Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier... Unlike Atiyah's paper, Hironaka's paper does not have a ...
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What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let ...
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34 votes
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Grothendieck's "List of classes of structures"

In Lawvere's article Comments on the Development of Topos Theory, the author writes: Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
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"Three great cocycles" in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
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30 votes
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Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
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29 votes
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Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
28 votes
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the ...
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25 votes
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Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
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25 votes
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The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor (http://www.ams.org/journals/jams/2012-25-03/S0894-0347-...
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Reference request for educational material In source format, for blind accessibility purposes

Introduction I am a blind undergraduate studen in mathematics. I use screen reading software, which uses synthesized speech to read aloud the contents of the screen, to read and write math. Due to ...
24 votes
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709 views

How much of the plane is 4-colorable?

In 1981, Falconer proved that the measurable chromatic number of the plane is at least 5. That is, there are no measurable sets $A_1,A_2,A_3,A_4\subseteq\mathbb{R}^2$, each avoiding unit distances, ...
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Exotic 4-spheres and the Tate-Shafarevich Group

The title is a talk given by Sir M. Atiyah in a conference with the following abstract: I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
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23 votes
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Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
23 votes
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Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
23 votes
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771 views

Characteristic classes for $E_8$ bundles

$\DeclareMathOperator\B{B}\DeclareMathOperator\SU{SU}$Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow \SU(\mathbb C^{248})$ and form the ...
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What is the covering density of a very thin annulus? Is it $\frac{\pi\sqrt{51\sqrt{17}-107}}{16}$?

Take some very small $\epsilon>0$, and consider the annulus/ring given by the set $\{(r,\theta)\ |\ 1-\epsilon\le r\le1\}\subset \mathbb{R}^2$. We wish to place translated copies of this annulus ...
21 votes
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Does every 5-celled animal tile the plane?

An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) One may ...
21 votes
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Is the mapping class group of $\Bbb{CP}^n$ known?

In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...
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A mysterious paper of Stallings that was supposed to appear in the Annals

In Stallings's paper Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389. he briefly discusses how to prove "several generalizations" of Brown's ...
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Reference request for a proof of the two-square Theorem

One can show (see below for a sketch of a proof) that every odd prime number $p$ can be written in exactly $(p+1)/2$ different ways as $$p=a\cdot b+c\cdot d$$ with $a,b,c,d\in\mathbb N$ satisfying $\...
19 votes
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Cartan–Oka vanishing in one variable without $\overline{\partial}$?

This is a literature question, about possible proofs of some very basic results in complex analysis. Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
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univariate integer version of Hilbert's 17th problem

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
19 votes
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How is this group theoretic construct called?

Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be $$\psi(g,h) = |g|+|h|-|gh|$$ Then $\psi:G\times G \...
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19 votes
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How to approach the Mazur-Wiles paper on Iwasawa theory?

I would like to read and understand the Mazur-Wiles paper on Iwasawa theory: "Class Fields of Abelian Extensions of $\Bbb Q$". What would be the right way to approach this paper? Currently, my ...
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19 votes
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Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...
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19 votes
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The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
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Homotopy groups of spheres and differential forms

The only infinite homotopy groups of spheres are $\pi_n(\mathbb{S}^n)$ and $\pi_{4n-1}(\mathbb{S}^{2n})$. This is a well known result of Serre. In both cases the nontriviality of these groups can be ...
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Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then $$\frac{|\{n \le x : \...
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local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...
17 votes
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Who first noticed the duality for finite groups?

A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short ...
17 votes
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An integral in Gradshteyn and Ryzhik

Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
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Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as: Theorem 1.2 (MacPherson). Let $(X,S)...
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Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
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How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
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Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
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Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
17 votes
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Characterization of Fréchet-Urysohn spaces using sequential continuity at a point

A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$. $$x_n\to a \qquad \Rightarrow \qquad f(...
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Joyal's topos in which $[0,1]$ fails to be compact

Some time around 1977, André Joyal constructed a topos (actually a locale, i.e., a localic topos, necessarily non-spatial) in which the closed unit interval $[0,1]$ fails to be compact. There are ...
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Reference request: Milnor rank of spheres

Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
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Reference request for Grothendieck's work on "Integration with values in a topological group"

Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
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16 votes
1 answer
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The homology of the orbit space

Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper). Is there a way to understand the homology ...
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Average value of j-invariant at infinity

Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$: $$ \...
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What is the geometric intuition behind Wilf-Zeilberger theory?

This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
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16 votes
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An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$

This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
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Survey of known results on equivariant transversality

Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant ...
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Grothendieck 's question - any update?

This question is migrated from math.stackexchange. I ask because it is still unclear to me and I did not receive an answer. I was reading Barry Mazur's biography and come across this part: ...
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Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
15 votes
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How much smoothness does the tennis ball theorem need?

The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are ...
15 votes
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Extending Kahler metric from a neighborhood of a divisor to the whole manifold

Let $X$ be a smooth complex projective variety with an ample line bundle $L$, and let $D\subset X$ be a smooth divisor. Suppose in an analytic neighborhood $U$ of $D$ there is a Kahler form $\omega$ ...
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References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology

Let me ask several related questions on discretization of classical field theory: In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...

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