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Group cohomology with coefficients in a permutation module (follow-up)

This is a follow-up question to Group cohomology with coefficients in a permutation module (which was for the case of a normal subgroup) Let $A$ be an Abelian group, and let $G$ be a finite group ...
Dominic Else's user avatar
1 vote
0 answers
966 views

Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?

Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...
user64742's user avatar
  • 111
4 votes
0 answers
2k views

Interplay between Algebraic and Differential Geometry

Apologies to start with if this question is 'too soft' or not really research level. I'm a graduate student interested in both algebraic and differential geometry, although very much a novice with the ...
A. Thomas Yerger's user avatar
20 votes
4 answers
1k views

Are surjections $[n]\to [k]$ more common than injections $[k]\to [n]$?

Here's an interesting inequality involving binomial coefficient and Stirling numbers of the second kind that I believe holds for all $n,k$: $$ k^n {n \choose k} \leq n^k {n \brace k} $$ On the left-...
Filip Nikšić's user avatar
53 votes
4 answers
8k views

Why is Quantum Field Theory so topological?

I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
A Physical newbie's user avatar
16 votes
2 answers
433 views

exponential functors on finite dimensional complex vector spaces

Denote by $Vect^{fin}_{\mathbb{C}}$ the category of finite dimensional complex vector spaces. We will call a pair $(F,\tau)$ that consists of a functor $F \colon Vect^{fin}_{\mathbb{C}} \to Vect^{fin}...
Ulrich Pennig's user avatar
3 votes
0 answers
217 views

Is the canonical divisor on a minimal arithmetic surface of genus $\ge 1$ linearly equivalent to a purely horizontal effective divisor?

Let $X\rightarrow S$ be an arithmetic surface (this means, in the language of Qing Liu's book, $X$ is a regular integral scheme of dimension 2, projective and flat over a Dedekind scheme $S$ of ...
Will Chen's user avatar
  • 10k
6 votes
1 answer
111 views

Rohklin’s formula and $\beta$ expansions

I've been reading K. Dajani and C. Kraaikamp's paper From greedy to lazy expansions and their driving dynamics. In the last page previous to the references, they mention that the entropy of the map $...
Rafael Alcaraz Barrera's user avatar
8 votes
1 answer
345 views

1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors

Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what ...
Krishna's user avatar
  • 561
10 votes
0 answers
241 views

What is the preimage of a braid in a covering space branched over the braid?

For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...
cduston's user avatar
  • 145
19 votes
2 answers
2k views

Quiver representations and coherent sheaves

I've heard that under certain assumptions on an algebraic variety $X$ there exist a quiver $Q$ for which there is an equivalence $$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q))$$ between the ...
user avatar
5 votes
1 answer
473 views

Classification of the quotients of the ring Z/4 [X]

Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
Martin Brandenburg's user avatar
2 votes
0 answers
63 views

Determining subgroup of finite group of Lie type from characteristic polynomials

Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
Watson Ladd's user avatar
  • 2,419
21 votes
2 answers
806 views

Do Betti numbers beyond the first have a "number of cuts" interpretation?

I have heard stated the following Theorem. If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you ...
Qfwfq's user avatar
  • 22.7k
7 votes
0 answers
527 views

Why study Bogomolov's T-Stability

Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-...
user avatar
4 votes
0 answers
319 views

Higher genus Cohen-Jones-Segal's conjecture?

Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism $$(ev,\Phi):\...
Nati's user avatar
  • 1,971
1 vote
2 answers
311 views

group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$

I am trying to understand Kloosterman sums and their estimates (e.g. from [1], which does not prove) $$ \Big| S(m,n;c) \Big| = \Big| \sum_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, ...
john mangual's user avatar
  • 22.6k
7 votes
0 answers
147 views

Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$

What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient? I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
wonderich's user avatar
  • 10.3k
3 votes
0 answers
290 views

Paper by Moser on commuting circle diffeomorphisms and simultaneous Diophantine approximations

I am reading Moser's paper On commuting circle mappings and simultaneous Diophantine approximations and I found it hard because it is my first time that I seriously have to read a paper. It is a local ...
Boris's user avatar
  • 39
8 votes
1 answer
658 views

Fourier transform that is almost a brick wall - but why?

Let $$g(x) := \sqrt{1+x^2},$$ and $$h(x) := g^{-3/2}(x) \exp(-i2\pi g(x)).$$ I can observe that the Fourier transform $|H(f)|$ is almost flat if $|f|<1$, and $H(f)\approx 0, \; |f|>1$. This ...
Nicki's user avatar
  • 129
1 vote
1 answer
178 views

The order of the system normalizer in a finite solvable group

Definition: Let $G$ be a finite solvable group and $\Sigma \in \text{H}(G)$, the set of Hall systems of $G$. The normaliser of $\Sigma$ is defined as $$ N_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for ...
R Maharaj's user avatar
  • 366
3 votes
1 answer
871 views

Reference for etale cohomology on stacks

Is there good reference for general theory.of etale cohomology on stacks and more advanced topics? Thanks
Hao Yu's user avatar
  • 771
6 votes
1 answer
774 views

The prime gap 2 and the prime gap 4, are they equally common?

This question was posed today by a student of mine, and I have not been able to find any relevant references. Let $p_1, p_2, p_3, \ldots$ be the sequence of prime numbers listed in increasing order, ...
Andreas Holmstrom's user avatar
8 votes
2 answers
269 views

Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$

Given two commutative rings $A$ and $B$, any map of rings $A\to B$ will automatically preserve the commutative structure. This is to say, the forgetful functor $\operatorname{CRing}\to \operatorname{...
A Rock and a Hard Place's user avatar
1 vote
2 answers
297 views

The algebra of regular functions of a quasi-affine toric variety

Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with ...
Anonymous's user avatar
  • 413
60 votes
9 answers
5k views

Publishing conjectures

One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was ...
5 votes
2 answers
295 views

Bounding a graph invariant

We are given a graph $G=(V,E)$, which has clique number $k$. The graph invariant in question is given by $$q_{\mathrm{a}}(G)=\min_T \min_{A\subset T} |T|-|A|$$ where $T$ is a transversal of the ...
A Simmons's user avatar
  • 193
1 vote
0 answers
78 views

Family of functions which satisfies $f(\boldsymbol{x}) = 0$ if $\nabla f(\boldsymbol{x})=0$? [closed]

I have a Lagrangian of which I want to find the supremum in the primal variable $\boldsymbol{x}$: $\mathscr{L}(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})^T\boldsymbol{a} + \boldsymbol{\...
Danilo Socovan's user avatar
2 votes
0 answers
200 views

About the representation theory of $SL_2$ in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic, let $G=SL_2(k)$ acting naturally on $V=k^2$, and let $S^r V$ be the $r$-th symmetric power of the standard representation $V$. Is ...
Dupont's user avatar
  • 21
2 votes
0 answers
135 views

A gradient trajectory connecting boundary components in an annulus

In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in ...
aglearner's user avatar
  • 14k
52 votes
3 answers
2k views

What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent ...
Steve McCormick's user avatar
4 votes
0 answers
136 views

The class of all iterated antiderivatives of rational functions

Consider the following property of a function $f$: There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function. Question 1: Is there a name in the ...
Andreas Holmstrom's user avatar
4 votes
1 answer
172 views

DF-spaces and F spaces

It is well known that when $E$ is a $DF$-space and $F$ is a Fréchet space, the space $\mathcal{L}_{b} (E,F)$ is Fréchet. The converse, that is the fact that $\mathcal{L}_{b} (F,E)$ would be $DF$, is ...
James's user avatar
  • 103
11 votes
1 answer
518 views

What is the consistency strength of "Every set is a member of a transitive model"?

Recall that $\kappa$ is a worldly cardinal if $V_\kappa$ is a model of $\sf ZFC$. While every worldly cardinal is a strong limit cardinal, it is not necessarily regular. The point being that the short ...
Asaf Karagila's user avatar
  • 38.1k
4 votes
1 answer
177 views

Question on standard filtration Ex 3.7(a) James Humphreys "Representations of Semisimple Lie Algebras in the BGG Category O"

Exercise 3.7(a) James Humphreys's "Representations of Semisimple Lie Algebras in the BGG Category O" Let $V \in \mathcal{O}$ be a module which admits a standard filtration. Suppose that there is a ...
Learner's user avatar
  • 41
7 votes
1 answer
236 views

Largest inscribed triangle with a given vertex

It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a ...
Xiaosheng Mu's user avatar
3 votes
1 answer
425 views

Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix?

Is there a formula in radicals for the eigenvalues, or at least the largest eigenvalue, of an $n\times n$ symmetric matrix, in terms of the entries? For what $n$ is there a formula? There obviously ...
Bruce Blackadar's user avatar
6 votes
1 answer
2k views

Minimizing KL divergence: the asymmetry, when will the solution be the same?

The KL divergence between two distribution $p$ and $q$ is defined as $$ D( q \| p)\int q(x)\log \frac{q(x)}{p(x)} dx $$ and is known to be asymmetry: $D(q\|p)\neq D(p\|q)$. If we fix $p$ and try to ...
Sung-En Chiu's user avatar
1 vote
1 answer
315 views

How does $RCA_0$ achieve weak completeness?

Few days ago I asked about $WKL_0$ and the role of binary trees to provide for completeness for first order theories, and the question was nicely answered by Joel David Hamkins: Does $WKL_0$ plus CON(...
Frode Alfson Bjørdal's user avatar
4 votes
0 answers
2k views

What is the definition of "geometric analysis"? [closed]

Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone ...
Zhexiu Tu's user avatar
10 votes
0 answers
216 views

Cospectral mate of rhombic dodecahedron

I am wondering if the following pair of cospectral graphs was previously known. The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'): As far as I know, it was previously ...
David Roberson's user avatar
1 vote
1 answer
86 views

If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$

Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take $$XY := \{xy: x \in X,\, y \in Y\}.$$ We call a set $I \subseteq H$ an ideal of $H$ ...
Salvo Tringali's user avatar
9 votes
1 answer
450 views

For an intersecting family of $m$ sets there are at least $2m$ sets that are contained in at least one of them

Let $F$ be a finite family of non-empty sets such that any two of them intersect. Consider the set $F'$ consisting of all sets that are a subset of at least one element of $F$. Prove $|F'|\geq 2|F|$. ...
Gorka's user avatar
  • 1,825
16 votes
0 answers
246 views

Gap two Sierpinski set?

Is it consistent to have a set of reals $X$ of size $\aleph_3$ such that for every $Y \subseteq X$, $Y$ has measure zero iff $|Y| \leq \aleph_1$?
Ashutosh's user avatar
  • 9,771
4 votes
1 answer
199 views

Irreducible Hurwitz Factorization of A Complex Polynomial

I've decided to repost this question, which originally appeared on MSE, here. It is part of my series of open problems for enthusiasts and, while I understand this crowd is focused on professionals, ...
JMJ's user avatar
  • 263
19 votes
2 answers
568 views

Sequences with 3 letters

For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2'...
user35593's user avatar
  • 2,286
5 votes
2 answers
482 views

A question about the logarithmic complex and Morgan's paper

I have a question about the Morgan's paper ``The algebraic topology of smooth complex varieties''. Let $\Bbbk=\mathbb{C}$. Given a smooth non singular variety $X$ with normal crossing divisor $D$, the ...
Cepu's user avatar
  • 1,424
7 votes
1 answer
642 views

Negatively curved manifolds with many totally geodesic submanifolds

I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of ...
Clark's user avatar
  • 179
6 votes
2 answers
409 views

Mathematical physics applications in present-day image processing

During the past few years several important areas of image processing and image classification or generation became dominated by convolutional neural networks. I'm interested if there are any methods ...
nikkou's user avatar
  • 161
4 votes
1 answer
166 views

Compactness modulo symmetries of critical NLS solution

I was referred to the paper C. Kenig and F. Merle, "Global well-posedness, scattering, and blow-up for the energy critical, focusing non-linear Schrodinger equation in the radial case", in which one ...
Matt Rosenzweig's user avatar

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