All Questions
5,076 questions with no upvoted or accepted answers
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Hopficity of Baumslag-Solitar groups
I am struggling to find the exact source (with proofs) of the following ''well-known'' statement:
the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...
- 3,181
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Aspherical fibrations and group epimorphisms
Let $\mathsf{Top}$ denote the category of pointed spaces having the pointed homotopy type of a pointed CW-complex. Let $\mathsf{Grp}$ denote the category of groups. It is well documented that for ...
- 35.9k
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308
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Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
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358
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Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
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174
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A model where the uniformity of the meager ideal is strictly below the almost disjointness number
I'm looking for a model satisfying the inequality described in the title. Recall that the uniformity of the meager ideal, denoted $\operatorname{non}(\mathcal M)$ (or $\operatorname{non}(\mathcal B)$) ...
- 1,882
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251
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Exponential sums over integers with a fixed number of prime divisors
Are there bounds in the literature on sums of the form
$$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$
for $\alpha$ on minor arcs (i.e., not very close ...
- 20.2k
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274
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Does a generalization of Tietze's extension theorem hold for set-valued functions?
Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
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393
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When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
- 12.5k
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278
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Which field extensions do not affect Chow groups?
Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...
- 16.9k
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331
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Connections between spectral geometry and critical point/Morse theory
I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
- 217
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489
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Category of metric spaces
Is there a standard/good reference text that does category of metric spaces?
Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- ...
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A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
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890
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How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
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Lagrangian subgroups/submanifolds, 2d topological boundary and 3d "non-abelian" Chern–Simons theory
This post is meant to ask for proper references to fill a gap in the literature.
My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
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353
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Quantum Hamiltonian reduction and Quantum Airy structure
I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section ...
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545
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Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
- 519
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373
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Examples for a conjecture of Beilinson
Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in ...
- 5,901
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254
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Decomposition of linear groups into free products
I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
- 5,901
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189
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Motivic homotopy theory and Noether problem
Let $G$ be a finite group, and let $V$ be a faithful representation of $G$. The Noether problem asks whether $V/G$ is rational (stably rational, retract rational) or not.
To construct ...
- 161
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718
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Andrei Suslin's works
Andrei Suslin, a well known mathematician, died 10 July 2018. (https://en.wikipedia.org/wiki/Deaths_in_2018) I believe it may be appropriate to give an overview of his work on this site. Personally, ...
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192
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Where is it shown that a countable self-injective ring is semilocal?
In Lawrence, John. "A countable self-injective ring is quasi-Frobenius." Proceedings of the American Mathematical Society (1977): 217-220. the first line is this:
...
- 1,509
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569
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A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 42k
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184
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Demazure’s principle for tower of reductive groups and analogy with Teichmuller tower
Grothendieck in his Esquisse states (page 6/7 of the English translation) that the “principle” that the Teichmuller tower, i.e., the system of profinite fundamental groupoids $\hat{T}_{g,\nu}$ of ...
user122285
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170
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Hyperelliptic locus is a $K(\pi,1)$
It is said in many papers that the hyperelliptic locus $\mathcal{H}_g\subseteq \mathcal{M}_g$ is a $K(\pi,1)$. (in the sense of orbifolds). This is justified by saying that it can be constructed as an ...
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230
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Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
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305
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Moschovakis' discovery of E-recursion
E-recursion is a notion of generalized computability theory which seeks to extend computations to allow arbitrary sets as inputs. In contrast with e.g. $\alpha$-recursion, it disallows unbounded ...
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470
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Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
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240
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Reference request: integral formula for $\sum_{\text{roots }\lambda}e^{-|\lambda|^2}$
Consider a polynomial $f(z)=c\prod_m(z-\lambda_m)\in\mathbb{C}[z]$. I am mostly interested in the case where this actually lies in $\mathbb{R}[z]$, but that is not essential. I wanted to find a nice ...
- 56.9k
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146
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A characterization of Moishezon manifolds via sections of $L^k$ with $k\to \infty$
Let $X$ be a smooth compact complex manifold of dimension $n$. Suppose $L$ is a line bundle on $X$ such that $dim(H^0(X,L^k))>c\cdot k^n$ for $c>0$ and $k>>0$.
Question. Is it true that $...
- 14.3k
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261
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Reference request: $H^* X$-module structure on the Mayer–Vietoris coboundary
Is the following presumed folklore fact written anywhere?
Let $E^*$ a multiplicative cohomology theory. Then the coboundary map in the Mayer–Vietoris sequence of an excisive triad $(X;U,V)$ preserves ...
- 2,995
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371
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Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it ...
- 18.4k
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271
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Cancellation in a sum of Möbius evaluated along a quadratic form
Let $Q(x,y)$ be an indefinite binary quadratic form. Suppose $0 < B < \sqrt{A} $ are such that $B \gg \sqrt{A}$.
Is it true one can save an arbitrary power of log from the trivial bound in
$$...
- 2,283
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998
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Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
- 435
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418
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Equivariant obstruction theory done wrong
Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...
- 35.9k
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409
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The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
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230
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Clozel's unpublished manuscript
I'm looking for Clozel's unpublished manuscript
L. Clozel, Modular properties of automorphic representations I: Applications of
the Selberg trace formula (1993)
cited in Urban's Eigenvarieties ...
- 3,799
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332
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Is the perfectness of Fourier-Mukai kernels proved by Toen?
In Toen's paper The homotopy theory of dg-algebras and derived Morita theory, Theorem 8.15, he essentially proved the following result.
Let $X$ and $Y$ be two smooth and proper schemes over $k$. ...
- 9,019
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910
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Grothendieck's motivation of crystalline cohomology
Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...
- 7,377
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632
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Relative Chow groups
Most cohomologies have the notion of cohomology with support on a closed subspace, and also cohomology with compact support. In general, for any morphism $f\colon Y\to Z$ the inverse image fits into a ...
- 2,871
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602
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Topological entropy and periodic sequences of a subshift
Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$.
...
- 2,135
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298
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An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
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299
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On an unpublished result of Magidor
In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and $2^{\aleph_\omega}=\aleph_{\...
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383
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Existence of universal extension between two modules?
I need a reference for the following fact:
Let $R$ be (for simplicity) an algebra over the field $k$, let $A, B$ be $R$-modules. Let $E = Ext^1_R(B, A)$. There is a natural isomorphism between $...
- 525
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275
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Complete list of exceptions and efficient algorithm for Waring's problem
2 weeks ago, Samir Siksek https://arxiv.org/abs/1505.00647 proved the more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, ...
- 91
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414
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In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
- 2,719
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257
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Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles
I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
- 191
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360
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Phillips-Sarnak conjecture in higher dimension
The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...
- 245
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319
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From Frege to Gödel - German equivalent?
I know this question does not quite fit here, but I felt it could best be answered here. I recently stumbled upon the book From Frege to Gödel, which is a sourcebook containing some of the most ...
- 213
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251
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Decay rate of measures on Cantor set
I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...
- 91
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275
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Proving regularity properties from forcing axioms
It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...
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