theory of Numbers (v. 2, p. 434; v. 3, p. 224) mention only the case of $xy+xz+yz=N$ . Does anyone here know a general result or reference?
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Here is a proof of the conjecture. I will refer several times to the book Cassels: Rational quadratic forms (Academic Press, 1978). 1. Let $p$ be a prime such that $p\nmid a$ . Using the invertible linear change of variables over $\mathbb{Z}_p$ $$x'=ax+bz,\qquad y'=y+(c/a)z,\qquad z'=(1/a)z,$$ we have $$x'y'-(abc)z'^2=axy+byz+czx.$$ Therefore, the quadratic forms $axy+byz+czx$ and $xy-(abc)z^2$ are equivalent over $\mathbb{Z}_p$ . By symmetry, we draw the same conclusion when $p\nmid b$ or $p\nmid c$ (note that $p$ cannot divide all of $a,b,c$ ). 2. For $p>2$ , we see that $axy+byz+czx$ is equivalent to $x^2-y^2-(abc)z^2$ over $\mathbb{Z}_p$ . Following the notation and proof of the first Corollary on p.214, we infer that $U_p\subset\theta(\Lambda_p)$ . For $p=2$ , we infer the same by the second Corollary of p.214. Now, combining the Corollary on p.213 with Theorem 1.4 on p.202, we conclude that the genus of $axy+byz+czx$ contains precisely one proper equivalence class. 3. By the conclusions of the previous two points, the quadratic forms $axy+byz+czx$ and $xy-(abc)z^2$ are properly equivalent. As $xy-(abc)z^2$ clearly represents all integers, the same is true of $axy+byz+czx$ . Remark. The crux of the proof are the Corollary on p.213 and Theorem 1.4 on p.202. The first statement relies on the Hasse principle (cf. Lemma 3.4 on p.209 and its proof). The second statement is a straightforward application of strong approximation for the spin group.
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341,959 |
I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and continued to be studied for other reasons. Here is a prominent example I know of: Lie theory : It is my understanding that Lie introduced Lie groups with the idea that they would help in solving differential equations (I guess, by consideration of the symmetries of these equations). While symmetry techniques for differential equations to some extent continue to be studied (see differential Galois theory ), they remain far from the mainstream of DE research. But of course Lie theory is nonetheless now seen as a central topic in mathematics. Are there some other examples along these lines?
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I quote at length from the Wikipedia essay on the history of knot theory: In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra. Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and Thomas Kirkman. James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also developed a strong interest in knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss' linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated by an electric current along the other component. Maxwell also continued the study of smoke rings by considering three interacting rings. When the luminiferous æther was not detected in the Michelson–Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels.
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342,393 |
The following seems like an extremely basic algebraic topology question, but it's not something I ever learned, nor does it look familiar to the algebraic topologists I've asked. Let $f:X\to Y$ be a map, inducing $f^*:H^*(Y)\to H^*(X)$ . Hence the image $R$ of $f^*$ is a subring of $H^*(X)$ . Is there a natural way to factor $f$ as $X \to Z \to Y$ , such that $f^*$ factors as $H^*(Y) \twoheadrightarrow R \hookrightarrow H^*(X)?$ I have a particular $X,Y$ in mind (the inclusion of one compact complex manifold into another, each with even-degree cohomology) but I'm hoping phrases like "Postnikov tower", "cofibrant replacement", "mapping cone" will serve to give a general answer.
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No. Consider the Hopf map $\eta:S^3\to S^2$ . If there were such a space $Z$ , it would have $\widetilde H^*(Z)=0$ , so at the very least $Z$ would be stably trivial, forcing $\eta$ to be stably trivial; but it’s not.
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342,590 |
As it is reasonable to think the work of mathematicians will be developed/made in their offices of universities (or in eventual seminars or conferences),
here are the colleagues, books and journals, connection to databases and blackboards. My belief is that a great part of mathematicians continue, somehow, their work outside working hours of their professional environment the university. In fact I think they have enough resources in their homes for this purpose and that they communicate with collaborators or colleagues while they are in the continuation (progress/attempts) of their research in their homes. I even evoke periodic meetings of nearby collaborators to study and work in specific problems. Question. Is it reasonable to think that the professional mathematician does research in mathematics outside the office of his/her university? Typically, under what conditions? Many thanks. The secondary question is a general overview of this situation and scenario, in case that the work outside of their professional enviroment is remarkable and can be characterized. I don't know if there are well-known examples of proofs of theorems due to mathematicians having an origin at home, coffee shops...I say research sessions/working day outside their offices. Thus an answer for the question under what conditions? should be pedagogical and informative, so that your colleagues and the general public can to know how the research in mathematics is done outside of university and get good results (and if there are general advices to schedule research sessions, remarkable preferences or tips to research in mathematics outside your office of your university).
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Mathematics differs from most other professions in that the only "resources" which are really needed are paper and pencil. (Even these are not strictly necessary, one can use sand and stick as the ancients did. Some can do even without sand, as the examples of famous blind mathematicians show). As a result, the working habits of mathematicians widely wary. I know one study of this variety of working habits: J. Hadamard's book An essay on the psychology of invention in the mathematical field .
It contains in particular the results of a poll that he made among his friends mathematicians. (For example it cites a famous story by Poincare how he invented authomorphic functions, while boarding a bus). Many similar examples are known from the memoirs of mathematicians, or books like Littlewood's Miscellany. The idea of the uniformization theorem came to Klein when he was recuperating from asthma in a seaside resort (described in his book History of mathematics in XIX century). He was so excited that interrupted his vacation and rushed home to write a paper. Banach had a habit of working in a cafe. Feynman (not exactly a mathematician but close to it) recalls that he used to work in a topless bar at some time. If I remember correctly the so-called "tropical geometry" was invented by a group of mathematicians in a Rio-de-Janeiro beach, perhaps this is just
a legend. There is no opposition between "an office' and "home". Many mathematicians have offices at their homes, with books, computers, etc. Some of my friends
have even blackboards in their home offices. A blackboard saves paper and it is more convenient for conversations. It is just a question of habit and convenience, where one prefers to work. Some people can have nicer office at home than at the university. Some people whom I know prefer home because smoking is prohibited on most US campuses:-) Also, in some countries many mathematicians do not have convenient isolated offices. This was the case in Soviet Union, for example. Many of them also did not have convenient offices at home. In the beginning of my career, I remember proving most of my results while walking. I had regular walks with my adviser in a park near the university (my adviser shared an "office" in the university with 6 or 7 people, so we rarely discussed mathematics in his office). When I moved to the US and obtained a convenient office, I still remember proving several theorems while walking my dog. I even think that walking
stimulates mental activity, especially walking in a nice environment, in a park or a forest. Many people used to work in a library if a library with convenient working space was available. Nowadays computers replace books, which makes the choice of a working place even more flexible. I also know mathematicians who come to their office at 9 and work till 5, and do not work on Saturdays and Sundays. My impression is that this is a minority, but I am not aware of any statistics.
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342,632 |
I am far from an expert in this area.
I'd be grateful if someone could explain in what sense $\mathop{SL}(2,q)$ is
"very far from abelian," to quote Emanuele Viola ?
Why does Theorem 1 (below) justify this "far from abelian" claim? (Snapshot from blog here .) Earlier related MO questions: Measures of non-abelian-ness How nearly abelian are nilpotent groups? .
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Another measure of how far a finite group $G$ is from commutative is the "commuting probability", which goes back at least as far as W.H. Gustafson, and certainly predates the work of Gowers. This is just the probability that a pair of elements of $G$ commute, where the uniform distribution is put on $G \times G$ . This turns out to be $\frac{k(G)}{|G|}$ , where $k(G)$ is the number of conjugacy classes of $G$ . It has been noted by several authors, including in recent years P. Lescot, and long ago, (implicitly) by E. Wigner, that this is somewhat related to the smallest degree $d$ of a non-linear complex irreducible character of $G$ , though as Derek Holt's example in comments illustrates, the influence can wane if $G$ is far from perfect and has many linear characters. However, when $G$ is perfect, we have $1 +(k(G)-1)d^{2} < |G|$ , so that $k(G) < \frac{|G|}{d^{2}}+1,$ and the commuting probability of $G$ is bounded above by something only slightly larger than $\frac{1}{d^{2}}.$ As noted in the question the smallest non-linear complex irreducible character degree of ${\rm SL}(2,q)$ is $\frac{q-1}{2}$ when $q$ is odd, so leading to a upper bound for the commuting probability of something close to $\frac{4}{q^{2}}$ for ${\rm SL}(2,q).$ Another approach to this in the case $G = {\rm PSL}(2,q)$ is to note that ${\rm PSL}(2,q)$ always has at most $q+1$ complex irreducible characters (equality is achieved when $q$ is even). In this case, we have $|G| = \frac{q(q-1)(q+1)}{2}$ if $q$ is odd, and the commuting probability for $G$ is less than $\frac{2}{q(q-1)}$ when $q$ is odd
(and is equal to $\frac{1}{q(q-1)}$ when $q$ is even). The same inequalities hold for ${\rm SL}(2,q)$ . Note that this gives that the commuting probability of $G = {\rm PSL}(2,q)$ is bounded above by something like $c|G|^{\frac{-2}{3}}$ for a small fixed constant $c.$ On the other hand, Bob Guralnick and I proved (using the classification of finite simple groups) that for any finite group $G$ with $F(G) = 1$ , the commuting probability of $G$ is at most $|G|^{-\frac{1}{2}}$ , so the bound which holds for ${\rm PSL}(2,q)$ is significantly smaller than the general bound we obtained. Later edit: To be more precise, the arithmetic mean (say $\mu_{d}(G)$ ) of the complex irreducible character degrees of $G$ is quite strongly related to the commuting probability ${\rm cp}(G)$ of $G$ . The Cauchy-Schwartz inequality gives $\sum_{\chi \in {\rm Irr}(G)} \chi(1) \leq \sqrt{k(G)|G|}$ so that $\mu_{d}(G){\rm cp}(G) \leq \sqrt{{\rm cp}(G)}$ and hence ${\rm cp}(G) \leq \frac{1}{\mu_{d}(G)^{2}}.$
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342,966 |
I have seen a plethora of job advertisements in the last few years on mathjobs.org for academic positions in data science . Now I understand why economic pressures would cause this to happen, but from a traditional view of university organization, but how does data science fit in? I would have guessed that at most, a research group having to do with something labeled "data science" could be formed as an interdisciplinary project between applied mathematicians, statisticians, and computer scientists, with corporate funding. But I don't see why it is a fundamentally distinct intellectual endeavor, prompting mathematics hires specifically in data science. The first time I heard the term “data science,” it was said that they wanted to take PhD’s who had experience in statistically analyzing large data sets, and train them for a few weeks to apply these skills to marketing and advertising. Now just a few years later people want to hire professors of this. Question: What about data science is particularly interesting from a mathematical point of view?
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I will stay away from the academic politics of hiring "professors of data science", but if I interpret the question more specifically as "does data science offer problems of mathematical interest", I might refer to Bandeira's list of 42 Open Problems in Mathematics of Data Science. (The full list from 2016 is here , and Bandeira's home page links to solutions of some of these.)
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343,132 |
Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been uncomfortable with their existence, although this discomfort is mostly rooted in a personal aesthetic ideal: I see technical digressions on cardinals, universes, and transfiniteness as a stain on otherwise clean mathematical theories. In the literature on $\infty$ -categories, a great deal of attention appears to be given to so-called presentable $\infty$ -categories. As a reminder, we say an $\infty$ -category $\mathcal{C}$ is presentable if it has all small colimits, and there exists a regular cardinal $\kappa$ so that $\mathcal{C}$ can be realised as the category of $\kappa$ -small $\operatorname{Ind}$ -objects of some small $\infty$ -category. While in most mathematical areas I am interested in, size contraints are of only minor interest, in higher category theory, they appear to arise a lot, usually in the form of requiring certain $\infty$ -categories to be presentable. As an example, in Lurie's Higher Algebra , the word 'presentable $\infty$ -category' appears hundreds of times. Being such a common occurrence, I can no longer sweep these constraints under the rug: I have to face reality. Having swallowed this set-theoretic pill, the solution appears simple: I shall make myself remember the definition of presentability, and apply the truths as decreed by Lurie and others only to those $\infty$ -categories which either Google tells me are presentable, or which I have verified to be presentable myself. Several problems arise, however, and they are the motivation for the questions (or rather, three perspectives on a single question) that I wish to ask. I fail to truly understand the definition. I have only a limited conception of what a cardinal really is, let alone 'regular cardinal'. I have neither feeling for the definition, nor for determining whether a given $\infty$ -category satisfies it. Question 1. What is the intuitive idea behind presentability? What does it mean to presentable? How do I recognise if a given $\infty$ -category is presentable? I fail to truly understand the motivation behind the definition. I am not well-versed enough in the concepts involved to understand what goes wrong if we do not assume presentability, nor do I understand why presentability is defined the way it is. What if we drop 'regular' in 'regular cardinal'? Why not realise $\mathcal{C}$ as $\operatorname{Pro}$ -objects? Question 2. Why is the definition of presentable the way it is, and not something slightly different? I fail to see the beauty in the definition. Although this one is purely subjective, I hope someone feels what I feel. There are these beautifully clean theorems in higher category theory — theorems which have nothing to do with size — that convince me straight away that $\infty$ -categories truly are natural and intrinsically simple things, but on top of that one has size constraints all over the place. They simply feel in dissonance with an otherwise flawless theory. Question 3 (optional). In your opinion, why is presentability a natural, and aesthetic definition?
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Presentable $\infty$ -categories can be understood without every having to think about cardinals. An $\infty$ -category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$ , where $C$ is a small $\infty$ -category, $R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$ , and $\mathcal{P}(C,R)$ is the full subcategory of $\mathrm{PSh}(C)$ spanned by $F$ such that $\mathrm{Map}(f,F)$ is an isomorphism of $\infty$ -groupoids for all $f\in R$ . That's it. The conditions that $C$ is small and $R$ is a set allow you to show that the inclusion $\mathcal{P}(C,R)\to \mathrm{PSh}(C)$ admits a left adjoint, which implies that $\mathcal{P}(C,R)$ is complete and cocomplete, which is something you definitely want. "Presentable" should be thought of in terms of "presentation", analogous to presentations of a group. In some sense $\mathcal{P}(C,R)$ is "freely generated under colimits by $C$ , subject to relations $R$ ". More precisely, there is an equivalence between (i) colimit preserving functors $\mathcal{P}(C,R)\to D$ to cocomplete $\infty$ -category $D$ , and (i) a certain full subcategory of all functors $F\colon C\to D$ that "send relations to isomorphisms" (precisely: those $F$ such that $\widehat{F}(f)$ is iso for all $f\in R$ , where $\widehat{F}\colon \mathrm{PSh}(C)\to D$ is the left Kan extension of $F$ along $C\to \mathrm{Psh}(C)$ ). So its easy to construct colimit preserving functors from presentable categories (and all such functors turn out to be left adjoints).
|
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|
343,264 |
Nash embedding theorem states that every smooth Riemannian manifold can be smoothly isometrically embedded into some Euclidean space $E^N$ . This result is of fundamental importance, for it unifies the intrinsic and extrinsic points of view of Riemannian geometry, however, it is less clear that it is also useful. Most if not all results in differential geometry that I am aware of seem to have been obtained using the intrinsic point of view so avoiding any recurse to Nash embedding theorem. Can you mention results that used in their proof the Nash embedding in an essential way, or results whose proof was considerably simplified by the Nash embedding result? If not can you explain why Nash theorem is less useful and powerful than expected?
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The Nash embedding theorem is an existence theorem for a certain nonlinear PDE ( $\partial_i u \cdot \partial_j u = g_{ij}$ ) and it can in turn be used to construct solutions to other nonlinear PDE. For instance, in my paper Tao, Terence , Finite-time blowup for a supercritical defocusing nonlinear wave system , Anal. PDE 9, No. 8, 1999-2030 (2016). ZBL1365.35111 . I used the Nash embedding theorem to construct discretely self-similar solutions to a supercritical defocusing nonlinear wave equation $-\partial_{tt} u + \Delta u = (\nabla F)(u)$ on (a backwards light cone in) ${\bf R}^{3+1}$ that blew up in finite time. Roughly speaking, the idea was to first construct the stress-energy tensor $T_{\alpha \beta}$ and then find a field $u$ that exhibited that stress-energy tensor; the stress-energy tensor $T_{\alpha \beta} = \partial_\alpha u \cdot \partial_\beta u -\frac{1}{2} \eta_{\alpha \beta} ( \partial^\gamma u \cdot \partial_\gamma u + F(u))$ was close enough to the quadratic form $\partial_i u \cdot \partial_j u$ that shows up in the isometric embedding problem that I was able to use the Nash embedding theorem (applied to a backwards light cone, quotiented by a discrete scaling symmetry) to resolve the second step of the argument. The field $u$ had to take values in quite a high dimensional space - I ended up using ${\bf R}^{40}$ - because of the somewhat high dimension needed in the target Euclidean space for the Nash embedding theorem to apply. Also, there is a major indirect use of the Nash embedding theorem: the Nash-Moser iteration scheme that was introduced in order to prove this theorem has since proven to be a powerful tool to establish existence theorems for several other nonlinear PDE, though in many cases it turns out later that with some trickery one can avoid this scheme. For instance the original proof by Hamilton of the local existence for Ricci flow in Hamilton, Richard S. , Three-manifolds with positive Ricci curvature , J. Differ. Geom. 17, 255-306 (1982). ZBL0504.53034 . relied on Nash-Moser iteration, though a later trick of de Turck in DeTurck, Dennis M. , Deforming metrics in the direction of their Ricci tensors , J. Differ. Geom. 18, 157-162 (1983). ZBL0517.53044 . allowed one to avoid using this scheme. (For Nash embedding itself, a somewhat similar trick of Gunther in Günther, Matthias , Isometric embeddings of Riemannian manifolds, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 1137-1143 (1991). ZBL0745.53031 . can be used to also avoid applying Nash-Moser iteration.)
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|
343,955 |
Given a spherically symmetric potential $V: {\bf R}^d \to {\bf R}$ , smooth away from the origin, one can consider the Newtonian equations of motion $$ \frac{d^2}{dt^2} x = - (\nabla V)(x)$$ for a particle $x: {\bf R} \to {\bf R}^d$ in this potential well. Under the spherical symmetry assumption, one has conservation of angular momentum (as per Noether's theorem, or Kepler's second law), and using this one can perform a "symplectic reduction" and reduce the dynamics to an autonomous second order ODE of the radial variable $r = |x|$ as a function of an angular variable $\theta$ ; see for instance https://en.wikipedia.org/wiki/Kepler_problem#Solution_of_the_Kepler_problem . In general (assuming an attractive potential and energy not too large), the energy surfaces of this ODE are closed curves, and this leads to the radial variable $r$ depending in a periodic fashion on the angular variable $\theta$ , provided that one lifts the angular variable from the unit circle ${\bf R}/2\pi {\bf Z}$ to the universal cover ${\bf R}$ . In the special case of the inverse square law $V(x) = -\frac{GM}{|x|}$ , it turns out that the period of the map $\theta \mapsto r$ is always equal to $2\pi$ , which means in this case that the orbits are closed curves in ${\bf R}^d$ (whereas for almost all other potentials, with the exception of the quadratic potentials $V(x) = c |x|^2$ , the orbits exhibit precession). Indeed, as was famously worked out by Newton (by a slightly different method), the calculations eventually recover Kepler's first law that the orbits under the inverse square law are ellipses with one focus at the origin. The calculations are not too difficult - basically by applying the transformation $u = 1/r$ one can convert the aforementioned ODE into a shifted version of the harmonic oscillator - but they seem rather "miraculous" to me. My (rather vague) question is whether there is a "high level" (e.g., symplectic geometry) explanation of this phenomenon of the inverse square law giving periodic orbits without precession. For instance, in the case of quadratic potentials, the phase space has the structure of a toric variety ${\bf C}^d$ (with the obvious action of $U(1)^d$ ), and the Hamiltonian $\frac{1}{2} |\dot x|^2 + \frac{c}{2} |x|^2$ is just one linear component of the moment map, so the periodicity of the orbits in this case can be viewed as a special case of the behaviour of general toric varieties. But I wasn't able to see a similar symplectic geometry explanation in the inverse square case, as I couldn't find an obvious symplectic torus action here (in large part because the period of the orbits varies with the orbit, as per Kepler's third law). Is the lack of precession just a "coincidence", or is there something more going on here? For instance, is there a canonical transformation that transforms the dynamics into a normal form that transparently reveals the periodicity (similar to how action-angle variables reveal the dynamics on toric varieties)?
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The gravitational or Coulomb potential has a "hidden" symmetry (hidden in the sense that it does not follow from the rotational symmetry). The resulting integral of the motion (the Runge-Lenz vector ) prevents space-filling orbits in classical mechanics (all orbits are closed), and introduces a degeneracy of the energy levels in quantum mechanics (energy levels do not depend on the azimuthal quantum number). The hidden symmetry raises the rotational symmetry group from three to four dimensions, so from SO(3) to SO(4). A geometric interpretation in four-dimensional momentum space of the SO(4) symmetry is given on page 234 of Lie Groups, Physics, and Geometry by Robert Gilmore. Historically, this interpretation goes back to V. Fock in Zur Theorie des Wasserstoffatoms [Z. Phys. 98, 145-154 (1935)]. The elliptic motion of the coordinate corresponds to a circular motion of the momentum. The circle in $\mathbb{R}^3$ is promoted to a circle in $\mathbb{R}^4$ by a projective transformation. SO(4) transformations in $\mathbb{R}^4$ rotate circles into circles, which then project down to circular momentum trajectories in the physical space $\mathbb{R}^3$ . You probably know that planets go around the sun in elliptical orbits. But
do you know why? In fact, they’re moving in circles in 4 dimensions.
But when these circles are projected down to 3-dimensional space, they
become ellipses! [ John Baez , animation by Greg Egan .
]
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|
343,958 |
Let $M$ be a smooth compact manifold of dimension $n$ , and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$ , embedded in $M$ . The embedding induces maps on $\pi_1$ . If $\pi_1(\partial U) \to \pi_1(M)$ is injective, does this imply that $\pi_1(U) \to \pi_1(M)$ is injective? If true, can you direct me to a reference or a short proof? EDIT: I reformulated the question adding compactness of M and U to rule out the counterexample given in an answer.
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The answer is 'yes' by Britton's lemma (see wikipedia and, more generally, Serre's book Trees and Scott and Wall's article 'Topological methods in group theory'). Since $M$ and $U$ are smooth and compact, $\partial U$ has a product neighbourhood $N(\partial I)\cong \partial U\times I$ . Cutting along $\partial U$ realises $M$ as a graph of spaces , with vertices corresponding to $\pi_0(M-\partial U)$ and edge spaces corresponding to $\pi_0(\partial U)$ . Van Kampen's theorem now asserts that the graph of spaces structure on $M$ induces a graph of groups structure on $\pi_1M$ . Note that part of the definition of a graph of groups is that the edge maps should be injective -- this is where your hypothesis that $\pi_1(\partial U)$ injects comes in. Finally, Britton's lemma (or, more precisely, its generalisation to the context of graphs of groups) implies that the natural maps from the vertex groups to the fundamental group inject, which is the fact you are asking for.
|
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|
344,219 |
Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at the moment we don't have a more satisfactory version." Is this still a fair description of the situation?
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My opinion, and that of many other people although not of everyone, is that the "correct" notion is that of stable ∞-category . Now, this is not a category in the strictest sense, rather a generalization of the notion of category known as an (∞,1)-category , or ∞-category for short, where to any pair of objects $x,y$ there is an associated homotopy type $\mathrm{Map}_{\mathcal{C}}(x,y)$ , usually called the mapping space . You can get a category from that datum by taking the connected components $\pi_0\mathrm{Map}_{\mathcal{C}}(x,y)=:[x,y]$ . The resulting category is called the homotopy category $h\mathcal{C}$ , and can be seen as the best approximation you can give of an ∞-category using an ordinary category. You can talk about limits and colimits in an ∞-category, and in fact pretty much all of classical category theory goes through in this more general setting without problems (although with the occasional very important modification). Then you can say that an ∞-category $\mathcal{C}$ is stable if it satisfies the two following conditions: It has a zero object (i.e. an object $0$ such that $\mathrm{Map}(x,0)$ and $\mathrm{Map}(0,x)$ are contractible for every $x\in\mathcal{C}$ ). It has all pullbacks and pushouts and a square (i.e. a diagram of the form $[1]\times [1]\to\mathcal{C}$ ) is cartesian iff it is cocartesian. As you can see, it is a fairly simple definition. It can be rephrased in a few equivalent ways, some of which are rather easy to check. This notion has a few very important properties: For every stable ∞-category $\mathcal{C}$ , the homotopy category $h\mathcal{C}$ has a canonical triangulated structure. All triangulated categories that actually show up in mathematical practice usually come equipped with a specific stable enrichment (i.e. a stable ∞-category whose homotopy category is the triangulated category you were thinking about). In a few cases, the stable ∞-category is actually easier to define. There are examples of triangulated categories that do not come from a stable ∞-category. All the examples tend to look unnatural, and we would very much like a definition that excludes them. In stable ∞-categories, a lot of the theorems that one would expect to be naively true for triangulated categories are actually true. For example, cones are functorial, and you can define the algebraic K-theory of a stable ∞-category (while you cannot do so for a triangulated category!), obtaining the expected results (e.g. the algebraic K-theory of the stable ∞-category of perfect complexes over a ring is exactly the algebraic K-theory of the ring). More abstractly, stable ∞-categories work well in families. For example, the functor sending a scheme $X$ to the stable ∞-category of perfect complexes over $X$ is a fppf sheaf (for an appropriate notion of sheaf of ∞-categories). This is not true for the corresponding triangulated categories!
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|
344,713 |
We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is: $$
S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j
$$ We have observed numerically that $S_n \approx 2 n e^{-n}$ . We would like to establish whether this conjecture is true. More precisely, we would like to show that $S_n= \Theta(n e^{-n})$ . The sum is quite hard to evaluate numerically because the summands grow in absolute value initially and then decrease toward zero. The largest term is exponential in $n$ while the sum is conjectured to converge to zero exponentially fast. Any references or ideas are welcome!
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I have obtained a formula for the generating function of your sequence. Let $S_n$ be defined as in the quesion. We extend the definition to $n = 0$ by demanding $0^0 = 0$ , hence $S_0 = 0$ . Consider $S(t) = \sum_{n\geq 0} S_n t^n$ . I will work out a formula for $S(t)$ . \begin{eqnarray*}
S(t) &=& \sum_{n=0}^\infty \sum_{j = 0}^n\frac{(-e)^{-j}}{j!}(n - j)^j t^n\\
&=& \sum_{j = 0}^\infty \frac{(-e)^{-j}}{j!}t^j\sum_{n = 0}^\infty n^jt^n\\
&=& \sum_{j = 0}^\infty \frac{(-e^{-1}t)^j}{j!}\frac{tA_j(t)}{(1 - t)^{j + 1}}\\
&=& \frac{t}{1 - t}\sum_{j = 0}^\infty A_j(t)\frac{(\frac{e^{-1}t}{t - 1})^j}{j!}\\
&=& \frac{t}{1 - t}\frac{t - 1}{t - e^{e^{-1}t}}\\
&=& \frac{t}{e^{e^{-1}t} - t}.
\end{eqnarray*} Explanation for the calculation: The key step is $\sum_{n\geq 0} n^jt^n = \frac{tA_j(t)}{(1 - t)^{j + 1}}$ , where $A_j(t)$ is the Eulerian polynomial . We then use the generating function $\sum_{j \geq 0}A_j(t)\frac{x^j}{j!} = \frac{t - 1}{t - e^{(t - 1)x}}$ . It is then reasonable to make the change of variable $T = e^{-1}t$ , which leads to $S_n = e^{-n}S'_n$ , where the sequence $(S'_n)_n$ has generating function $\frac{eT}{e^T - eT}$ . This at least gives a numerically better way to calculate the numbers $S_n$ . And the question becomes to prove that $S'_n = 2n + \frac{2}{3} + o(1)$ , where $S'_n$ is the $n$ -th coefficient of the Taylor expansion of the function $\frac{T}{e^{T - 1} - T}$ at $T = 0$ . I'm however not able to proceed further... One may try to subtract $\sum_{n \geq 0}(2n + \frac{2}{3})T^n$ , which is a rational fraction, and try to show that the difference has Taylor coefficients tending to $0$ . But this doesn't seem to help much. Also the $n$ -th Taylor coefficient could be calculated via a residue at $0$ , but again doesn't seem to help much. Another observation is that $T = 1$ is a pole of the function $\frac{T}{e^{T - 1} - T}$ . Experimentally, it seems that the parameter $e$ is quite important. For this parameter, we indeed have $S'_n = 2n + \frac{2}{3} + o(1)$ , as Henri Cohen stated in the comment. In fact, the $o(1)$ is also exponentially decreasing. If $e$ is changed to anything larger, then the sequence $(S'_n)_n$ increases much faster; if it is changed to anything smaller, then negative terms appear.
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|
344,968 |
Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character. Consider the following combinatorial property of $\Lambda$ : for all triple $(j,k,\ell)$ $$\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} \ge 0.$$ It is a consequence of a more general result involving subfactor planar algebra and fusion category (see here Corollary 7.5, see also this answer ). Question : Is this combinatorial property already known to finite group theorists? If yes: What is a reference? If no: Is there a group theoretical elementary proof? In any case : Are there other properties of the same kind? To avoid any misunderstanding, let us see one example. Take $G=A_5$ , its character table is: $$\left[ \begin{matrix}
1&1&1&1&1 \\
3&-1&0&\frac{1+\sqrt{5}}{2}&\frac{1-\sqrt{5}}{2} \\
3&-1&0&\frac{1-\sqrt{5}}{2}&\frac{1+\sqrt{5}}{2} \\
4&0&1&-1&-1 \\
5&1&-1&0&0
\end{matrix} \right] $$ Take for example $(j,k,\ell) = (2,4,5)$ , then $\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} = \frac{5}{3} \ge 0$ .
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By standard manipulations with the group algebra, your sum has a combinatorial/probabilistic interpretation that makes its nonnegativity clear. The element $ \frac{1}{|G|} \sum_{ g\in G} [g hg^{-1} ]$ in the group algebra is conjugacy invariant, and so acts by scalars on each irreducible representation. Because its trace on a representation with the character $\chi$ is $ \frac{1}{|G|} \sum_{ g\in G} \chi( g hg^{-1} ) = \frac{1}{|G|} \sum_{ g\in G} \chi( h )= \chi(h)$ , its unique eigenvalue must be $\frac{\chi(h)}{\chi(1)}$ . Hence for $h_1,h_2,h_3$ three elements of the group, $$ \left( \frac{1}{|G|} \sum_{ g\in G} [g h_1g^{-1} ]\right) \left( \frac{1}{|G|} \sum_{ g\in G} [g h_2g^{-1} ]\right) \left( \frac{1}{|G|} \sum_{ g\in G} [g h_3g^{-1} ]\right) $$ acts on this representation with eigenvalue $\frac{ \chi(h_1) \chi(h_2) \chi(h_3)}{\chi(1)^3}$ . Now the group algebra, as a module over itself, is the sum over irreducible characters $\chi$ of $\chi(1) $ copies of the representation with character $\chi$ . Hence the trace of this element on the group algebra is $$\sum_{\chi} \chi(1) \cdot \chi(1) \cdot \frac{ \chi(h_1) \chi(h_2) \chi(h_3)}{\chi(1)^3}= \sum_{\chi} \frac{ \chi(h_1) \chi(h_2) \chi(h_3)}{\chi(1)}.$$ On the other hand, the trace of an element of the group algebra on itself is the order of the group times the coefficient of $[1]$ . The coefficient of $[1]$ in this particular element is $\frac{1}{ |G|^3}$ times the number of $g_1,g_2,g_3$ such that $g_1 h_1 g_1^{-1} g_2 h_2 g_2^{-1} g_3 h_3 g_3^{-1} =1$ . This gives the combinatorial interpretation $$\sum_{\chi} \frac{ \chi(h_1) \chi(h_2) \chi(h_3)}{\chi(1)} = \frac{1}{ |G|^2} \left| \{ g_1,g_2,g_3 \in G \mid g_1 h_1 g_1^{-1} g_2 h_2 g_2^{-1} g_3 h_3 g_3^{-1} =1 \}\right|$$ from which non-negativity is clear. I would guess this is probably in the group theory literature somewhere but I wouldn't know where.
|
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345,275 |
Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power. Of course, Lebesgue and Poincaré knew each other, they even met on several occasions and shared a common close friend, Émile Borel. However, it seems Lebesgue never wrote to Poincaré and, according to Lettres d’Henri Lebesgue à Émile Borel , note 321, p. 370 … la seule information, de seconde main, que nous avons sur l’intérêt
de Poincaré pour la « nouvelle analyse » de Borel, Baire et Lebesgue the only second-hand information we have on Poincaré's interest in the "new analysis" of Borel, Baire and Lebesgue is this, Lebesgue to Borel, 1904, p. 84: J’ai appris que Poincaré trouve mon livre bien ; je ne sais pas
jusqu’à quel point cela est exact, mais j’en ai été tout de même très
flatté ; je ne croyais pas que Poincaré sût mon existence. I learned that Poincaré finds my book good; I do not know to what extent that is accurate, but I nevertheless was very flattered; I did not believe that Poincaré knew of my existence. See also note 197, p. 359 Nous ne connaissons aucune réaction de Poincaré aux travaux de Borel,
Baire et Lebesgue. We do not know any reaction of Poincaré to the works of Borel, Baire and Lebesgue. To my mind this situation is totally unexpected, almost incredible: the Lebesgue integral and measure theory are major mathematical achievements but Poincaré, the ultimate mathematical authority at this time, does not say anything??? What does it mean? So, please, are you aware of any explicit or implicit statement by Poincaré on the Lebesgue integral or measure theory? If you are not, how would you interpret Poincaré’s silence? Pure disinterest? Why? Discomfort? Why? Something else? This question is somewhat opinion-based, but The true method of forecasting the future of mathematics is the study
of its history and current state. according to Poincaré and his silence is a complete historical mystery, at least to me.
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It has nothing to do with the conflict with Borel which developed later, and one can find a pretty explicit answer in the aforementioned letters of Lebesgue to Borel. (These letters were first published in 1991 in Cahiers du séminaire d’histoire des mathématiques ; selected letters with updated commentaries were also published later by Bru and Dugac in an extremely interesting separate book .) In letter CL (May 30, 1910) Lebesgue clearly states: Poincaré m'ignore; ce que j'ai fait ne s'écrit pas en formules. Poincaré ignores me, [because] what I have done can not be written in formulas. EDIT In interpreting this statement of Lebesgue I trust the authority of Bru and Dugac who in "Les lendemains de l'intégrale" accompany this passage with a footnote (missing in the 1991 publication) stating that Dans [the 1908 ICM address] Poincaré ne semble pas considérer l'intégrale de Lebesgue comme faisant partie de "l'avenir des mathématiques", puisqu'il ne mentionne pas du tout la théorie des fonctions de variable réelle de Borel, Baire et Lebesgue. In [the 1908 ICM address] Poincaré does not seem to consider the Lebesgue integral as a part of the "future of mathematics", as he does not mention at all the theory of functions of a real variable of Borel, Baire and Lebesgue. I would rather interpret the meaning of "formulas" in the words of Lebesgue in a more straighforward and naive way. It seems to me that he was referring to the opposition which was more recently so vividly revoked by Arnold in the form of "mathematics as an experimental science" vs "destructive bourbakism". By the way, it is interesting to mention that the first applications of the Lebesgue theory were - may be surprisingly - not to analysis, but to probability (and the departure point of Borel's Remarques sur certaines questions de probabilité , 1905 is clearly and explicitly the first edition of Poincaré's "Calcul des probabilités"). Poincaré had taught probability for 10 years and remained active in this area (let me just mention "Le hasard" that appeared first in 1907 and then was included as a chapter in "Science et méthode", 1908 and the second revised edition of "Calcul des probabilités", 1912), and still he makes no mention of Lebesgue's theory. This issue has been addressed, and there are excellent articles by Pier ( Henri Poincaré croyait-il au calcul des probabilités? , 1996), Cartier ( Le Calcul des Probabilités de Poincaré , 2006, the English version is a bit more detailed) and Mazliak ( Poincaré et le hasard , 2012 or the English version ). To sum them up, [Poincaré's] seemingly limited taste for new mathematical techniques, in particular measure theory and Lebesgue’s integration, though they could have provided decisive tools to tackle numerous problems (Mazliak) is explained by his approach of a physicist and not of a mathematican (Cartier) to these problems.
|
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|
345,383 |
What is the meaning of the text inside this AMS logo? The image is from here , and the logo seems to have been frequently used until the 80's. The text is ΑΓΕΩΜΕΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ but Google translate and Googling have not helped. Added: (07-11-2019) Who is the author (designer/ suggester) of this AMS logo?
|
ἀγεωμέτρητος μη εἰσίτω - Let no one untutored in geometry enter here According to tradition this text was displayed in the
entrance of Plato's Academy. (The tradition is of a late data, see this critical discussion. ) Nicolaus Copernicus choose to print it on the cover
page of the book De Revolutionibus . image from 1938 (left) and from 2015 (right) De Revolutionibus
|
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|
345,388 |
In computability theory, what are examples of decision problems of which it is not known whether they are decidable ?
|
An integer linear recurrence sequence is a sequence $x_0, x_1, x_2, \ldots$ of integers that obeys a linear recurrence relation $$x_n = a_1 x_{n-1} + a_2 x_{n-2} + \cdots + a_d x_{n-d}$$ for some integer $d\ge 1$ , some integer coefficients $a_1, \ldots, a_d$ , and all $n\ge d$ . The following problem is sometimes known as "Skolem's problem": Given $d$ , $a_1, \ldots, a_d$ , $x_0, \ldots x_{d-1}$ , does there exist $n$ such that $x_n=0$ ? It is unknown whether the above problem is undecidable. For more information, see Terry Tao's blog post on the subject.
|
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"source": [
"https://mathoverflow.net/questions/345388",
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|
345,515 |
Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy classes of paths from $p$ to $x \in X$ . Is there an alternative construction of $\varphi$ that avoids choosing a basepoint?
|
I think that homotopy-theorists often fall into the habit of working mainly with based spaces, even when they don't need to. It can be instructive to notice when the use of a basepoint is unnecessary, even artificial. But it's also important to notice the parts of the subject where the use of a basepoint is necessary. This (the topic of universal covering spaces) is one of those parts. By "universal covering space" of a connected manifold $M$ I assume we mean a simply connected manifold $\tilde M$ with covering map $p:\tilde M\to M$ . (By "simply connected" I mean connected and having trivial $\pi_1$ for one, hence any, basepoint. The empty space is not connected.) There is always a universal covering space, and to explain how to make one we usually start by picking a point $x\in M$ . Any two universal covering spaces, no matter how they are constructed, are related by an isomorphism, by which I mean a diffeomorphism that respects the projection to $M$ . But this isomorphism is not unique, because for any such $(\tilde M,p)$ there is a group of isomorphisms $\tilde M\to \tilde M$ (i.e. deck transformations), a nontrivial group except in the case when $M$ itself is simply connected. Suppose that there were a way of making a universal covering space $\tilde M$ that did not depend on a choice of basepoint (or any other arbitrary choice), and suppose that for $x\in M$ there was a canonical isomorphism between this $\tilde M$ and the one determined by $x$ . But this would imply that when we use two points $x\in M$ to make two universal covering spaces of $M$ then there is a canonical isomorphism between these. Every homotopy class of paths from $x$ to $y$ in $M$ (homotopy with endpoints fixed) determines an isomorphism between the two covering spaces, and every isomorphism arises from exactly one such homotopy class. So if we had a canonical isomorphism we would have a canonical homotopy class of paths from $x$ to $y$ . And surely we don't. (That's not rigorous, because what does "canonical" mean? But surely if one had an actual recipe for making an $\tilde M$ for $M$ without first making some arbitrary choice then for any diffeomorphism $h:M_1\cong M_2$ the choice of canonical path classes in $M_1$ would be related by $h$ to the corresponding choice in $M_2$ . In particular this would be the case for a reflection $S^1\to S^1$ that fixes two points $x$ and $y$ but of course does not fix any class of paths from $x$ to $y$ .)
|
{
"source": [
"https://mathoverflow.net/questions/345515",
"https://mathoverflow.net",
"https://mathoverflow.net/users/126543/"
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|
345,801 |
Let $f$ be a continuous function defined on the closed interval $[0,1]$ . Clearly $f$ is bounded and attains its bounds. Then my question is how often can $f$ take a value in its range countably many times? Formally let $Z(f,p)=\{x\in \text{Image}(f):|f^{-1}(x)| = p\}$ . I would like to know large $Z(f,\aleph_0)$ can be? I conjecture that we can find $f$ with $|Z(f,\aleph_0)|=2^{\aleph_0}$ . Note that I can find f with $Z(f,2^{\aleph_0}) = \aleph_0$ which I think makes the conjecture plausible: We know that the zero set of a continuous function on $[0,1]$ can be any closed subset as we can take $f$ to be the distance to the closed set. If we let the closed set be the cantor ternary set $C$ on $[0,1]$ we can create a function on $[0,1]$ with uncountably infinite zero set. By taking suitable scalings of the function on intervals $[\frac{1}{2^{2n+1}}, \frac{1}{2^{2n}}]$ with ramps in between we can find a function $f$ s.t. $Z(f)$ is countably infinite. Hence we have $Z(f,2^{\aleph_0}) = \aleph_0$ .
|
How about Devil's staircase (a.k.a. Cantor function) $C(x)$ , but with every horizontal segment replaced by a rescaled zigzag $Z(x)$ ? By a zigzag I mean, for example: $$ Z(x) = \tfrac{2}{\pi} \arcsin(\sin(2\pi x)) .$$ The function $f$ is formally defined as: $$ f(x) = \begin{cases} C(x) & \text{if $x$ has no $1$ in ternary expansion,} \\ C(x) + 2^{-n} Z(3^n(x - a_{n,k})) & \text{if $x \in [a_{n,k}, a_{n,k}+3^{-n}]$,} \end{cases} $$ where $a_{n,k}$ , $k = 1, 2, \ldots, 2^n$ , is the enumeration of left endpoints of maximal line segments of length exactly $3^{-n}$ on which $C(x)$ is constant. For every $y_0$ which is not a dyadic rational there is exactly one $x$ with no $1$ in ternary expansion such that $f(x) = C(x) = y_0$ , and for each $n = 1, 2, \ldots$ the line $y = y_0$ intersects exactly one zigzag of $f$ of horizontal length $3^{-n}$ . EDIT: To clarify the definition of $f$ , write $$ x = \sum_{n = 1}^\infty \frac{x_n}{3^n} $$ for the ternary expansion of $x$ , and let $K \in \{1, 2, \ldots, \infty\}$ be the position of first digit $1$ . The Cantor function $C(x)$ is equal to $$ C(x) = \sum_{n = 1}^K \frac{\lceil x_n/2 \rceil}{2^n} . $$ The function $f$ is defined by $$ f(x) = \begin{cases} C(x) & \text{if $K = \infty$,} \\ C(x) + 2^{-K} Z\biggl(\sum_{n = 1}^\infty \dfrac{x_{K+n}}{3^n}\biggr) & \text{otherwise.} \end{cases} $$ Here is the plot of $f$ : To see that $f$ is continuous, it is enough to observe that $f - C$ is an infinite series of continuous "zigzag" functions with disjoint supports and decreasing supremum norms. The series thus converges uniformly, and consequently $f - C$ is continuous. Regarding the level sets: Suppose that $y$ is not a dyadic rational, with binary digits $y_n$ : $$ y = \sum_{n = 1}^\infty \frac{y_n}{2^n} . $$ Then $C(x) = y$ has exactly one solution (namely: an $x$ with $x_n = 2 y_n$ ), and this will also be a solution of $f(x) = y$ . All other solutions $x$ necessarily have $K < \infty$ . For such an $x$ , we have $|f(x) - C(x)| \le 2^{-K}$ , that is, $|y - C(x)| \le 2^{-K}$ . Since $y$ is not a dyadic rational, it follows that $x_n = 2 y_n$ for $n = 1, 2, \ldots, K - 1$ , and of course $x_K = 1$ . Therefore, $$ 2^{-K} Z\biggl(\sum_{n = 1}^\infty \frac{x_{K+n}}{3^n}\biggr) = f(x) - C(x) = y - C(x) = \frac{y_K - 1}{2^K} + \sum_{n = K + 1}^\infty \frac{y_n}{2^n} . $$ The above equation clearly has exactly two solutions ( $Z$ is essentially two-to-one). It follows that $f(x) = y$ has one solution with $K = \infty$ and two solutions corresponding to every finite $K$ . By the way, level sets corresponding to dyadic rationals are countable, too, by a very similar argument.
|
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"source": [
"https://mathoverflow.net/questions/345801",
"https://mathoverflow.net",
"https://mathoverflow.net/users/7113/"
]
}
|
345,826 |
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question. Let $m$ be the Lebesgue measure in $\mathbb{R}$ . A measurable set $E\subseteq\mathbb{R}$ has density $d$ at $x$ if $$\lim_{h\to 0} \frac{m(E\cap [x-h, x+h])}{2h} $$ exists and equals $d$ . Denote by $\phi(E)$ , $\{x\in\mathbb{R} : d(x, E)=1\}$ . The family of all measurable sets $E$ such that $E\subseteq\phi(E)$ is a topology on $\mathbb{R}$ , henceforth denoted by $(X, \mathcal{T})$ or just X if confusion is unlikely. Clearly $\mathcal{T}$ is stronger that the usual topology $(\mathbb{R}, \mathcal{E})$ , that is, $\mathcal{E}\subseteq\mathcal{T}$ . This topology is called the density topology in $\mathbb{R}$ . Some properties of the density topology. FACT 1 The Borel subsets of $X$ are precisely the measurable sets. Every Borel subset of $X$ is a $G_{\delta}$ . Every regular open set is a Euclidean $F_{\sigma \delta}$ . $X$ satisfies the countable chain condition. $X$ is neither separable nor first countable, but every subspace of $X$ is Baire. FACT 2 The following conditions on a subset $Y$ of $X$ are equivalent: $Y$ is a nullset (i.e. has measure zero) $Y$ is a nowhere dense $Y$ is a first category $Y$ is closed discrete. My question is the following : Suppose $G$ is an additive subgroup of $\mathbb{R}$ of positive Lebesgue outer measure such that $G$ is of the first category in $(\mathbb{R}, \mathcal{E})$ . How can I conclude that $G$ is dense in $(\mathbb{R}, \mathcal{T})$ ? Remember that the Lebesgue inner measure of $E\subseteq \mathbb{R}$ is defined as $$m_{*}(E)=\sup\{m(C) : C\subseteq E, C\hspace{0.1cm} \text{is}\hspace{0.1cm}\mathcal{E}-\text{closed} \} $$ In general, we have the following characterization for dense subsets in the density topology on $\mathbb{R}$ . Theorem. A subset $D$ of $\mathbb{R}$ is $\mathcal{T}$ -dense in $\mathbb{R}$ iff $m_{*}(\mathbb{R}\setminus D)=0$ . Proof. Suppose that $D$ is $\mathcal{T}$ -dense in $\mathbb{R}$ , then $\text{int}_{\mathcal{T}}(\mathbb{R}\setminus D)=\emptyset$ . Let $C$ be a closed set of $(\mathbb{R}, \mathcal{E})$ such that $C\subseteq \mathbb{R}\setminus D$ , in particular $C$ is $\mathcal{T}$ -closed, then $\text{int}_{\mathcal{T}}(\overline{C}^{\mathcal{T}})=\text{int}_{\mathcal{T}}(C)\subseteq \text{int}_{\mathcal{T}}(\mathbb{R}\setminus D)=\emptyset$ , by FACT 2 , $m(C)=0$ , then $m_{*}(\mathbb{R}\setminus D)=0$ . Now, suppose that $D$ is not $\mathcal{T}$ -dense, then there is $A\in \mathcal{T}\setminus \{\emptyset \}$ such that $A\cap D=\emptyset$ , so $A\subseteq \mathbb{R}\setminus D$ , therefore $m(A)=0$ , contradiction (because every non-empty $\mathcal{T}$ -open subset of $\mathbb{R}$ has positive measure).
|
How about Devil's staircase (a.k.a. Cantor function) $C(x)$ , but with every horizontal segment replaced by a rescaled zigzag $Z(x)$ ? By a zigzag I mean, for example: $$ Z(x) = \tfrac{2}{\pi} \arcsin(\sin(2\pi x)) .$$ The function $f$ is formally defined as: $$ f(x) = \begin{cases} C(x) & \text{if $x$ has no $1$ in ternary expansion,} \\ C(x) + 2^{-n} Z(3^n(x - a_{n,k})) & \text{if $x \in [a_{n,k}, a_{n,k}+3^{-n}]$,} \end{cases} $$ where $a_{n,k}$ , $k = 1, 2, \ldots, 2^n$ , is the enumeration of left endpoints of maximal line segments of length exactly $3^{-n}$ on which $C(x)$ is constant. For every $y_0$ which is not a dyadic rational there is exactly one $x$ with no $1$ in ternary expansion such that $f(x) = C(x) = y_0$ , and for each $n = 1, 2, \ldots$ the line $y = y_0$ intersects exactly one zigzag of $f$ of horizontal length $3^{-n}$ . EDIT: To clarify the definition of $f$ , write $$ x = \sum_{n = 1}^\infty \frac{x_n}{3^n} $$ for the ternary expansion of $x$ , and let $K \in \{1, 2, \ldots, \infty\}$ be the position of first digit $1$ . The Cantor function $C(x)$ is equal to $$ C(x) = \sum_{n = 1}^K \frac{\lceil x_n/2 \rceil}{2^n} . $$ The function $f$ is defined by $$ f(x) = \begin{cases} C(x) & \text{if $K = \infty$,} \\ C(x) + 2^{-K} Z\biggl(\sum_{n = 1}^\infty \dfrac{x_{K+n}}{3^n}\biggr) & \text{otherwise.} \end{cases} $$ Here is the plot of $f$ : To see that $f$ is continuous, it is enough to observe that $f - C$ is an infinite series of continuous "zigzag" functions with disjoint supports and decreasing supremum norms. The series thus converges uniformly, and consequently $f - C$ is continuous. Regarding the level sets: Suppose that $y$ is not a dyadic rational, with binary digits $y_n$ : $$ y = \sum_{n = 1}^\infty \frac{y_n}{2^n} . $$ Then $C(x) = y$ has exactly one solution (namely: an $x$ with $x_n = 2 y_n$ ), and this will also be a solution of $f(x) = y$ . All other solutions $x$ necessarily have $K < \infty$ . For such an $x$ , we have $|f(x) - C(x)| \le 2^{-K}$ , that is, $|y - C(x)| \le 2^{-K}$ . Since $y$ is not a dyadic rational, it follows that $x_n = 2 y_n$ for $n = 1, 2, \ldots, K - 1$ , and of course $x_K = 1$ . Therefore, $$ 2^{-K} Z\biggl(\sum_{n = 1}^\infty \frac{x_{K+n}}{3^n}\biggr) = f(x) - C(x) = y - C(x) = \frac{y_K - 1}{2^K} + \sum_{n = K + 1}^\infty \frac{y_n}{2^n} . $$ The above equation clearly has exactly two solutions ( $Z$ is essentially two-to-one). It follows that $f(x) = y$ has one solution with $K = \infty$ and two solutions corresponding to every finite $K$ . By the way, level sets corresponding to dyadic rationals are countable, too, by a very similar argument.
|
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"https://mathoverflow.net/questions/345826",
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|
346,264 |
This article describes the discovery by three physicists, Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago, and Peter Denton of Brookhaven National Laboratory, of a striking relationship between the eigenvectors and eigenvalues of Hermitian matrices, found whilst studying neutrinos. Their result has been written up in collaboration with Terence Tao here . From what I understand, although very similar results had been observed before, the link to eigenvector computation had not been explicitly made prior to now. For completeness here is the main result, "Lemma 2" from their paper: Let $A$ be a $n × n$ Hermitian matrix with eigenvalues $\lambda_i(A)$ and normed eigenvectors $v_i$ .
The elements of each eigenvector are denoted $v_{i,j}$ . Let $M_j$ be the $(n − 1) × (n − 1)$ submatrix of $A$ that results from deleting the $j^{\text{th}}$ column and the $j^{th}$ row, with eigenvalues $\lambda_k(M_j)$ . Lemma 2 . The norm squared of the elements of the eigenvectors are related to the
eigenvalues and the submatrix eigenvalues, $$|v_{i,j}|^2\prod_{k=1;k\neq i}^n(\lambda_i(A)-\lambda_k(A))=\prod_{k=1}^{n-1}(\lambda_i(A)-\lambda_k(M_j))$$ I was wondering what are the mathematical consequences of this beautiful result? For example are there any infinite dimensional generalisations? Does it affect matrix algorithms or proofs therein? What about singular values ?
|
The OP asks about generalisations and applications of the formula in arXiv:1908.03795 . $\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same result as in the 2019 paper has been derived for normal matrices (with possibly complex eigenvalues) --- rather than just for Hermitian matrices: On the eigenvalues of principal submatrices of normal, hermitian and symmetric matrices , by Peter Nylen, Tin-Yau Tam & Frank Uhlig (1993): theorem 2.2 (with the identification $b_{ij}=|u_{ij}|^2$ made at the very end of the proof). A further generalisation to a signed inner product has been given in On the eigenvalues of principal submatrices of J-normal matrices (2011). In that case $b_{ij}=\epsilon_i\epsilon_j|u_{ij}|^2$ , with $\epsilon_i=\pm 1$ the signature of the inner product: $(x,y)=\sum_i \epsilon_i x_i^\ast y_i$ . $\bullet$ Concerning applications: in the 1993 paper the theorem is used to solve the following problem: when does a normal $n\times n$ matrix $A$ with principal sub-matrices $M_j$ exist, given the sets of distinct (complex) eigenvalues $\lambda_i(A)$ of $A$ and $\lambda_k(M_j)$ of $M_j$ . The answer is that the matrix $B$ with elements $$b_{ij}=\frac{\prod_{k=1}^{n-1}(\lambda_i(A)-\lambda_k(M_j))}{\prod_{k=1;k\neq i}^n(\lambda_i(A)-\lambda_k(A))}$$ should be unistochastic , meaning that $b_{ij}=|u_{ij}|^2$ , where the matrix $U$ with elements $u_{ij}$ is the eigenvector matrix of $A$ . Since the 1993 paper is behind a paywall, I reproduce the relevant page: A brief Mathematica file to test the formula is here. Addendum: following up on the trail pointed out by Alan Edelman on Tao's blog: this 1966 paper by R.C. Thompson, "Principal submatrices of normal and Hermitian matrices" , has the desired formula in the general case of normal matrices as Equation (15). where $\theta_{ij}=|u_{ij}|^2$ when all eigenvalues $\mu_\alpha$ of $A$ are distinct (the $\xi_{ij}$ 's are eigenvalues of $M_i$ ). The older papers mentioned in the comments below do not seem to have an explicit formula for $|u_{ij}|^2$ . Because this appears to be the earliest appearance of the eigenvector/eigenvalue identity, it might be appropriate to refer to it as "Thompson's identity", $^*$ as a tribute to professor Robert Thompson (1931-1995). It would fit in nicely with this quote from the obituary: Among Thompson's many services to research was his help in dispelling
the misinformed view that linear algebra is simple and uninteresting.
He often worked on difficult problems, and as much as anyone, he
showed that core matrix theory is laden with deeply challenging and
intellectually compelling problems that are fundamentally connected to
many parts of mathematics. $^*$ "Thompson's identity" to distinguish from Thompson's formula
|
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"https://mathoverflow.net/questions/346264",
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|
346,290 |
Given $(a_n \in \mathbb{N})$ , when is there a monad $T$ on $\mathrm{FinSet}$ such that $$
| T(n) | = a_n\quad\forall n\in \mathbb{N}\:?
$$
|
The OP asks about generalisations and applications of the formula in arXiv:1908.03795 . $\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same result as in the 2019 paper has been derived for normal matrices (with possibly complex eigenvalues) --- rather than just for Hermitian matrices: On the eigenvalues of principal submatrices of normal, hermitian and symmetric matrices , by Peter Nylen, Tin-Yau Tam & Frank Uhlig (1993): theorem 2.2 (with the identification $b_{ij}=|u_{ij}|^2$ made at the very end of the proof). A further generalisation to a signed inner product has been given in On the eigenvalues of principal submatrices of J-normal matrices (2011). In that case $b_{ij}=\epsilon_i\epsilon_j|u_{ij}|^2$ , with $\epsilon_i=\pm 1$ the signature of the inner product: $(x,y)=\sum_i \epsilon_i x_i^\ast y_i$ . $\bullet$ Concerning applications: in the 1993 paper the theorem is used to solve the following problem: when does a normal $n\times n$ matrix $A$ with principal sub-matrices $M_j$ exist, given the sets of distinct (complex) eigenvalues $\lambda_i(A)$ of $A$ and $\lambda_k(M_j)$ of $M_j$ . The answer is that the matrix $B$ with elements $$b_{ij}=\frac{\prod_{k=1}^{n-1}(\lambda_i(A)-\lambda_k(M_j))}{\prod_{k=1;k\neq i}^n(\lambda_i(A)-\lambda_k(A))}$$ should be unistochastic , meaning that $b_{ij}=|u_{ij}|^2$ , where the matrix $U$ with elements $u_{ij}$ is the eigenvector matrix of $A$ . Since the 1993 paper is behind a paywall, I reproduce the relevant page: A brief Mathematica file to test the formula is here. Addendum: following up on the trail pointed out by Alan Edelman on Tao's blog: this 1966 paper by R.C. Thompson, "Principal submatrices of normal and Hermitian matrices" , has the desired formula in the general case of normal matrices as Equation (15). where $\theta_{ij}=|u_{ij}|^2$ when all eigenvalues $\mu_\alpha$ of $A$ are distinct (the $\xi_{ij}$ 's are eigenvalues of $M_i$ ). The older papers mentioned in the comments below do not seem to have an explicit formula for $|u_{ij}|^2$ . Because this appears to be the earliest appearance of the eigenvector/eigenvalue identity, it might be appropriate to refer to it as "Thompson's identity", $^*$ as a tribute to professor Robert Thompson (1931-1995). It would fit in nicely with this quote from the obituary: Among Thompson's many services to research was his help in dispelling
the misinformed view that linear algebra is simple and uninteresting.
He often worked on difficult problems, and as much as anyone, he
showed that core matrix theory is laden with deeply challenging and
intellectually compelling problems that are fundamentally connected to
many parts of mathematics. $^*$ "Thompson's identity" to distinguish from Thompson's formula
|
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"https://mathoverflow.net/questions/346290",
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|
346,612 |
We began an introductory course on Differential Geometry this semester but the text we are using is Kobayashi/Nomizu, which I'm finding to be a little too advanced for an undergraduate introductory course in DG. There are also no graded homeworks, quizzes, or exams so a text with solved problems would be preferred. Textbook recommendations for introductory DG books is not a new question here, but I was specifically looking for books that follow a similar formalism as Kobayashi/Nomizu.
|
Yikes, that's brutal - Kobayashi-Nomizu is an excellent reference text, but using it in a first course on the subject is a bit like learning English from the Oxford English Dictionary. For instance: chapter 2 is about connections on principal bundles, chapter 3 is about linear / affine connections, and chapter 4 is about the special case of Riemannian connections; this is conceptually an elegant way to build the theory, but pedagogically it is exactly backwards. I second the suggestions in the comments to at least start with curves and surfaces. If the course has to go beyond that then it gets tough - there are good books about curves and surfaces, and there are good books about connections on vector bundles, but there aren't many that do both subjects in a unified way. In fact the only example that I know is Loring Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes, which covers both branches of the subject and bridges the gap with explicit calculations involving Riemannian connections on surfaces in $\mathbb{R}^3$ . It has a modest number of problems at the end of each chapter, and they're generally pretty good if not numerous.
|
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"https://mathoverflow.net/questions/346612",
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|
346,645 |
This integral is needed to obtain the joint
distribution of the sample variances of a random sample from a bivariate
Gaussian distribution. For details on the joint distribution of the sample
means, sample variances, and sample correlation, please see Sections 14.11 to
14.13 of Kendall's book (2nd edition): Kendall, Maurice G. , The Advanced Theory of Statistics. Vol. I, Philadelphia: J. B. Lippincott Co., XI, 457 p. (1944). ZBL0063.03214 . The integral is $$ I\left( n,c\right)
=\int_{-1}^{1}\exp\left( cr\right) \left( 1-r^{2}\right) ^{\left(
n-4\right) /2}dr,$$ where $c\neq0$ is a constant, and $n\geq4$ is (a constant and) an integer
(and is the sample size). When $n-4$ is even, we can use ``integration by parts'' to get $I\left( n,c\right) $ after
some tedious computations. In general, we can use the expansion $$\left(
1-r^{2}\right) ^{\left( n-4\right) /2}=\sum_{k=0}^{\infty}a_{k,n}r^{2k},$$ compute each $$s_{k}=\int_{-1}^{1}\exp\left( cr\right) a_{k,n}r^{2k}dr,$$ and
then compute $\sum_{k=0}^{\infty}s_{k}$ . For this method, we may not have an
explicit, analytic expression for $I\left( n,c\right) $ . On the other hand, we can use the expansion $$\exp\left( cr\right)
=\sum_{k=0}^{\infty}\frac{c^{k}r^{k}}{k!},$$ compute $$b_{2k}=2\int_{0}^{1}%
\frac{c^{2k}}{\left( 2k\right) !}r^{2k}\left( 1-r^{2}\right) ^{\left(
n-4\right) /2}dr,$$ and compute $\sum_{k=0}^{\infty}b_{2k}$ . Note that each $b_{2k}$ relates to the Beta function (upon a change of variable $r^{2}%
\mapsto\tau$ ). But what does $\sum_{k=0}^{\infty}b_{2k}$ look like? Could someone please point me to some references on whether $I\left(
n,c\right) $ has an explicit, analytic expression, or on whether such an expression can be obtained by one of the attempts mentioned above? Dose this integral involve special functions? Thanks.
|
Yikes, that's brutal - Kobayashi-Nomizu is an excellent reference text, but using it in a first course on the subject is a bit like learning English from the Oxford English Dictionary. For instance: chapter 2 is about connections on principal bundles, chapter 3 is about linear / affine connections, and chapter 4 is about the special case of Riemannian connections; this is conceptually an elegant way to build the theory, but pedagogically it is exactly backwards. I second the suggestions in the comments to at least start with curves and surfaces. If the course has to go beyond that then it gets tough - there are good books about curves and surfaces, and there are good books about connections on vector bundles, but there aren't many that do both subjects in a unified way. In fact the only example that I know is Loring Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes, which covers both branches of the subject and bridges the gap with explicit calculations involving Riemannian connections on surfaces in $\mathbb{R}^3$ . It has a modest number of problems at the end of each chapter, and they're generally pretty good if not numerous.
|
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"source": [
"https://mathoverflow.net/questions/346645",
"https://mathoverflow.net",
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|
346,903 |
Let $(X_\alpha)_{\alpha \in A}$ be a family of boolean random variables $X_\alpha: \Omega \to \{0,1\}$ on a probability space $\Omega = (\Omega, {\mathcal F}, {\mathbf P})$ . Let ${\mathcal S}$ be a family of boolean sentences that each involve finitely many of the $X_\alpha$ . Suppose that each sentence $S \in {\mathcal S}$ is almost surely satisfied by the $(X_\alpha)_{\alpha \in A}$ . Can one then "repair" the random variables by locating further random variables $(\tilde X_\alpha)_{\alpha \in A}$ with each $\tilde X_\alpha$ almost surely equal to $X_\alpha$ , such that the $\tilde X_\alpha$ surely satisfy all the sentences $S \in {\mathcal S}$ ? If $|A| \leq \aleph_0$ (that is to say there are at most countably many random variables) then the task is easy, for then the set of sentences $S$ is also at most countable, and (because the countable union of null events is null) there is a single null event $N$ outside of which the $X_\alpha$ already surely satisfy all the sentences $S$ . In particular there is a deterministic choice $X_\alpha^0 \in \{0,1\}$ of boolean inputs that satisfy all the sentences, and if one sets $\tilde X_\alpha$ to equal $X_\alpha$ outside of $N$ and $X_\alpha^0$ in $N$ , we obtain the claim. If $|A| \leq \aleph_1$ (that is to say $A$ has at most the cardinality of the first uncountable ordinal) and $\Omega$ is complete, then a slight variant of the above argument also works. We may well order $A$ so that every element $\alpha$ has at most countably many predecessors. We then use transfinite induction to recursively select $\tilde X_\alpha$ almost surely equal to $X_\alpha$ , with the property that for all (not just almost all) sample points $\omega \in \Omega$ , the tuple $(\tilde X_\beta(\omega))_{\beta \leq \alpha}$ may be extended to a tuple $(x_\beta)_{\beta \in A}$ solving all the sentences $S \in {\mathcal S}$ . Indeed, if such variables $\tilde X_\beta$ have already been constructed for all $\beta < \alpha$ , then the random variable $X_\alpha$ will already have this property outside of a null set $N_\alpha$ (here we use the fact that the set of tuples in the metrisable space $\{0,1\}^{\{ \beta: \beta \leq \alpha\}}$ that can be extended is the continuous image of a compact set and is thus closed and measurable). For each $\omega \in N_\alpha$ , there exists at least one choice of $\tilde X_\alpha(\omega)$ that will obey the required extension property, thanks to the compactness theorem; using the axiom of choice to arbitrarily define $\tilde X_\alpha$ on this null set, we obtain a $\tilde X_\alpha$ with the required properties (it is measurable because $\Omega$ is assumed complete), and then the entire tuple $(\tilde X_\alpha)_{\alpha \in A}$ will surely satisfy all the sentences $S \in {\mathcal S}$ . [It may be possible to drop the completeness hypothesis here by appealing to a measurable selection theorem; I have not thought about this carefully.] Another illustrative case where the answer is affirmative is if $A$ is arbitrary and ${\mathcal S}$ is just the collection of equality sentences $X_\alpha = X_\beta$ for $\alpha,\beta \in A$ . Thus we have $X_\alpha=X_\beta$ almost surely for each $\alpha,\beta$ , and we wish to modify each $X_\alpha$ on a null set to create new random variables $\tilde X_\alpha$ such that $\tilde X_\alpha = \tilde X_\beta$ . Note that for each $\omega \in \Omega$ it is not necessarily the case (even after deleting a null set) that all the $X_\alpha(\omega)$ are equal to each other (e.g., suppose $A=\Omega=[0,1]$ and $X_\alpha(\omega) = 1_{\alpha=\omega}$ ), but nevertheless the problem is easily solved in this case by arbitrarily selecting one element $\alpha_0$ of $A$ and defining $\tilde X_\alpha := X_{\alpha_0}$ . However, I do not have a good intuition as to whether the answer to this question is affirmative in general, even if one assumes good properties on the probability space $\Omega$ (e.g., that it is a standard probability space). The appearance of the cardinal $\aleph_1$ hints that perhaps the answer is sensitive to undecidable axioms in set theory. (For my ultimate application I would eventually like to replace the boolean space $\{0,1\}$ with the interval $[0,1]$ or other Polish spaces, and the sentences $S$ with closed conditions involving finitely many or countably many of the variables at a time, but the Boolean case already seems nontrivial and captures much of the essence of the problem.) EDIT: The following "near-counterexample" may also be suggestive. Set $\Omega = [0,1]$ , let $A = 2^{[0,1]}$ be the power set of $\Omega$ , and let $\mathcal{S}$ be the set of sentences $X_\alpha = X_\beta$ where $\alpha,\beta \subset [0,1]$ differ by at most one point. If one sets $X_\alpha(\omega) := 1_{\omega \in \alpha}$ , then one morally has that the $X_\alpha$ almost surely satisfy all the sentences in $S$ , but that there is no way to repair the $X_\alpha$ to random variables $\tilde X_\alpha$ that surely satisfy the equations as this would force $\tilde X_{[0,1]} = \tilde X_\emptyset$ while $X_{[0,1]}=1$ and $X_\emptyset = 0$ . However this is not actually a counterexample because most of the $X_\alpha$ are non-measurable. (Removed due to errors)
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In Terry's answer, he shows that his original question reduces to the question of whether, given a $\sigma$ -algebra $\mathcal F$ on some set $X$ and a measure $\mu$ on $(X,\mathcal F)$ , there is a ``splitting'' of the quotient algebra $\mathcal F / \mathcal N$ , where $\mathcal N$ denotes the ideal of $\mu$ -null sets. In this context, a splitting is a Boolean homomorphism $\Phi: \mathcal F / \mathcal N \rightarrow \mathcal F$ such that $\Phi([A]) \in [A]$ for all $A \in \mathcal F$ . (Some authors call this a lifting instead of a splitting.) When some such $\Phi$ exists, let us say that $(X,\mathcal F,\mu)$ has a splitting. I did some digging on this question this afternoon, and found two very good sources of information: David Fremlin's article in the Handbook of Boolean Algebras (available here ) and a survey paper by Maxim Burke entitled "Liftings for noncomplete probability spaces" (available here ). I'll summarize some of what I found below to supplement what Terry mentions in his answer. He mentions already that it is independent of ZFC whether $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting: $\bullet$ (von Neumann, 1931) Assuming $\mathsf{CH}$ , $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting. $\bullet$ (Shelah, 1983) There is a forcing extension in which $([0,1],\text{Borel},\text{Lebesgue})$ has no splitting. Also mentioned already is the fact that if we expand the $\sigma$ -algebra in question from the Borel sets to all Lebesgue-measurable sets, then the situation is more straightforward: $\bullet$ (Maharam, 1958) If $(X,\mu)$ is a complete probability space, then $(X,\mu\text{-measurable},\mu)$ has a splitting. Now on to some not-yet-mentioned results.
First, it's worth pointing out that one can obtain splittings with nice extra properties. $\bullet$ (Ioenescu-Tulcea, 1967) Let $G$ be a locally compact group, and let $\mu$ denote its Haar measure. Then $(G,\mu\text{-measurable},\mu)$ has a translation-invariant splitting (which means $\Phi([A+c]) = \Phi([A])+c$ for every $\mu\text{-measurable}$ set $A$ ). Once again, shrinking our $\sigma$ -algebra from all $\mu$ -measurable sets to only the Borel sets causes problems. $\bullet$ (Johnson, 1980) There is no translation-invariant splitting for $([0,1],\text{Borel},\text{Lebesgue})$ . Thus, interestingly, Shelah's consistency result becomes a theorem of $\mathsf{ZFC}$ if we insist on the splitting being translation-invariant (with respect to mod- $1$ addition). More generally: $\bullet$ (Talagrand, 1982) If $G$ is a compact Abelian group and $\mu$ is its Haar measure, then there is no translation-invariant splitting for $(G,\text{Borel},\mu)$ . What stood out to me most in Fremlin and Burke's articles is how many questions seem to be wide open. Open question: Is it consistent that every probability space has a lifting? If yes, this would give a consistent positive answer to Terry's original question. Open question: Is it consistent with $2^{\aleph_0} > \aleph_2$ that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting? (Carlson showed that it is consistent to have $2^{\aleph_0} = \aleph_2$ and for $([0,1],\text{Borel},\text{Lebesgue})$ to have a splitting. Specifically, he showed that this holds whenever one adds precisely $\aleph_2$ Cohen reals to a model of $\mathsf{CH}$ .) Open question: Does Martin's Axiom (or $\mathsf{PFA}$ , or $\mathsf{MM}$ ) imply that $([0,1],\text{Borel},\text{Lebesgue})$ has a splitting? Open question: What of the same question in other well-known models of set theory (the random model, Sacks model, Laver model, etc.)?
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347,334 |
Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$ ) of genus $g$ for some $g>1$ . I'm interested in knowing how Riemann proved that $M_g$ has dimension $3g-3$ . A modern proof involves deformation theory and Riemann-Roch theorem. In particular, one needs the notion of sheaf cohomology, which was not available at Riemann's time. How did Riemann prove that $M_g$ has dimension $3g-3$ for $g>1$ ? I would appreciate if someone could provide some reference to the original proof by Riemann.
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Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays.
He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere. He computes this dimension in two ways. By Riemann-Roch this dimension is $2d-g+1$ , for a fixed Riemann surface. (Indeed, Riemann-Roch says that
the dimension of the space of such functions with $d$ poles fixed
is $d-g+1$ (when $d\geq 2g-1$ which we may assume), but these poles can be moved, so one has to add $d$ parameters). On the other hand, such a function has $2(d+g-1)$ critical points by Riemann-Hurwitz.
Generically, the critical values are distinct, and can be arbitrarily assigned, and this gives the dimension of the set of all such maps on all Riemann surfaces
of genus $g$ .
So the space of all Riemann surfaces of genus $g$ must be of dimension $$2(d+g-1)-(2d-g+1)=3g-3.$$ Riemann-Roch is proved in section 5 of Part I and and the dimension of the moduli space is counted in section 12 of Part I of the paper cited in F. Zaldivar's answer. Remark. Indeed, Riemann did not know about sheaves, cohomology and Serre duality. Neither he knew the general definition of a Riemann surface (which is due to Weyl). But one should take into account that all these notions were developed for the purpose to explain and digest what Riemann wrote in this paper. Remark 2. A pair $(S,f)$ , where $S$ is a Riemann surface, and $f$ a meromorphic function from $S$ to the Riemann sphere is called "a Riemann surface spread over the sphere'' (Uberlagerungsflache). All such pairs can be constructed in the
following way: choose critical values of $f$ and make some cuts between them
so that the remaining region on the sphere is simply connected. Then take $d$ copies of this region (they are called sheets) stack them over the sphere, and paste them together along the cuts. You obtain a surface $S$ together with a map $f$ , the "vertical" projection onto the sphere. Parameters are critical values. Riemann did not have any exact definition of "Riemann surface", he just explained this procedure of gluing as a visualization tool. For him, $S$ is a "class of algebraic curves $F(x,y)=0$ under birational equivalence". Until the work of Weyl, these pairs $(S,f)$ were called Riemann surfaces, and only Weyl defined exactly what $S$ is. Nowadays $S$ is called a Riemann surface, and a pair $(S,f)$ a "Riemann
surface spread over the sphere".
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347,540 |
There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways: A total characterisation, where somehow "all of the information" about a field has been uncovered. A negative result, rendering the field somehow irrelevant. A possible example for 1 might be the classification of finite simple groups. Examples for 2 might be Goedel's theorem effectively halting Hilbert's programme, or results showing e.g. certain large cardinal axioms to be inconsistent undermining work which assumes it. What are some other examples of results "closing" a field? Are there examples of a small number of papers "completing" a field in the sense of 1 above? (Apologies for many scare quotes!)
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Let me preface this by saying that this is just my own account, based on various conversations I've had over the years with many mathematicians, of the following example. In 1976, William Thurston proved that a closed smooth manifold has a codimension one foliation if and only if it has zero Euler characteristic. Moreover, every codimension one distribution in the tangent bundle is homotopic to an integrable one. While history is always more complicated, at least at the folklore level, this result is said to have caused a mass exodus of people working in the theory of foliations. You can read about Thurston's point of view on this, which reflects the history being more complicated, in his note Proof and Progress in Mathematics. Of course, it's absurd to conclude that this "closed" the theory of foliations. Rather, what I've understood to be the case is that he proved a theorem which was largely expected to be false, and this rendered a nascent industry of building an obstruction theory for co-dimension one foliations largely irrelevant. Nonetheless, I've been told by many people who know way more about this story than I do that graduate students were actively encouraged to avoid the theory of foliations around this time; the general impression being that Thurston was cleaning up the subject.
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347,590 |
There are various differentiations/derivatives. For example, Exterior derivative $df$ of a smooth function $f:M\to \mathbb{R}$ Differentiation $Tf:TM\to TN$ of a smooth function between manifolds $f:M\to N$ Radon-Nikodym derivative $\frac{d\nu}{d\mu}$ of a $\sigma$ -finite measure $\nu$ Fréchet derivative $Df$ of a function between Banach spaces $f:V\to W$ What is the most general definition of differentiation or derivative?
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That's a real can of worms. There are tons of different notions of differentiability for functions lacking classical smoothness: the Gateaux derivative, the weak derivative, the distributional derivative, the directional derivative, the subgradient (for convex functions), Clarke's generalized gradient, Hadamard differentiability, Bouligand differentiability, the metric derivative and the upper gradient to name a few. All these notions can't be ordered by "generality" in any way I am aware of. And there are even notions of differentiability for set-valued maps... I think the reason for the large number of notions is that differentiability has distinct motivations: local approximation by a simpler structure (e.g. by linear maps (add some sort of continuity if you like)), measuring the rate of change (add a notion of direction if you like), inverting the process of integration in some sense, finding descent directions for optimization, an algebraic ruleset for polynomials… Oh, and you may want to have a look at this answer which contains 12(!) more notions of differentiability.
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348,126 |
In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of Hilbert, von Neumann established the link between quantum theory and functional analysis that has lasted. Moreover, partly through von Neumann's own contributions (which are on a par with those of Bohr, Einstein, and Schrodinger), the precision that functional analysis has brought to quantum theory has greatly benefited the foundational debate. However, it is simultaneously a loser's history: starting with Dirac and continuing with Feynman, until the present day physicists have managed to bring quantum theory forward in utter (and, in my view, arrogant) disregard for the relevant mathematical literature. As such, functional analysis has so far failed to make any real contribution to quantum theory as a branch of physics (as opposed to mathematics), and in this respect its role seems to have been limited to something like classical music or other parts of human culture that adorn life but do not change the economy or save the planet. On the other hand, like General Relativity, perhaps the intellectual development reviewed
in this paper is one of those human achievements that make the planet worth saving. To balance this interesting debate, if there actually exists real reasons to disagree with above bolded sentence of Klaas Landsman, let me ask the following: What are the real contributions of functional analysis to quantum theory as a branch of physics? Here "real" should be understood in the sense underlying the above paragraph. This question was asked on physics.stackexchange and on PhysicsOverflow .
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I'm can't claim to have studied the relevant history in a lot of detail, but count me a skeptic of Landsman's claim. Let's take this little paper and the companion that it cites as a test case, which I hope we can all agree is "real physics". The authors are clearly well versed in the calculus of variations and the representation theory of Lie groups. Both of these subjects are heavily intertwined with functional analysis - functional analysis is even foundational for the former. Are we to believe that these physicists were entirely ignorant of the subject? Or is the argument that functional analysis only influenced them indirectly through its contact with those mathematical applications? I think Landsman's argument makes an error common among pure mathematicians about how mathematics is actually applied to the sciences. We tend to think about theorems, because those are the main objects of study in our work, but for consumers of mathematics it is the definitions that are important. The role of theorems is to validate the correctness and importance of definitions, and sometimes provide tools for manipulating them. The definitions of functional analysis - (un)bounded linear operators, Hilbert spaces, states, and so on - appear all over the place in quantum mechanics. And many of the big open problems in theoretical physics call primarily for definitions rather than theorems: Is there a measure space on which path integrals make sense? What is the correct notion of Dirac operator on the loop space of a manifold? Is there a gauge theory which includes both gravity and the standard model? And so on.
|
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|
348,810 |
Suppose that $G, H$ are finitely generated groups such that $H$ is isomorphic to a finite index subgroup of $G$ and vice versa. Does it follow that $G$ is isomorphic to $H$ ? I am sure that the answer is negative but cannot find an example. I am mostly interested in the case of finitely presented groups. The assumption of finite index is, of course, necessary, otherwise one can take any two nonabelian free groups of finite rank. Here is what I know: Given a pair of groups $G, H$ as above, there is, of course, a sequence of isomorphic proper subgroups of finite index $$
... G_n\lneq G_{n-1}\lneq ...\lneq G_1\lneq G
$$ One can rule out the existence of such a sequence when $G$ is nonelementary hyperbolic, but this does not say much. PS. Noam's example makes me feel rather silly since I have seen such groups in vivo : The affine Coxeter groups $\tilde{B}_n, \tilde{C}_n$ , $n\ge 3$ .
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Simple counterexample: $G$ is the square of an infinite dihedral group,
consisting of symmetries of the ${\bf Z}^2$ lattice
of the form $(x,y) \mapsto (\pm x + a, \pm y + b)$ with $a,b \in \bf Z$ ;
and $H$ is the index- $2$ subgroup where $a \equiv b \bmod 2$ .
Then $H$ has index- $2$ subgroup consisting of the symmetries $(x,y) \mapsto (\pm x + 2a, \pm y + 2b)$ ,
and this subgroup is isomorphic with $G$ , but $G \not \cong H$ .
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|
349,461 |
Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant interests that one wouldn’t guess one from the other? (Best if the two interests are not endpoints of a continuum , as may have been the case of past universalists like Euler or Cauchy. For this reason, maybe best restrict to post-1850 or so?) The point of asking is that it seems not so rare, but you don’t normally learn it other than by chance. Edit: Now CW, works best with “one mathematician per answer” (and details of actual achievement , e.g. “war work on radar” may have been creative for some but maybe not all who did it). $\,\!$
|
John von Neumann was the first person to come to my mind. He published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones. […] In a short list of facts about his life he submitted to the National Academy of Sciences, he stated, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."
|
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350,060 |
The following families of polytopes have received a lot of attention: permutahedra , associahedra , cyclohedra , ... My question is simple: Why? As I understand, at least the latter two were initially constructed by their face lattice representing certain combinatorial objects (e.g. ways to insert parentheses into a string).
So I assumed that representing these structures as a face lattice was of some use. But then people got interested in realizing these objects geometrically, and it turns out that, e.g. the associahedron can be realized in many ways.
Was this surprising?
Is there something to be learned from that fact?
On the other hand, for the permutahedron the realization came probably first, so is there anything deep to learn from its combinatorial structure? Further, there seem to exist connections to algebra, e.g. homotopy theory. I cannot wrap my head around these connections.
For me, these polytopes are just further examples of polytopes, nothing else. So what's up?
Do they have some extremal properties?
Are they especially symmetric (i.e. are they interesting for their symmetries)?
Does the geometric point of view make apparent some hidden combinatorial properties of the underlying structures (e.g. the cyclohedron is said to be "useful in studying knot invariants")?
What justifies this interest?
|
Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way. Think about it this way - why care about sequences like $\{n!\}$ , Fibonacci or Catalan numbers ? The honest answer is "because they come up all the time". Now, once you know these sequences, you may want to understand the underlying structures (permutations, trees, Dyck paths, triangulations, etc.) You may then want to understand connections between structures (e.g. bijections), algebraic or geometric interpretations (e.g. group representations, volumes of polytopes), etc. Once you have developed some kind of structures you may want to understand the relations between different structures, whether your bijections are structure-preserving, etc. That's how you develop the theory starting with just numbers! In general, basic objects in combinatorics tend to lack structure. Adding structures is always welcome as they present a deeper understanding of the underlying objects (and sometimes even just numbers). It's what allows to employ and further develop tools from other parts of Combinatorics and other fields. This is the setup in which one can understand results such as Kuperberg's proof of the number of ASMs or the Adiprasito-Huh-Katz theorem , but it doesn't have to be so spectacular. Sometimes even a weak structure can lead to unexpected connections and generalizations unforeseen otherwise. In summary, "these polytopes are just further examples of polytopes" is a misunderstanding of the context in the same way as Fibonacci and Catalan numbers are not "just numbers". Viewed in context, permutahedra and associahedra exhibit structures of combinatorial objects invisible otherwise.
|
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|
350,082 |
I'm looking for some suggestions on how we might calculate cycles of a specific length in an edge-weighted graph. For example, imagine my phone tells me that I need to walk three miles today. It might be nice if the phone could calculate a few tours of appropriate length, ideally avoiding things like walking up and down the same streets repeatedly. A previous posting reviewed some algorithms for finding fixed-length cycles in unweighted graphs. However, I'm not aware of anything concerning the edge-weighted case. A simple algorithm for a road network might work as follows. Assume that I want a cycle of length $x$ that starts and ends at a source vertex $s$ . Now use Dijkstra's algorithm to produce a shortest path tree. Next, find the node $v$ in this tree whose distance from $s$ is close to $x/2$ . Our solution then involves walking from $s$ to $v$ and then back to $s$ again. While this algorithm is polynomial, the solution is not great as we will end up walking up and down the same streets (so its not really a cycle at all...). Another option might be to convert a weighted graph into an unweighted one. Assuming all edge-weights are integer, an edge of length $x$ is replaced by a chain of $x-1$ vertices, as the following demonstrates: How to convert an edge weighted graph into an unweighted graph We can then apply methods used for the unweighted graphs. An issue with this approach is that it could massively increase the size of the graph. Given that existing approaches for simple graphs can be quite expensive, this doesn't seem like a good idea. Does anyone have any ideas/thoughts?
|
Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way. Think about it this way - why care about sequences like $\{n!\}$ , Fibonacci or Catalan numbers ? The honest answer is "because they come up all the time". Now, once you know these sequences, you may want to understand the underlying structures (permutations, trees, Dyck paths, triangulations, etc.) You may then want to understand connections between structures (e.g. bijections), algebraic or geometric interpretations (e.g. group representations, volumes of polytopes), etc. Once you have developed some kind of structures you may want to understand the relations between different structures, whether your bijections are structure-preserving, etc. That's how you develop the theory starting with just numbers! In general, basic objects in combinatorics tend to lack structure. Adding structures is always welcome as they present a deeper understanding of the underlying objects (and sometimes even just numbers). It's what allows to employ and further develop tools from other parts of Combinatorics and other fields. This is the setup in which one can understand results such as Kuperberg's proof of the number of ASMs or the Adiprasito-Huh-Katz theorem , but it doesn't have to be so spectacular. Sometimes even a weak structure can lead to unexpected connections and generalizations unforeseen otherwise. In summary, "these polytopes are just further examples of polytopes" is a misunderstanding of the context in the same way as Fibonacci and Catalan numbers are not "just numbers". Viewed in context, permutahedra and associahedra exhibit structures of combinatorial objects invisible otherwise.
|
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|
350,138 |
In the Soviet times there was a famous Encyclopedia of Mathematics. I think it is still familiar to every Russian mathematician maybe except very young ones, and yours truly is in possession of all 5 volumes. Browsing it recently (with no real purpose) I came across a certain peculiarity. In the article "Kervaire invariant" by M.A.Shtan'ko there was a claim that the Kervaire invariant is nontrivial in all dimensions $2^k-2$ for $2\le k\le 7$ - yes, including 126. Which, for what I know, is still an open problem Kervaire invariant: Why dimension 126 especially difficult? . In this article, the credit for the $k=6$ and $k=7$ cases (lumped together) was given to M. Barratt, M. Mahowald, and A. Milgram, but with no actual reference. To be fair, the absence of references is understandable (in the original article, that is) because it was written in 1978 while a complete proof for dimension 62 was only published six years later https://web.math.rochester.edu/people/faculty/doug/otherpapers/barjoma.pdf It is possible that back in 1978 the result was just announced. But, what happened to the 126? And to Milgram? The simplest possible explanation is that a proof for 126 was also announced but later retracted. However this is by no means the only possibility, so I am curious what really happened. Besides, those MO folks who know more about the subject then myself might wonder what the attempted proof was like. After a bit of search I found a reference which may be relevant (hopefully). In "Some remarks on the Kervaire invariant problem from the homotopy point of view" by M.E.Mahowald (1971) there is Theorem 8 attributed to Milgram and after it the following Remark: "It can be shown that $\theta_4^2=0$ and thus Milgram's theorem implies $\theta_6$ exists". If I get it right this indeed means a nontrivial Kervaire invariant in dimension 126, so probably there is a mistake somewhere in this argument. (But even if it is so, damned if I have a clue who has made it: Milgram, Mahowald, or somebody else.) I have to admit that a few things about this story look suspicious. To begin with,
A. Milgram died in 1961 so it should probably be R. J. Milgram if any. In the introduction to "The Kervaire invariant of extended power manifolds " J. Jones stated explicitly that the 62 case is solved by Barratt and Mahowald but not published yet while in the higher dimensions the problem is open, in contradiction to what Shtan'ko wrote the same year. In a couple of papers between 1978 and 1981 I spotted references like [Barratt M. G., Mahowald M., The Arf invariant in dimension 62, to appear] but no traces whatsoever of 126 and Milgram. (Besides this article of Mahowald from almost a decade before.) I am at a loss what to make of all this. It would be nice if someone can set it straight - at the very least, I want to know if Shtan'ko made it up. By the way, an English translation of this Encyclopedia article can be found here https://www.encyclopediaofmath.org/index.php/Kervaire_invariant Only, the year is written 1989 instead of 1978 (a second edition, apparently).
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I found the following remark in Zhouli Xu's paper "The strong Kervaire invariant problem in dimension 62" : In [19], R. J. Milgram claims to show that under the same condition as in Theorem 1.1, one has $θ_{n+2}$ exists. If this were true, then we would have that $\theta_6$ exists. However, Milgram’s argument fails because of a computational mistake
[8]. The paper containing the mistake is R. J. Milgram, "Symmetries and operations in homotopy theory" Amer. Math. Soc. Proc.
Symposia Pure Math., 22(1971), 203-211 and the other reference is private communication with Robert Bruner.
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350,473 |
Let $G$ be a finite group, and $\rho_1, \rho_2: G\to GL_n(\mathbb C)$ be two representations. Suppose that $\rho_1$ and $\rho_2$ are equivalent (i.e. conjugate over $\mathbb C$ ), and suppose that both groups $\rho_1(G)$ , $ \rho_2(G)$ belong to $GL(n,\mathbb Z)$ . Is it true that these two groups are conjugate in $GL(n,\mathbb Z)$ ? If not, is this at least true in the case when $G$ is a symmetric group $S_n$ and the representation $\rho$ is irreducible? The motivation for this question is the following: I know that all complex irreducible representations of $S_n$ can be defined over integers. I wonder whether there is somehow a canonical choice.
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The smallest counterexample involving irreducible representations of symmetric groups is the $2$ -dimensional irreducible module for $\mathbb{C}S_3$ . It can be defined over the integers as the submodule $U = \langle e_2-e_1, e_3-e_1\rangle_\mathbb{Z}$ of the natural integral permutation module $\langle e_1, e_2, e_3 \rangle_\mathbb{Z}$ . Then $U \otimes_\mathbb{Z} \mathbb{C}$ is irreducible and affords the ordinary character labelled $(2,1)$ . The dual $U^\star = \mathrm{Hom}_{\mathbb{Z}}(U,\mathbb{Z})$ is isomorphic to the quotient of $\langle e_1,e_2,e_3 \rangle$ by the trivial submodule $\langle e_1+e_2+e_3\rangle$ . The corresponding homomorphisms $\rho, \rho^\star : S_3 \rightarrow \mathrm{GL}_2(\mathbb{Z})$ are such that $\rho(S_3)$ and $\rho^\star(S_3)$ are conjugate in $\mathrm{GL}_2(\mathbb{C})$ but not in $\mathrm{GL}_2(\mathbb{Z})$ . To prove the final claim: if the representations are $\mathbb{Z}$ -equivalent then the modules $U \otimes_\mathbb{Z} \mathbb{F}_3$ and $U^\star \otimes_\mathbb{Z} \mathbb{F}_3$ are isomorphic. The first has a trivial submodule spanned by $$(e_2-e_1) +(e_3-e_1) = e_1+e_2+e_3;$$ the quotient by this submodule is the sign module. The second is its dual, with the factors in the opposite order. Since both are indecomposable, they are not isomorphic.
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350,571 |
It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology theories? Is it simply that they are more computable? Is there a good a priori reason that this is an important class of cohomology theories to study?
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There is a sort of a priori reason why one would consider the cohomology theory $MU$ , without first knowing of its connection to manifold geometry, to formal groups, … .
Since complex-oriented cohomology theories are those cohomology theories with a ring map from $MU$ , perhaps having a sufficiently strong interest in $MU$ would suffice to then motivate them. $\newcommand{\co}{\colon\thinspace}\newcommand{\F}{\mathbb F}\newcommand{\Z}{\mathbb Z}\newcommand{\A}{\mathcal A}\newcommand{\from}{\leftarrow}$ There are a variety of constructions in homotopy theory that allow one to delete a class, where "class" can have different meanings and "delete" different levels of finesse.
For instance, if you have a map $f\co S^n \to X$ which on homology sends the fundamental class $\iota_n \in H_n S^n$ to some class $f_* \iota_n \in H_n X$ , then the mapping cone $C(f)$ has the property that $f_* \iota_n$ pushes forward to $0$ in $H_n C(f)$ .
In fact, if $f_* \iota_n$ is not torsion, then $H_* C(f)$ will be exactly $(H_* X) / f_* \iota_n$ , as one can see by writing out the homology long exact sequence.
By the same token, if $f_* \iota_n$ is $m$ –torsion, the same long exact sequence will produce a fresh class in $H_{n+1} C(f)$ , due to the death of $m \iota_n$ under $f_*$ .
This is a general truism in homotopy theory: if a class has been deleted "twice", then you get a fresh class one degree higher. For a slight twist on this same idea, consider the mod– $p$ cohomology of a space, which is related to its homology by the universal coefficient sequence $$ 0 \to \operatorname{Ext}(H_{n-1}(X; \Z), \F_p) \to H^n(X; \F_p) \to \operatorname{Hom}(H_n(X; \Z), \F_p) \to 0. $$ A torsion-free class in $H_n(X; \mathbb Z)$ contributes a class only to $H^n(X; \F_p)$ , but a $p$ –power–torsion class contributes classes both to $H^n(X; \F_p)$ and $H^{n+1}(X; \F_p)$ .
There is actually a process by which one can marry these pairs of classes and reconstruct integral cohomology: the Bockstein spectral sequence has signature $$H^*(X; \F_p) \otimes \F_p[b] \Rightarrow H^*(X; \Z_p),$$ it converges for connected $X$ of finite type, and its differentials perform exactly these marriages.
Its construction relies on the facts that $\Z_p$ is $p$ –complete and that it participates in the short exact sequence $$0 \to \Z_p \xrightarrow{p \cdot -} \Z_p \to \F_p \to 0,$$ and for us it suffices to leave its explanation at that. The first differential in the Bockstein spectral sequence takes the form of a stable, additive homomorphism $$\beta\co H^n(X; \F_p) \to H^{n+1}(X; \F_p).$$ In fact, this map is natural in $X$ , and such natural transformations in general are called stable cohomology operations .
The Steenrod algebra is the collection of all stable mod– $p$ cohomology operations, given by $$\A^* := [H\F_p, H\F_p]_* = H^*(H\F_p; \F_p).$$ This associative algebra is calculable from first principles [1]: $$\A^* = \F_p\langle \beta, P^n \mid j \ge 1 \rangle \, / \, (\text{various relations}),$$ where the angle brackets indicate that these generators do not commute; where $P^n$ is the " $p$ th power operation", which sends a cohomology class of degree $2n$ to its $p$ th power [2]; and where $\beta$ is the same Bockstein operation as above. Taking this calculation for granted, one can then make a further calculation of a cousin of these operations: $$H^*(H\Z_p; \F_p) = [H\Z_p, H\F_p]_*.$$ Starting with same the quotient sequence $$H\Z_p \xrightarrow{p} H\Z_p \to H\F_p,$$ and applying $[-, H\F_p]_*$ , the multiplication-by- $p$ map induces zero on cohomology, and hence the going-around map participates in a short-exact sequence of $\A$ –modules $$0 \from [H\Z_p, H\F_p]_* \from \A^* \from [H\Z_p, H\F_p]_{*+1} \from 0.$$ The first map is onto, hence our $\A^*$ –module of interest is cyclic; the second map presents it as a submodule of $\A^*$ generated by a class in degree $1$ ; and, since $\beta$ is the only operation in degree $1$ , we learn $$[H\Z_p, H\F_p]_* = \A^* / (\beta \cdot \A^*).$$ In this sense, $\beta$ witnesses the double-quotient of $H\Z_p$ by $p$ , in that the quotient appears on the left- and on the right-hand sides of $[H\F_p, H\F_p] = [H\Z_p / p, H\Z_p / p]$ .
By removing the quotient from the left and instead studying $[H\Z_p, H\Z_p/p] = [H\Z_p, H\F_p]$ , we avoid killing $p$ twice, and $\beta$ disappears. However , longer monomials in which $\beta$ appears in the middle still survive, and they contribute other odd-degree cohomology operations.
One might wonder whether every such odd-degree cohomology operation belongs to some spectrum $X$ with a non-nilpotent endomorphism $v\co \Sigma^n X \to X$ whose quotient recovers $H\F_p$ and which admits an associated Bockstein spectral sequence. It turns out that a version of this is true: $[MU, H\F_p]_*$ is (almost [3]) the quotient of $\A^*$ with $\beta$ fully deleted.
In this sense, $MU$ is the "maximally unquotiented" version of $H\F_p$ within even-concentrated ring spectra.
Even more than this, there are multiple theorems along these lines, coming at the same problem from different angles. The study of the mod– $p$ cohomology of $MU$ (and its various features, including its relation to that of $H\F_p$ ) is originally due to Milnor. The homotopy of $MU$ is given by $$\pi_* MU = \Z[x_1, x_2, \ldots],$$ and the generators $x_n$ (again, almost [3]) iteratively play the role of $v$ in the above fantasy resolution of $H\F_p$ . Starting with the $p$ –local sphere, one can iteratively remove its odd-degree homotopy while retaining even-concentrated homology.
Priddy showed that this ultimately leads to a spectrum called $BP$ , which is an indivisible $p$ –local summand of $MU$ . Starting with the $p$ –local sphere, one can iteratively remove its odd-degree homotopy by $A_\infty$ –algebra maps.
This leads to a sequence of spectra $X(n)_{(p)}$ with $X(\infty)_{(p)} = MU_{(p)}$ .
The original study of this sequence of spectra is due to Ravenel, then taken up by Hopkins, Devinatz, and Smith, and this perspective in terms of iterated quotients is due to Beardsley. Each of these objects begins with some desirable properties: the sphere spectrum has pleasant homology (but very knotty homotopy), and the Eilenberg–Mac Lane spectrum has pleasant homotopy (but knotty co/homology).
By trying to correct the unpleasant part, one keeps ending up at (a chunk of) $MU$ , which has even-concentrated homotopy, even-concentrated homology, even-concentrated co/operations, … .
All I mean to point out by this is that $MU$ is an extremely natural object to bump into, especially if one has a preference for even-concentration, whether due to a preference for commutativity over graded-commutativity or due to an aversion toward unnecessarily killing classes twice. –––––––––– [1] - Here I'm restricting to odd primes, but you can say all these same words with slightly different formulas at $p = 2$ . [2] - $P^n$ does other, more mysterious things in degrees other than $2n$ . [3] - The precise statement is that the $p$ –localization $MU_{(p)}$ splits as a sum of shifts of copies of a ring spectrum $BP$ .
This new spectrum has homotopy given by $$\pi_* BP = \Z_{(p)}[v_1, v_2, \ldots],$$ $[BP, H\F_p]$ is the submodule of $\A^*$ where $\beta$ is totally deleted, and these $v_j$ s are the desired self-maps $v$ .
(In terms of $\pi_* MU$ , $v_j$ is equivalent to $x_{p^j}$ modulo decomposables.)
Although $BP$ has all these nice properties, it depends on the prime $p$ , and $MU$ is to be thought of as the best integral object capturing all of them at once.
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351,019 |
It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. ( https://www.jstor.org/stable/3072368?origin=crossref )
When we add two numbers by elementary addition, we choose a basis $b$ for example $b=2$ which corresponds to the cyclic group $C_2$ . Suppose we have words $w_1,w_2$ of (possibly different) lengths from this group $C_2$ , how do we add them to get a new word $w$ in elementary addition? For example: $2=10_2=w_1$ , $3=11_2=w_2$ . Consider these as $w_1$ and $w_2$ . Adding these numbers we get $5=101_2$ so the new word $w=101$ . (1) But how exactly is the process of adding these two "words" from $C_2 = \{0,1\}$ in group theoretic means? (2) Is this "elementary addition" also possible for example for a non-cylcic group such as the Klein Four group? (3) We also assign a number to such a word (b-adic expansion). Is this assignment also possible for the Klein Four group? Thanks for your help. Edit : In view of the plot given below, I decided to put the tag "fractals" to this question.
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I think the point is that, forgetting the final carry, the group of $n$ -digit binary words is isomorphic to $C_{2^n}$ . In the simplest case, the group of 2-digit binary words is isomorphic to $C_4$ , which is built as a nontrivial extension $$ 0 \to C_2 \to C_4 \to C_2 \to 0 $$ The 2-cocycle you mention is the one corresponding to this extension. In general, $C_{2^n}$ is built up as an iterated extension of $C_2$ 's in the same way, with each carry being the associated 2-cocycle.
If we want to avoid forgetting the final carry, we can take the limit of the whole system to get the 2-adics $\mathbb{Z}_2$ . The natural numbers $\mathbb{N}$ sit inside this as the submonoid of "finite words" (words whose digits are eventually 0 as we read right to left)
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351,558 |
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$ , and the polynomial given by: $$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$ I've found with Sage that for every $1\leq m \leq n \leq 80$ this polynomial has the property that all of its roots are real (negative, of course). It seems these roots are not nice at all. For example for $m=3$ and $n=10$ , one has $$P(t) = 120t^3 + 135 t^2 + 30t+1$$ and the roots are: $$ t_1 = -0.8387989...$$ $$ t_2 = -0.2457792...$$ $$ t_3 = -0.0404217...$$ Is it true that all roots of $P_{m,n}(t)$ are real?
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If you have two polynomials $f(x)=a_0+a_1x+\cdots a_mx^m$ and $g(x)=b_0+b_1x+\cdots+b_nx^n$ , such that the roots of $f$ are all real, and the roots of $g$ are all real and of the same sign, then the Hadamard product $$f\circ g(x)=a_0b_0+a_1b_1x+a_2b_2x^2+\cdots$$ has all roots real. This was proven originally in E. Malo, Note sur les équations algébriques dont toutes les racines
sont réelles, Journal de Mathématiques Spéciales, (4), vol. 4 (1895) One can make stronger statements, such as the result by Schur that says that $\sum k!a_kb_k x^k$ will only have real roots, under the same conditions. Schur's theorem combined with the fact that $\{1/k!\}_{k\geq 0}$ is a Polya frequency sequence, implies Malo's theorem. I'm not sure what the best reference to learn the theory of real rooted polynomials, and the associated operations that preserve real rootedness is. One textbook I know that discusses some of these classical results is Marden's "Geometry of Polynomials".
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351,617 |
Is it true that any finite graph has a $K_n$ minor, where $n$ is a minimal vertex degree?
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No. The edge-graph of the icosahedron is regular of degree five, but does not have a $K_5$ minor because it is planar ( Kuratowski's theorem ).
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351,640 |
This question is cross-posted from academia.stackexchange.com where it got closed with the advice of posting it on MO. Kevin Buzzard's slides ( PDF version ) at a recent conference have really unsettled me. In it, he mentions several examples in what one would imagine as very rigorous areas (e.g., algebraic geometry) where the top journals like Annals and Inventiones have published and never retracted papers which are now known to be wrong. He also mentions papers relying on unpublished results taken on trust that those who announced them indeed have a proof. He writes about his own work: [...] maybe some of my work in the p-adic Langlands philosophy relies
on stuff that is wrong. Or maybe, perhaps less drastically, on stuff
which is actually correct, but for which humanity does not actually
have a complete proof. If our research is not reproducible, is it
science? If my work in pure mathematics is neither useful nor 100
percent guaranteed to be correct, it is surely a waste of time. He says that as a result, he switched to formalizing proofs completely, with e.g. Lean , which guarantees correctness, and thus reusability forever. Just how widespread is the issue? Are most areas safe, or contaminated? For example, is there some way to track the not-retracted-but-wrong papers? The answer I accepted on academia.stackexchange before the closure gives a useful general purpose method, but I'd really appreciate more detailed area-specific answers. For example, what fraction of your own papers do you expect to rely on a statement " for which humanity does not actually have a complete proof " ?
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"Are most areas safe, or contaminated?" Most areas are fine. Probably all important areas are fine. Mathematics is fine. The important stuff is 99.99999% likely to be fine because it has been carefully checked. The experts know what is wrong, and the experts are checking the important stuff. The system works. The system has worked for centuries and continues to work. My talk is an intentionally highly biased viewpoint to get people talking. It was in a talk in a maths department so I was kind of trolling mathematicians. I think that formal proof verification systems have the potential to offer a lot to mathematicians and I am very happy to get people talking about them using any means necessary. On the other hand when I am talking to the formal proofs people I put on my mathematician's hat and emphasize the paragraph above, saying that we have a human mathematical community which knows what it is doing better than any computer and this is why it would be a complete waste of time formalising a proof of Fermat's Last Theorem -- we all know it's true anyway because Wiles and Taylor proved it and since then we generalised the key ideas out of the park. It is true that there are holes in some proofs. There are plenty of false lemmas in papers. But mathematics is robust in this extraordinary way. More than once in my life I have said to the author of a paper "this proof doesn't work" and their response is "oh I have 3 other proofs, one is bound to work" -- and they're right. Working out what is true is the hard, fun, and interesting part. Mathematicians know well that conjectures are important. But writing down details of an argument is a lot more boring than being imaginative and figuring out how the mathematical world works, and humans generally do a poorer job of this than they could. I am concerned that this will impede progress in the future when computers start to learn to read maths papers (this will happen, I guess, at some point, goodness knows when). Another thing which I did not stress at all in the Pittsburgh talk but should definitely be mentioned is that although formal proof verification systems are far better when it comes to reliability of proofs, they have a bunch of other problems instead. Formal proofs need to be maintained, it takes gigantic libraries even to do the most basic things (check out Lean's definition of a manifold, for example), different systems are incompatible and systems die out. Furthermore, formal proof verification systems currently have essentially nothing to offer the working mathematician who understands the principles behind their area and knows why the major results in it are true. These are all counterpoints which I didn't talk about at all. In the future we will find a happy medium, where computers can be used to help humans do mathematics. I am hoping that Tom Hales' Formal Abstracts project will one day start to offer mathematicians something which they actually want (e.g. good search for proofs, or some kind of useful database which actually helps us in practice). But until then I think we should remember that there's a distinction between "results for which humanity hasn't written down the proof very well, but the experts know how to fill in all of the holes" and "important results which humanity believes and are not actually proved". I guess one thing that worries me is that perhaps there are areas which are currently fashionable, have holes in, and they will become less fashionable, the experts will leave the area and slowly die out, and then all of a sudden someone will discover a hole which nobody currently alive knows how to fill, even though it might have been the case that experts could once do it.
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352,249 |
Prelude : In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem , as presented in Nash's well-known paper "The Imbedding Problem for Riemannian Manifolds" (Annals of Math, 1956), and to offer a nontrivial fix for the problem, as detailed in this erratum note prepared by John Nash . This topic is also discussed in this MO question . Of course, any mathematician who has been around long enough knows of many published proofs with significant gaps, some provably irreparable, and some perhaps authored by himself or herself. What makes the above situation striking--and discomforting to many of us--is the combination of the following three factors: (1) The theorem whose proof is found faulty is a major result that was published in 1950 or after, in a readily accessible source to experts in the field . (I chose the 1950 lower bound as a way of focusing on the somewhat recent past). (2) The gap detected is filled with a nontrivial fix that is publicly available and consented to by experts in the field (so we are not talking about gaps easily filled, or about gaps alleged by pseudomathematicians, or about false publicly accepted theorems, as discussed in this MO question ). (3) There is an interlude of 30 years or more between the publication of the proof and the detection of the gap (I chose 30 years since it is approximately the age difference between consecutive generations, even though the interlude is 42 years in the case of the Nash embedding theorem). Question to fellow mathematicians : what is the most dramatic instance you know of where all of the three above factors are present?
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In 1970, I. N. Baker published a proof of a basic result in holomorphic dynamics: a transcendental entire function cannot have more than one completely invariant domain. A completely invariant domain is an open connected set $D$ such that $f(z)\in D$ if and only if $z\in D$ . Baker "proved" a more general statement that: there cannot be two disjoint domains whose preimages are connected. The "proof" was a simple topological argument which occupied less than one page.
Since then this result has been used and generalized by extending his simple argument. In summer 2016 I was explaining Baker's argument to Julien Duval, he was somewhat slow in understanding and kept asking questions. Few weeks later he found a gap in the proof. It also took him some time to convince me that there is a gap indeed. Specialists were informed. Half a year later an amazing counterexample has been constructed in https://arxiv.org/abs/1801.06359 by Lasse Rempe-Gillen and David Sixsmith. This paper contains the full account of the story.
This is a counterexample to Baker's more general statement only, not to the highlighted theorem itself, which is now an important open question.
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352,255 |
In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$ . What is known about the relationship between the existence of such a function and large cardinal phenomena? Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$ . What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references. [1] Larson, Paul B. , The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031 .
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In 1970, I. N. Baker published a proof of a basic result in holomorphic dynamics: a transcendental entire function cannot have more than one completely invariant domain. A completely invariant domain is an open connected set $D$ such that $f(z)\in D$ if and only if $z\in D$ . Baker "proved" a more general statement that: there cannot be two disjoint domains whose preimages are connected. The "proof" was a simple topological argument which occupied less than one page.
Since then this result has been used and generalized by extending his simple argument. In summer 2016 I was explaining Baker's argument to Julien Duval, he was somewhat slow in understanding and kept asking questions. Few weeks later he found a gap in the proof. It also took him some time to convince me that there is a gap indeed. Specialists were informed. Half a year later an amazing counterexample has been constructed in https://arxiv.org/abs/1801.06359 by Lasse Rempe-Gillen and David Sixsmith. This paper contains the full account of the story.
This is a counterexample to Baker's more general statement only, not to the highlighted theorem itself, which is now an important open question.
|
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352,298 |
Sets are the only fundamental objects in the theory $\sf ZFC$ . But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The idea is that every statement that mathematicians care about is equivalent to some question about sets. An example of such an encoding is Kuratowski's definition of ordered pair, $(a,b) = \{\{a\},\{a,b\}\}$ , which can then be used to define the cartesian product, functions, and so on. I'm wondering how arbitrary the choice was to use sets as a foundation. Of course there are alternative foundations that don't use sets, but as far as I know all these foundations are still based on things that are quite similar to sets (for example $\sf HoTT$ uses $\infty$ -groupoids, but still contains sets as a special case of these). My suspicion is that we could instead pick almost any kind of mathematical structure to use as a foundation instead of sets and that no matter what we chose it would be possible to encode all of mathematics in terms of statements about those structures. (Of course I will add the caveat that there has to be a proper class of whichever structure we choose, up to isomorphism. I'm thinking of things like groups, topological spaces, Lie algebras, etc. Any theory about a mere set of structures will be proved consistent by $\sf ZFC$ and hence be weaker than it.) For concreteness I'll take groups as an example of a structure very different from sets. Can every mathematical statement be encoded as a statement about groups? Since we accept that it is possible to encode every mathematical statement as a statement about sets, it would suffice to show that set theory can be encoded in terms of groups. I've attempted a formalization of this below, but I would also be interested in any other approaches to the question. We'll define a theory of groups, and then ask if the theory of sets (and hence everything else) can be interpreted in it. Since groups have no obvious equivalent of $\sf ZFC$ 's membership relation we'll instead work in terms of groups and their homomorphisms, defining a theory of the category of groups analogous to $\sf{ETCS+R}$ for sets. The Elementary Theory of the Category of Sets, with Replacement is a theory of sets and functions which is itself biinterpretable with $\sf ZFC$ . We'll define our theory of groups by means of an interpretation in $\sf{ETCS+R}$ . It will use the same language as $\sf{ETCS+R}$ , but we'll interpret the objects to be groups and the morphisms to be group homomorphisms. Say the theorems of our theory are precisely the statements in this language whose translations under this interpretation are provable in $\sf{ETCS+R}$ . This theory is then recursively axiomatizable by Craig's Theorem . Naturally we'll call this new theory ' $\sf{ETCG+R}$ '. The theory $\sf{ETCS+R}$ is biinterpretable with $\sf ZFC$ , showing that any mathematics encodable in one is encodable in the other. Question: Is $\sf{ETCG+R}$ biinterpretable with $\sf ZFC$ ? If not, is $\sf ZFC$ at least interpretable in $\sf{ETCG+R}$ ? If not, are they at least equiconsistent?
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The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional details I found on MO). The key observation is the following: Theorem: The category of co-group objects in the category of groups is equivalent to the category of sets and partial map between then, or equivalently to the category of pointed sets.. (Thanks to Martin Brandenbourgh for pointing out the mistake in an earlier version) (According to the nLab , this is due to Kan, from the paper "On monoids and their dual" Bol. Soc. Mat. Mexicana (2) 3 (1958), pp. 52-61, MR0111035 ) Co-groups are easily defined in purely categorical terms (see Edit 2 below). The equivalence of the theorem is given by free groups as follows: if $X$ is a set and $F_X$ is the free group on X then Hom $(F_X,H)=H^X$ is a group, functorially in H, hence $F_X$ has a cogroup object structure. As functions between sets induce re-indexing functions: $H^X \rightarrow H^Y$ that are indeed group morphisms, morphisms between sets indeed are cogroup morphisms. Explicitly, $\mu:F_X \rightarrow F_X * F_X$ is the map that sends each generator $e_x$ to $e_x^L * e_x^R$ , and $i$ is the map that sends each generators to its inverse. An easy calculation shows that the generators and the units (Edit: that's were the mistake was) are the only elements such that $\mu(y)=y^L*y^R$ and hence that any cogroup morphism comes from a partial function between sets: each generator is sent either to a generator or to the unit. And with a bit more work, as nicely explained on this other MO answer , one can check that any cogroup object is of this form. As pointed out by Martin Brandenbourgh in his answer - once we have the category of set and partial maps, (or equivalently the category of pointed sets) one can easily characterize the category of sets and map in purely categorical language: this is the (non-full) subcategory containing all objects and whose maps are the $f:X \to Y$ such that the square obtained by adding the maps $0 \to X$ and $0\to Y$ (where $0$ is the initial object) is a pullback square. So let's call this the category of "cogroup objects and total maps between them" (were by total, I mean the map that satisfies this pullback condition) Now, as all this is a theorem of $\sf{ETCS}$ , it is a theorem of $\sf{ETCG}$ that all the axioms (and theorems) of $\sf{ETCS}$ are satisfied by the category of "cogroup objects and total maps between them" in any model of $\sf{ETCG}$ , which gives you the desired bi-interpretability between $\sf{ETCS}$ and $\sf{ETCG}$ . Adding supplementary axioms to $\sf{ETCS}$ (like R) does not change anything. In fact, one has more than bi-interpretability: the two theories are equivalent in the sense that there is an equivalence between their models. But one has a lot better: Corollary: Given $T$ a model of $\sf{ETCS}$ , then $Grp(T)$ is a model of $\sf{ETCG}$ . Given $A$ a model of $\sf{ETCG}$ , then the category $CoGrp(A)^{total}$ of cogroups object and full map between them is a model of $\sf{ETCS}$ . Moreover these two constructions are inverse to each other up to equivalence of categories. Edit: this an answer to a question of Matt F. in the comment to give explicit example of how axioms and theorems of $\sf{ECTS}$ translate into $\sf{ECTG}$ . So in $\sf{ECTS}$ there is a theorem (maybe an axioms) that given a monomorphism $S \rightarrow T$ there exists an object $R$ such that $T \simeq S \coprod R$ . In $\sf{ECTG}$ this can be translated as: given $T$ a cogroup object and $S \rightarrow T$ a cogroup monomorphism* then there exists a co-group $R$ such that $T \simeq S * R$ as co-groups**. *: It is also a theorem of $\sf{ECTG}$ that a map between cogroup is a monomorphism of cogroup if and only if the underlying map of objects is a monomorphisms. Indeed that is something you can prove for the category of groups in $\sf{ECTS}$ so it holds in $\sf{ECTG}$ by definition. ** : We can prove in $\sf{ECTG}$ (either directly because this actually holds in any category, or proving it for group in $\sf{ECTS}$ ) that the coproduct of two co-group objects has a canonical co-group structure which makes it the coproduct in the category of co-groups. Edit 2: To clarify that the category of cogroup is defined purely in the categorical language: The coproduct in group is the free product $G * G$ and is definable by its usual universal property. A cogroup is then an object (here a group) equipped with a map $\mu: G \rightarrow G * G$ which is co-associative, that is $\mu \circ (\mu * Id_G) = \mu \circ (Id_G * \mu)$ , and counital (the co-unit has to be the unique map $G \rightarrow 1$ ), that is $(Id_G,0) \circ \mu = Id_G$ and $(0,Id_G) \circ \mu = Id_G$ , where $(f,g)$ denotes the map $G * G \rightarrow G$ which is $f$ on the first component and $g$ on the other component, as well as an inverse map $i:G \rightarrow G$ such that $(Id_G ,i ) \circ \mu = 0 $ .
Morphisms of co-groups are the map $f:G \rightarrow H$ that are compatible with all these structures, so mostly such that $ (f * f) \circ \mu_H = \mu_G \circ f $ . If you have doubt related to the "choice" of the object $G * G$ (which is only defined up to unique isomorphisms) a way to lift them is to define "a co-group object" as a triple of object $G,G *G,G * G *G$ with appropriate map between them satisfying a bunch of confition (includings the universal property) and morphisms of co-group as triple of maps satisfying all the expected conditions. This gives an equivalent category.
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|
352,720 |
I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$ -dimensional Euclidean vector space. Then there exists another unit vector $x$ such that $$\sum_{i=1}^n |( x,u_i)|\geq \sqrt{n}.$$ I am looking for a reference for this conjecture. Also I will be happy to know what is known about it.
|
That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps: 1) Consider all sums $\sum_j \varepsilon_i u_i$ with $\varepsilon_i=\pm 1$ and choose the longest one. Replacing some $u_j$ with $-u_j$ if necessary, we can assume WLOG that it is $y=\sum_i u_i$ . Comparing $y$ with $y-2u_i$ (a single sign flip) we get $$
\|y\|^2\ge \|y-2u_i\|^2=\|y\|^2-4\langle y,u_i\rangle+4\|u_i\|^2
$$ whence $\langle y,u_i\rangle\ge 1$ for all $i$ . (That part is the main step in the solution of the plank problem). 2) Now we have $\|y\|^2=\sum_i\langle y,u_i\rangle\ge n$ , so for $x=\frac y{\|y\|}$ , we get $$
\sum_i\langle x,u_i\rangle=\sqrt{\sum_i\langle y,u_i\rangle}\ge \sqrt n
$$ The End :-)
|
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"https://mathoverflow.net/questions/352720",
"https://mathoverflow.net",
"https://mathoverflow.net/users/16183/"
]
}
|
353,508 |
I would like to know as curiosity how the editorial board or editors* of a mathematical journal evaluate the quality, let's say in colloquial words the importance, of papers or articles. Question. I would like to know how is evaluate the quality of an article submitted in a journal. Are there criteria to evaluate it? Many thanks. I think that it must be a difficult task to evaluate the quality of a mathematical paper due how is abstract (the high level of abstraction of research in mathematics) the work of professional mathematicians. Are there criteria to evaluate it, or is it just the experience, knowledges and good work of the people working as publishers? I'm curious about it but I think that this is an interesting and potentially useful post for other. As soon as I can I should accept an answer. If there is suitable references in the literature about how they do this work, feel free to refer the literature, answering this question as a reference request, and I try to search and read it from the literature. *I don't know what is the role of each person working in the edition of a mathematical journal.
|
Assume we are talking about a good journal with a large editorial board representing a wide scope of mathematical interests. I will describe both the role of the editors and the role of the referees. This is my personal viewpoint and others might have different opinion/experience. The role of the editors. Good journals can accept only about 20% of submitted papers. This is not easy to reject 80% of papers and it often results in rejecting really good papers. Everyone understands that. The procedure of evaluating the papers by the editors is more or less as follows: The editors have many years of research experience and (hopefully) developed a good mathematical taste. If the paper is close to the research interest of the editor then he or she can relatively easily identify the papers that are not particularly interesting. The reason for not being interesting can be based (for example) on the following criterion: The result is not well motivated. It is very technical and follows more or less standard arguments. The authors simply take a known result, and prove a new result by slightly modifying the given assumptions. Usually it means that they make the statement more complicated and in a sense more general. Often, they neither have interesting examples supporting such generalizations nor indication of possible applications. Unfortunately, most of the papers fell into this category. If the editor is sure that this is the case, then he or she rejects the paper without sending it to a referee. Then the authors usually get a rejection notice similar to this one: We regret that we cannot consider it, in part because at present
we have a large backlog of excellent articles awaiting publication.
We are thus forced to return articles that might otherwise be considered. If the editor is not sure about the quality of the paper, then they ask an expert (or several experts) for a quick opinion: I wonder if you could make a quick, informal assessment of it. Are the results strong enough to warrant sending the article to a referee? Because of our backlog, we like to send to referees only articles that appear to be of very good to outstanding quality. In that case the expert evaluating the paper is not asked to check all the proofs but to make a judgement based on the criterion explained above. This is an easy task for an expert. If the expert writes a negative opinion, then the authors receive a rejection notice often phrased the way as the rejection notice listed above. For top journals all experts have to write a positive opinion before the paper is sent to referees. If the experts' opinion is positive, then the paper is sent to a referee or to many referees. The most extreme case that I know of was a panel of 12 referees who took several years to evaluate the paper (this was when Thomas Hales proved the famous Kepler conjecture). For top quality journals all referees must write positive reports before the paper is accepted (once I received 6 reports, 4 positive and 2 not so positive and the paper was rejected). Let me also add that the editorial boards are structured basically in two different ways. (1) The authors are asked to choose an editor from the list of editors and submit the paper directly to them. Then the editor who receives the paper handles the submission process according to the rules explained above. (2) The authors submit the paper to the main editor or just to the journal and then the main editor either rejects the paper by themselves or he/she sends it to one of the editors from the editorial board and that editor applies the rules listed above. Of course some of the journals might have a slightly different approach than the one explained here. There is no a canonical solution and what I wrote is a somewhat a simplified version of the process that is applied in reality. The role of referees. A paper passed through an initial screening and it was sent to a referee. This is the most unpleasant part of the process. A referee spends a lot of time to read the paper, they are not paid for this job and since their work is anonymous, they do not get any recognition for what they do. What is the referee required to do? First of all, the referee has to assess originality of the results and whether the results are interesting enough. This part is the same as the one in the initial screening when the paper is sent to an expert for a quick opinion. Secondly, the referee is required to read the paper and check details. Let's be clear about that. Unless the paper is directly related to the research of the referee and he or she really wants to understand the details, there is no way the referee can check all details. Since I cannot speak for other people, let me say what I do in this situation.
My answer will only be a simplified version of the real process of the refereeing a paper, just a main idea of what I do. I go through the whole paper (or most of the paper) to have a good idea of what it is all about, to see a big picture not only of the meaning of the theorems, but also a big picture of the techniques used in the proof. Then I check carefully details of many/some arguments while for other arguments I briefly skim over. If the argument seems reasonable and believable to me I do not bother checking it very carefully. If all details that I check are correct and if all other arguments seem reasonable I am content. In this case, if I like the statement of the main result, I accept the paper. If however, some arguments seem fishy to me, then I check them carefully. This is a point where often I ask the authors for further clarifications. If I really cannot pass through the paper, because I think it has mistakes or if it is written in an unreadable way, I often reject the paper.
The biggest problem is when I am convinced that the result proved in a paper is of an outstanding quality, but the paper is very difficult and for that reason not very easy to read. Then, hmm... Then, it is not easy and I often struggle with making a right decision.
|
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"https://mathoverflow.net/users/142929/"
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|
353,682 |
Consider the following two-player pebble game. We have finitely
many stones on a finite linear track of squares. We take turns, and
the allowed moves are: move any one stone one square to the left, if that square is empty, or remove any one stone, or remove any two adjacent stones. Whoever takes the last stone wins. Question. What is the winning strategy? And which are the
winning positions? The game will clearly end always in finitely many moves, and so by
the fundamental theorem of finite games, one of the players will
have a winning strategy. So of course, I know that there is a
computable winning strategy by computing with the game tree, and we have a computable algorithm to answer any instance of the question. What I am hoping for is that there will be a simple-to-describe winning strategy. This is what I know so far: Theorem. It is a winning move to give your opponent a position
with an even number of stones, such that the stones in each successive pair stand
at even distance apart. By even-distance, I mean that there are an odd number of empty
squares between, so adjacent stones count as distance one, hence
odd. Also, I am only concerned with the even distance requirement within each successive pairs, not between the pairs. For example, it
is winning to give your opponent a position with stones at ..O...OO.O....O.....O........... We have distance 4 in the left-most pair, distance 2 in the next pair, distance 6 in the third pair, ignoring the distances in front and between the pairs. Proof. I claim that if you give your opponent a position like
that, then he or she cannot give you back a position like that, and
furthermore, you can give a position like that back again. If your
opponent removes a stone, then you can remove the other one in that
pair. If your opponent moves the lead stone on a pair, then you can
move the trailing stone. If your opponent moves the trailing stone
on a pair, then either you can move it again, unless that pair is
now adjacent, in which case you can remove both. And if your
opponent removes two adjacent stones, then they must have been from
different pairs (since adjacent is not even distance), which would
cause the new spacing to be the former odd number plus another odd
number plus 2, so an even number of empty squares between, and so
you can move the trailing end stone up one square to make an odd
number of empty squares between and hence an even distance between
the new endpoints. Thus, you can maintain this even-distance
property, and your opponent cannot attain it; since the winning
move is moving to the empty position, which has all even distances,
you will win. $\Box$ What I wonder is whether there is a similarly easy to describe
strategy that solves the general game.
|
The positions which are a win for the second player are those with: an even number of pebbles in odd-numbered squares, and an even number of pebbles in even-numbered squares. Indeed, from a position in this set $P$ , any move will be to a position not in that set, whereas from a position not in that set one can always make a move to a position in that set (if only the number of pebbles in the odd-numbered squares is odd, you can remove one, similarly for the even-numbered, and if both are odd, choose any pebble — that is not at the very leftmost square — and either more it to the left if that square is unoccupied or remove it along with the one to the left if it is). So, systematically moving to a position in $P$ (as described in the parenthesis in the previous paragraph) provides a winning strategy provided one starts with a position not in $P$ .
|
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|
354,115 |
I'd like to solve a differential equation $$ f^2(x) f''(x)=-x $$ where $f(x)$ is defined on $[0,1]$ and has a boundary condition $f(0)=f(1)=0$ . I somehow found out that the solution is fairly close to $f(x) = x^{1/3} \phi^{2/3}(\Phi^{-1}(1-x))$ where $\phi$ and $\Phi$ are pdf and cdf of a standard normal distribution, but it fails to solve the differential equation exactly. Thank for all comments! Based on the solution structure of Emden–Fowler Equation, I was able to identify the values of constants that satisfy the boundary conditions. The followings are the details: Define \begin{equation}
Z_R(\tau) \triangleq \sqrt{3} J_{1/3}(\tau) - Y_{1/3}(\tau)
, \quad
Z_L(\tau) \triangleq - \frac{2}{\pi} K_{1/3}(\tau)
\end{equation} where $J, Y, K$ are Bessel functions.
Further define \begin{equation}
\bar{\tau} \triangleq \inf\{ \tau > 0; Z_R(\tau) = 0 \} \approx 2.3834
, \quad
a \triangleq \frac{1}{ \bar{\tau}^{4/3} Z_R'(\bar{\tau})^2 } \approx 0.2910
, \quad
b \triangleq a \left( \frac{9}{2} \right)^{1/3} \approx 0.1763.
\end{equation} Then, the solution curve $\{ (x, f(x)) \}_{x \in [0,1]}$ is characterized by \begin{equation}
\left\{ \left( x_R(\tau), y_R(\tau) \right) \right\}_{\tau \in [0, \bar{\tau}]} \bigcup \left\{ \left( x_L(\tau), y_L(\tau) \right) \right\}_{\tau \in [0, \infty]}
\end{equation} where \begin{equation}
x_R(\tau) \triangleq a \tau^{-2/3}\left[ \left( \tau Z_R'(\tau) + \frac{1}{3} Z_R(\tau) \right)^2 + \tau^2 Z_R(\tau)^2 \right]
, \quad
y_R(\tau) \triangleq b \tau^{2/3} Z_R(\tau)^2.
\end{equation} \begin{equation}
x_L(\tau) \triangleq a \tau^{-2/3}\left[ \left( \tau Z_L'(\tau) + \frac{1}{3} Z_L(\tau) \right)^2 - \tau^2 Z_L(\tau)^2 \right]
, \quad
y_L(\tau) \triangleq b \tau^{2/3} Z_L(\tau)^2.
\end{equation} In addition to this analytic solution, I also obtained a numerical solution by repeatedly computing $$ f_{k+1}(x) \gets \left[ \left( f_k(x-2h) + f_k(x+2h) \right) + 4 \left(f_k(x-h)+f_k(x+h)\right) + \frac{8 x h^2}{f_k^2(x)} \right] \big/ 10 $$ on the grid $x \in \{2h,3h,\ldots,1-3h,1-2h\}$ for small $h$ with an initialization $f_0(x) \triangleq 0.5(1-(1-2x)^2)$ . The following figure shows these solutions:
|
Surprisingly, this case of the Emden-Fowler equation is explicitly solvable:
see formula (2.3.27) in A. Polyanin and V. Zaitsev, Handbook of exact solutions
of ordinary differential equations, Chapman & Hill, 2003. I copy the formula, without verifying it. Let $$Z=C_1J_{1/3}(\tau)+C_2Y_{1/3}(\tau),$$ or $$Z=C_1I_{1/3}(\tau)+C_2K_{1/3}(\tau),$$ where $J,Y$ are Bessel and $I$ , $K$ are modified Bessel functions.
Then $$x=a\tau^{-2/3}[(\tau Z^\prime+(1/3)Z)^2\pm\tau^2Z^2],\quad y=b\tau^{2/3}Z^2$$ satisfy $d^2y/dx^2=Axy^{-2}$ with $A=-(9/2)(b/a)^3.$ For the $+$ sign in $\pm$ take the first formula for $Z$ , and for the $-$ the second one. Remark. Emden-Fowler equation appears for the first time in the famous book by R. Emden, Gaskugeln (1907) and since then frequently arises in the study of stars and black holes.
|
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|
354,327 |
In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples: GroupNames: https://people.maths.bris.ac.uk/~matyd/GroupNames Finite groups of order ≤500, group names, extensions, presentations,
properties and character tables. Atlas of Finite Group Representations: http://brauer.maths.qmul.ac.uk/Atlas/v3/ This ATLAS of Group Representations has been prepared by Robert
Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker,
Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel
Abbott (in reverse alphabetical order, because I'm fed up with always
being last!). It currently contains information (including 5215
representations) on about 716 groups [mainly finite simple groups or almost simple]. Atlas of subgroup lattice of finite almost simple groups: http://homepages.ulb.ac.be/~dleemans/atlaslat/ This atlas contains all subgroup lattices of almost simple groups $G$ such that $S≤G≤Aut(S)$ and $S$ is a simple group of order less than 1
million appearing in the Atlas of Finite Groups by Conway et al. Some
simple groups and almost simple groups or order larger than 1 million
have also been included, but not in a systematic way. The L-functions and Modular Forms Database: https://www.lmfdb.org/ Welcome to the LMFDB, the database of L-functions, modular forms, and
related objects. These pages are intended to be a modern handbook
including tables, formulas, links, and references for L-functions and
their underlying objects [like field extensions and polynomial Galois groups]. The Inverse Symbolic Calculator: https://isc.carma.newcastle.edu.au/ The Inverse Symbolic Calculator (ISC) uses a combination of lookup
tables and integer relation algorithms in order to associate a closed
form representation with a user-defined, truncated decimal expansion
(written as a floating point expression). The lookup tables include a
substantial data set compiled by S. Plouffe both before and during his
period as an employee at CECM. If you know such a website on any area of mathematics, please put it as an answer (with a short description).
|
This catalogue of mathematical datasets could be of some interest to you - at least some of the entries are atlas-like websites. It includes several of the websites mentioned above, and I'm slowly adding more to it.
|
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|
354,402 |
If a mathematician specializes in a popular research area, then there are many job positions available, but at the same time, many competitors who are willing to get such job positions. For an esoteric research area, there are few competitors and job positions. There are very often pros and cons of such research areas. What are some pros and cons of specializing in esoteric research areas that many people may not know? Maybe it is a little hard to answer this question in full generality since circumstances vary. Hence, I especially want to listen to examples, personal experiences, and maybe urban legends. Now, if you are interested, let me tell my personal circumstance to give some context to this question. I am a student from outside of North America who just graduated from my undergraduate institution. Since I decided to study abroad, I applied to several North American universities last December and was admitted to some of them. Now I am wavering between two universities. Denote those universities X and Y. Among the specific research areas available at X (resp. Y), I am interested in two of them, say, A and B (resp. C and D).
I did some searching in those research areas, and I found out that there are many people who are researching C and D and some of them are in my country. However, there are only a few people interested in A and B and none of them are in my country. Based on my search, I think (with a little exaggeration) there are about 3 universities in the world where a graduate student can specialize in A. B is not as esoteric as A, but still, it seems there are not many people working on B. However, I think C and D are quite major research areas in my field of study.
[In this question, I used 'field of areas' as something in First-level areas of Mathematics Subject Classification and 'specific research areas' as something in Second or Third-level areas of it. ] At first glance, I prefer X over Y, because I was very interested in A. Also, this is partially (maybe totally) because X is considered more `prestigious’ than Y. However, I’m a little nervous about specializing in esoteric areas such as A and B, because of the number of job positions and this kind of problem .
Anyway, I think it is not a bad idea to ask a question at MO and to listen to the pros and cons of specializing in esoteric fields to make a better decision. Any personal stories or examples will be really helpful. Thanks in advance.
|
I think this is an important question, and something that is not talked about often enough. Just like we don't explain enough to undergraduates that the choice of their major has profound consequences for their career opportunities, we don't tell young mathematics researchers enough about the consequences of their choice of research area. (What's the analogue of the well-known joke about the English major serving food in your local restaurant?) Instead of answering your question, let me give a small analysis of research dynamics, drawn from my own (limited) experience. The below is not supposed to be exhaustive, but I'm hoping to show that there are underlying mechanisms that cause certain areas to be more popular than others. I'm hoping someone else will post an answer that gets closer to the core of your question, but I hope that this is at least somewhat useful. What makes interesting research? This may seem a little random when you're young, but there are actually some underlying mechanisms that are somewhat understandable. (Although admittedly I still haven't figured out a full answer to this question.) First of all, just like universities and journals, top mathematics research gains some prestige from age . This means that solutions to old conjectures are valuable, and a new theory is appreciated more when it says something about the mathematics that existed before. But there is a caveat: some areas are considered 'easy' or 'belonging to the past'. There are almost certainly areas that legitimately produce interesting mathematics but are not fashionable because they are considered 'too old'. Secondly, mathematicians like powerful ideas . If you have some new technique that seems applicable in many situations, this is valued more than an ad hoc argument. If you have a powerful machine that people can use with their eyes closed, that's going to be cited a lot. Thirdly, we prefer clean theorems . If you have a theorem without too many technical assumptions, then it's much easier to explain and motivate, and much easier for other people to use. Moreover, connections to other areas are important too: if you prove amazing theorems on a (mathematical) island, then that's not as interesting as when you prove something that relates to other work. A lot of the above seems driven by the fact that the mathematicians who evaluate your work do so on the basis of their own interests. This seems reasonable given that everyone's own expertise is limited, and their judgement biased towards their own work. What makes fashionable research? Areas of mathematics fall in and out of fashion, in part related to the criteria above. For example, if there is a new idea that seems to have potential, this can drive a research group until they feel the idea has been fully exploited. Especially new connections between different areas can spur a lot of activity, because now you have two communities working out these ideas. When this happens, you can suddenly see a lot of researchers thinking about very similar things, and this is both a blessing and a curse. Indeed, it makes it easier to explain your results and to find collaborators, but it also means you have to keep up with progress all the time and risk getting 'scooped'. I'm not entirely sure what the mechanisms for topics falling out of fashion are. One thing that can happen is that the main problem gets solved; for example I have the impression that the classification of finite simple groups ended an era of high activity in the area. But this doesn't always happen, because there could be other questions; for example the techniques for proving Fermat's Last Theorem are still very much alive in the Langlands programme. I imagine it's also possible for a research area to dry out without the main problem being solved, although I haven't been around long enough to know an example offhand. Note that timing is key: what is fashionable now may not be fashionable in the future, and sometimes researchers forecast the demise of their field. While it's hard to predict the future, it's probably a good idea to listen (critically) when people tell you something like this. Finally, there are also some political factor involved: if your area is well-represented in the editorial boards of top journals, then you're going to have an easier time publishing in those journals. As Henriksen points out in There are too many B.A.D. mathematicians , this does not always happen for the right reasons. So what to do next? Many research groups are driven by a handful of famous questions and a much larger collection of more technical questions. You could ask the research groups you're considering what the main goal in their work is, and then evaluate this by the criteria above to see how easy a time you would have selling your research. Usually the advice is to talk to some graduate students in the group to hear about their experiences, but in your case it might also be worthwhile to try to approach some postdocs¹ or senior researchers in the area. Another useful metric to look at is job placement: if an advisor has supervised many graduate students who did well on the academic market (short AND long term), this is useful information. Finally, I should remark that these things matter much more when you're young, because the mathematics job market is highly catered towards the researchers who are lucky enough to have early success. Once you get a tenure-track of tenured job, it becomes easier to switch areas again. This is especially true if you work in an area with connections to other areas, although there are certainly examples of famous researchers switching to completely unrelated fields. ¹Be aware that postdocs are in the most precarious stage of their career, so their answers will be a bit more cynical than those of (young and naive) grad students or (successful) tenure-track or tenured faculty. This can be useful for getting more 'real' advice, but be sure to recognise the context in which it is given.
|
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"source": [
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"https://mathoverflow.net",
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|
354,542 |
I am about to (hopefully!) begin my PhD (in Europe) and I have a question: how did you learn so much mathematics? Allow me to explain. I am training to be a number theorist and I have only some read Davenport's Multiplicative Number Theory and parts of Vaughan's book on the circle method. I have briefly seen some varieties from Fulton's algebraic curves and I may have read parts of homotopy and homology and differential geometry of smooth manifolds at the level of Hatcher and Lee. Yet, it seems that I am hopelessly ignorant of elliptic curves, modular forms and algebraic number theory. For example, if I were to try reading Deligne's proof of Weil's conjecture or Tate's thesis, it seems that I would have to do significant amounts of reading. When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about. That begs the questions: How much reading outside should I be doing outside my "area"? Is it a good idea to just focus narrowly on my thesis problem at this stage or is it more usual to be working on multiple problems at the same time? How and how often do you end up learning new areas? Sorry if the question is too vague: I just want to have a sense of how to go about being a good mathematician. Also, part of the reason I am asking this question is that when I go to seminars, I understand so little and I see some of my professors ask questions of the speakers even if they don't work in the same area.
|
The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally. First, there is no such thing as training to be a number theorist. There are many different kinds of number theorists, and very few of him are comfortable with all four of the works you mention here (Davenport, Vaughan, Deligne-Weil II, Tate' thesis). Very few analytic number theorists understand the proof of Weil II (though a lot more of them know something about how to use it). Very few algebraic number theorists are comfortable with all the standard argument in multiplicative number theory and the circle method (though a lot more know the key results about $L$ -functions). Of course the division into analytic and algebraic is already too coarse and simple. What you have is a lot of different number theorists with distinct but overlapping areas of knowledge. Analytic number theory specifically is one of the areas of Math famous for requiring relatively little knowledge (at least, when compared to other areas of Math ). If you like the stuff you read in Davenport and Vaughan, you're in luck! You may be a lot closer to the frontiers of research than you think. As for how exactly to get there, I agree with Timothy Chow that your adviser is the best person to figure this out. As to this phenomenon: When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about. Their knowledge may be less than you think. Or more precisely they know a broad overview of what the idea in a given field are and how they are used, but not the details. This might match the questions that someone with less experience in the area would ask them, but not be sufficient to write a good research paper in that field. However, it is by no means a parlor trick. This type of knowledge is very important because it suggests what research areas might be relevant for a given problem and thus who to talk to, what to read, etc. But it's not obtained from reading books! Probably the best way to attain this level of knowledge is attending seminar talks (and listening carefully, not being afraid to ask stupid questions, thinking about what the speaker is saying during and after the talk...) I think in general, a recipe for success on a particular problem or research sub-sub-area is to know (1) everything, or as much as possible, about the techniques that have been used to attack this problem before and (2) one relevant thing that hasn't been used to attack the problem before. The point being that you only need one new idea to make progress, but you will likely have to combine it with all or many of the previous ideas. So if you know which problem, or type of problem, you want to work on, you should learn diligently the topics of obvious relevance to that problem. For topics of unclear relevance, you do not need to learn everything to their fullest extent, as long as you do not completely abandon them - again, you only really need one new idea. Even (2) is not strictly necessary - plenty of progress has been made by applying the existing methods with a more clever strategy. But if you have a natural inclination to read and learn everything, you will probably find success as a mathematician by knowing at least a few things that your competitors don't. Focus on what seems relevant to your areas of greatest focus and ideally what seems fun and interesting as well. But there's no need to drive yourself insane.
|
{
"source": [
"https://mathoverflow.net/questions/354542",
"https://mathoverflow.net",
"https://mathoverflow.net/users/146669/"
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|
354,608 |
Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$ , $$
\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$ I can prove the result if they have a density, of if they have atoms, but not in the general case.
|
If $\epsilon \geqslant \tfrac{1}{n}$ , then $$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \sum_{i=1}^n \mathbb{P}(X, Y \in [\tfrac{i-1}{n}, \tfrac{i}{n}]) = \sum_{i=1}^n (\mathbb{P}(X \in [\tfrac{i-1}{n}, \tfrac{i}{n}]))^2 . $$ It follows that $$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \frac{1}{n} \biggl(\sum_{i=1}^n \mathbb{P}(X \in [\tfrac{i-1}{n}, \tfrac{i}{n}])\biggr)^2 = \frac{1}{n} \, . $$ If we choose $n$ so that additionally $\epsilon < \frac{1}{n-1}$ , then we obtain $$ \mathbb{P}(|X-Y|<\epsilon) > \frac{\epsilon}{\epsilon + 1} \, . $$ This leads to the desired result with $c = 1$ .
|
{
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"https://mathoverflow.net/questions/354608",
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|
354,655 |
Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$ ? A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \mathrm d t\right|\leqslant 2$ ? on Math StackExchange. However, this bound does not appear to be sharp so I was wondering how to find the maxima/minima of $$f(x)=e^x\int_x^{x+1}\sin(e^t) \,\mathrm d t$$ or at least how to prove $-1.4\le f(x)\le 1.4$ . Some observations, using the substitution $y=e^t$ : $$f(x)=e^x \int_{e^x}^{e^{x+1}} \frac{\sin(y)}y\,\mathrm dy=g(e^x),$$ where I have defined $$g(z)=z \int_z^{e z} \frac{\sin(y)}y\,\mathrm dy = z (\operatorname{Si}(e z)-\operatorname{Si}(z)).$$ ( $\operatorname{Si}$ is the Sine integral .) So the question reduces to: What are the maxima/minima of $g(z)$ for $z\geq 0$ ? Using the series of $\mathrm{Si}(z)$ , we get $$g(z)=\sum_{k=1}^\infty (-1)^{k-1} \frac{z^{2k}(e^{2k-1}-1)}{(2k-1)!\cdot(2k-1)}$$ and here is a plot of $g(z)$ : Also, notice that $g$ is analytic and $g'(z)=\sin (e z)-\sin (z)+\text{Si}(e z)-\text{Si}(z)$ which might help for the search of critical points (although I don't think that $g'(z)=0$ has closed form solutions).
|
Integrate by parts: \begin{align}
\int_x^{x+1}\sin(e^t)dt
& =\int_x^{x+1}e^{-t}d(-\cos(e^t)) \\
& =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-\int_x^{x+1}e^{-t}\cos e^{t}dt\\
& =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-\int_x^{x+1}e^{-2t}d\sin e^{t}\\
& =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-e^{-2(x+1)}\sin e^{x+1}\\
& \hphantom{={}}+e^{-2x}\sin e^x+2\int_x^{x+1}e^{-2t}\sin e^tdt.\end{align} From here we see that $e^x \int_x^{x+1}\sin(e^t)dt$ is bounded by $1+1/e+O(e^{-x})$ and $1+1/e\approx 1.368$ can not be improved, since both $\cos e^x$ and $-\cos e^{x+1}$ may be almost equal to 1:
if $e^x=2\pi n$ for large integer $n$ , then $e^{x+1}=2\pi e n$ , we want this to be close to $\pi+2\pi k$ , i.e., we want $en$ to be close to $\frac12+k$ . This is possible since $e$ is irrational. Moreover, $e$ is so special number that you may find explicit $n$ for which $en$ is nearly half-integer: $n=m!/2$ for large even $m$ works. Indeed, $e=\sum_{i=0}^{m-1}1/i!+1/m!+o(1/m!)$ yields $em!/2=\text{integer}+1/2+\text{small}$ .
|
{
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|
355,759 |
I was told that if we have an equivalence of categories $F : \mathcal{A} \rightarrow \mathcal{B}$ with $\mathcal{A}$ abelian, then it is not necessarily true that $\mathcal{B}$ is also abelian. I would like to know if there are nice examples of an abelian category $\mathcal{A}$ which is equivalent to a non-abelian category $\mathcal{B}$ . Furthermore, are there any conditions over $F$ or $\mathcal{B}$ so that we have " $F : \mathcal{A} \rightarrow \mathcal{B}$ is an equivalence and $\mathcal{A}$ is abelian implies $\mathcal{B}$ is abelian"?
|
What you were told is wrong, for we have the following: Proposition. If two categories are equivalent and one of them is abelian, then so is the other. A proof (and some related results) can be found in Satz 16.2.4 in H. Schubert, Kategorien II, Springer, 1970 (likewise in the English version https://www.amazon.com/Categories-Horst-Schubert/dp/3642653669 , under the same numbering).
|
{
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"https://mathoverflow.net/questions/355759",
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"https://mathoverflow.net/users/-1/"
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|
356,049 |
This is a follow-up on the following question . Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition). What is an example of a topological space $X$ with $X\not\cong \mathbb{R}$ but the monoids $\text{End}(X)$ and $\text{End}(\mathbb{R})$ are isomorphic?
|
No such space exists. We actually get the stronger statement that every isomorphism $\operatorname{End}(X) \stackrel\sim\to \operatorname{End}(\mathbf R)$ is induced by an isomorphism $X \stackrel\sim\to \mathbf R$ (unique by Observation 1 below). In contrast, in Emil Jeřábek's beautiful construction in this parallel post there is an 'outer automorphism' $\operatorname{End}(X) \stackrel\sim\to \operatorname{End}(X)$ that does not come from an automorphism $X \stackrel\sim\to X$ of topological spaces (it comes from an anti-automorphism of ordered sets). I will use the substantial progress by YCor and Johannes Hahn, summarised as follows: Observation 1 (YCor). For every topological space $X$ , the map $X \to \operatorname{End}(X)$ taking $x$ to the constant function $f_x$ with value $x$ identifies $X$ with the set of left absorbing¹ elements of $\operatorname{End}(X)$ . In particular, an isomorphism of monoids $\operatorname{End}(X) \stackrel\sim\to \operatorname{End}(Y)$ induces a bijection $U(X) \stackrel\sim\to U(Y)$ on the underlying sets. Observation 2 (Johannes Hahn). If $\operatorname{End}(X) \cong \operatorname{End}(\mathbf R)$ , then $X$ is $T_1$ . Since the closed subsets of $\mathbf R$ are exactly the sets of the form $f^{-1}(x)$ for $x \in \mathbf R$ , we conclude that these are closed in $X$ as well, so the bijection $X \to \mathbf R$ of Observation 1 is continuous. (The asymmetry is because we used specific knowledge about $\mathbf R$ that we do not have about $X$ .) To conclude, we prove the following lemma. Lemma. Let $\mathcal T$ be the standard topology on $\mathbf R$ , and let $\mathcal T' \supsetneq \mathcal T$ be a strictly finer topology. If all continuous maps $f \colon \mathbf R \to \mathbf R$ for $\mathcal T$ are continuous for $\mathcal T'$ , then $\mathcal T'$ is the discrete topology. Note that Observation 2 and the assumption $\operatorname{End}(X) \cong \operatorname{End}(\mathbf R)$ imply the hypotheses of the lemma, so we conclude that either $X = \mathbf R$ or $X = \mathbf R^{\operatorname{disc}}$ . The latter is clearly impossible as it has many more continuous self-maps. Proof of Lemma. Let $U \subseteq \mathbf R$ be an open set for $\mathcal T'$ which is not open for $\mathcal T$ . Then there exists a point $x \in U$ such that for all $n \in \mathbf N$ there exists $x_n \in \mathbf R$ with $|x - x_n| \leq 2^{-n}$ and $x_n \not\in U$ . Without loss of generality, infinitely many $x_n$ are greater than $x$ , and we can throw out the ones that aren't (shifting all the labels, so that $x_0 > x_1 > \ldots > x$ ). Up to an automorphism of $\mathbf R$ , we can assume $x = 0$ and $x_n = 2^{-n}$ for all $n \in \mathbf N$ . Taking the union of $U$ with the usual opens $(-\infty,0)$ , $(1,\infty)$ , and $(2^{-n},2^{-n+1})$ for all $n \in \mathbf N$ shows that $$Z = \big\{1,\tfrac{1}{2},\tfrac{1}{4},\ldots\big\}$$ is closed for $\mathcal T'$ . Consider the continuous function \begin{align*}
f \colon \mathbf R &\to \mathbf R\\
x &\mapsto \begin{cases}0, & x \leq 0,\\ x, & x \geq 1, \\ 2^nx, & x \in \big(2^{-2n},2^{-2n+1}\big], \\ 2^{-n}, & x \in \big(2^{-2n-1},2^{-2n}\big].\end{cases}
\end{align*} Then $f^{-1}(Z)$ is the countable union of closed intervals $$Z' = \bigcup_{n \in \mathbf N} \big[2^{-2n-1},2^{-2n}\big] = \big[\tfrac{1}{2},1\big] \cup \big[\tfrac{1}{8},\tfrac{1}{4}\big] \cup \ldots.$$ By the assumption of the lemma, both $Z'$ and $2Z'$ are closed in $\mathcal T'$ , hence so is the union $$Z'' = Z' \cup 2Z' \cup [2,\infty) = (0,\infty),$$ and finally so is $Z'' \cup (-Z'') = \mathbf R\setminus 0$ . Thus $0$ is open in $\mathcal T'$ , hence so is every point, so $\mathcal T'$ is the discrete topology. $\square$ ¹Elements $f$ such that $fg = f$ for all $g$ . (I would probably have called this right absorbing!)
|
{
"source": [
"https://mathoverflow.net/questions/356049",
"https://mathoverflow.net",
"https://mathoverflow.net/users/8628/"
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|
356,188 |
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$ , say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$ , and write $[x,z]$ for the set of points between $x$ and $z$ . Obviously, we have $x,z\in[x,z]$ ; $[x,z]=[z,x]$ ; $y \in [x,z]$ implies $[x,y] \subseteq [x,z]$ ; $w,y \in [x,z]$ implies: $w\in [x,y]$ iff $y \in [w,z]$ . My question: Has the family of objects with such an axiomatic “interval structure” $[\bullet,\bullet]:X \times X \to \mathcal{P}(X)$ satisfying approximately the above conditions been studied somewhere?
|
There is a wide body of work on this in connection with the classic De Bruijn–Erdős theorem . De Bruijn–Erdős Theorem. Every set of $n$ points in the
plane (not all lying on the same line) determine at least $n$ lines. There is a beautiful conjecture of Chen and Chvátal that the De Bruijn–Erdős theorem actually holds in every metric space, where lines are defined using the notion of betweenness that you describe. That is, given a metric space $M$ and two points $a,b \in M$ , the line determined by $a$ and $b$ is the set of points $c$ such that $c$ is between $a$ and $b$ , or $a$ is between $c$ and $b$ , or $b$ is between $a$ and $c$ . Note that this definition reduces to the usual notion of lines if we use the Euclidean distance. Chen-Chvátal Conjecture . Every set of $n$ points in a metric space (not all of which lie on the same line) determine at least $n$ lines. If you Google 'Chen-Chvátal Conjecture' you will find many results, including some that focus on the combinatorial aspects of betweenness.
|
{
"source": [
"https://mathoverflow.net/questions/356188",
"https://mathoverflow.net",
"https://mathoverflow.net/users/148575/"
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|
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