All Questions
5,076 questions with no upvoted or accepted answers
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Algebraic closure of a field in constructive mathematics
There is a short note by André Joyal called Les théorèmes de Chevalley-Tarski et remarque sur l'algèbre constructive (pp. 256-258). It is claimed there that there is a constructive version of the ...
- 4,459
13
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0
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797
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Borel-Weil-Bott, Langlands and Hitchin
Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$.
Given a dominant weight $\lambda$, one can construct a line ...
- 2,273
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883
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Arguments against Freiling's argument against Continuum Hypothesis
Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
13
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349
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Some $q-$analogues of $ \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}.$
Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} )$ and
let $ {{n}\brack{k}}_q$ denote a $q-$binomial coefficient.
I am interested in $q-$analogues of the identity $ \sum\...
- 5,622
13
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524
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Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
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476
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Singular cohomology of $BG$ and Borel cohomology of $G$
Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
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872
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Geometric meaning of the black hole horizon
It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...
- 16.5k
13
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282
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Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles
Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...
- 5,504
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680
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Singular chains generated by manifolds with corners --- does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
- 18.7k
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1k
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Source of a formula for tensor product multiplicities?
This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
- 52.9k
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615
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The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
- 3,184
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375
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Complex manifold which is algebraic away from codimension \ge 2
If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space?
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13
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591
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Generalization of Witten's computation of the volume of moduli space
Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.
There ...
- 18.7k
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943
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Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
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543
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Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
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493
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Proofs of Serre's theorem on simply-connected finite CW complexes
A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than ...
12
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558
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God's number for higher dimensional Rubik's cubes
In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...
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431
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Rational points of weighted projective spaces
[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.]
Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
- 19.6k
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556
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A measure of non-uniformity of a vector/probability distribution?
In the course of a research project about discrete probability distributions, my coauthors and I keep seeing some quantity appear, and I would like to understand whether it has been studied or has a ...
- 1,372
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452
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Books on exotic structures
The second half of the XX-th century has witnessed an explosion of results on the existence of smooth structures on topological manifolds. Following various sources in Wikipedia, a rough timeline goes ...
- 14.8k
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402
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
- 1,161
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195
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Where is it shown that homotopy sheaves form a higher stack?
Many references on infinity categories etc. advertise that one application is that it's an appropriate setting to glue (the appropriate replacement for) derived categories of sheaves.
What's the ...
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447
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Biased vs unbiased lax monoidal categories
There are two principal ways to define a monoidal category:
The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes ...
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12
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186
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Where can I find a copy of Bérard-Bergery's lecture notes on quaternionic manifolds?
In the 1970's, Bérard-Bergery proved certain results on quaternionic Kähler manifolds, some of which are explained in the book Einstein Manifolds by Besse. Several times, Besse's book references a set ...
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313
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For a Banach space $X$, when is $X$ homeomorphic to $X \setminus A$?
$\mathbb{R}^n\not\cong\mathbb{R}^n\setminus\{0\}$ are not homeomorphic is a triviality from Algebraic Topology. On the other hand, if $X$ is an infinite dimensional Banach space, then $X \cong X\...
- 43.2k
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265
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Galois groups of classical differential equations
I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following:
Is there a book or article devoted (either partially or completely) to ...
- 4,745
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729
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Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
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551
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Goodwillie's notes from MSRI Lecture Series
Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
12
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695
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"To operate the machine, it is not necessary to raise the bonnet."
The quotation in the title is attributed to Frank Adams and appears in several places:
In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not necessary ...
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424
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Reference for class of involutions containing longest element of finite Coxeter group?
Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
- 52.9k
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383
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Is the quotient map of the action of homeomorphisms on embeddings well-behaved?
It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the $C^\...
- 8,167
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279
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Entropy in elimination theory, or a brief remark by Gelfand-Kapranov-Zelevinsky
In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in $A$...
- 902
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747
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Cohomology and impossible figures
In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": http://upcommons.upc.edu/revistes/...
- 16.5k
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211
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Classes for which the Spectrum determines a Convex Shape
Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
- 2,222
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1k
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A conceptual proof of Jacobi's product formula for $\Delta$ ?
Let $\Delta$ be the unique normalized cusp form of weight 12 and level $1$. Then Jacobi's
wel-known formula states:
$$\Delta(z) = q \prod_{n=1}(1-q^n)^{24},$$
where $q=e^{2 i \pi z}$.
For a graduate ...
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456
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Enumeration of Standard Young Tableau of bounded height
First for some notation
$$ l(\lambda) = \text{ number of parts in a partition } \lambda \vdash n$$
$$ f_{\lambda} = \text{number of standard Young tableau of shape } \lambda\vdash n$$
The number $f_{...
- 771
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716
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Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
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851
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Compact Symplectic Fano (strongly monotone) manfiolds
What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?
We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that
$[c_1(...
- 28.9k
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552
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References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
- 12.8k
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346
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Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
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428
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Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
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315
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
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291
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Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
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374
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Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.4k
11
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389
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Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
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- 505
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540
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Outline of the unpublished proof of Erdős-Sós conjecture
In this post, it was mentioned that a long time ago, Ajtai, Kolmós, Simonovits, and Szemerédi announced a proof that for sufficiently large $k$, every $k$-vertex tree $T$ is a subgraph of every graph $...
- 3,499
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368
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Category theory book with lots of examples
A student of mine is writing her Bachelor's thesis in which she has to use several categorical concepts (co/equalizers, co/products, subobjects, (regular) monomorphisms, etc.) in a couple of different ...
11
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344
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Tauberian Theorem for 1-parameter groups of operators
The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
- 5,405
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248
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Almost isometric manifolds are diffeomorphic
I am looking for a reference to the following statement.
(It should be known --- I saw it before, don't remember where; search by keywords did not help.)
Let $f\colon M\to N$ be a homeomorphism ...
- 45k
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486
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Roadmap to homotopical group theory
I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
- 211