All Questions
152,895
questions
0
votes
0
answers
203
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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 ...
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 ...
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 ...
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-...
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 ...
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}...
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 ...
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 $...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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-...
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
$$(ev,\Phi):\...
1
vote
2
answers
311
views
group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$
I am trying to understand Kloosterman sums and their estimates (e.g. from [1], which does not prove)
$$ \Big| S(m,n;c) \Big| = \Big| \sum_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, ...
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{...
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 ...
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 ...
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 $$ N_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for ...
3
votes
1
answer
871
views
Reference for etale cohomology on stacks
Is there good reference for general theory.of etale cohomology on stacks and more advanced topics?
Thanks
6
votes
1
answer
774
views
The prime gap 2 and the prime gap 4, are they equally common?
This question was posed today by a student of mine, and I have not been able to find any relevant references.
Let $p_1, p_2, p_3, \ldots$ be the sequence of prime numbers listed in increasing order, ...
8
votes
2
answers
269
views
Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$
Given two commutative rings $A$ and $B$, any map of rings $A\to B$ will automatically preserve the commutative structure. This is to say, the forgetful functor $\operatorname{CRing}\to \operatorname{...
1
vote
2
answers
297
views
The algebra of regular functions of a quasi-affine toric variety
Let $k$ be an algebraically closed field of characteristic zero and let $X$
be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with ...
60
votes
9
answers
5k
views
Publishing conjectures
One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was ...
5
votes
2
answers
295
views
Bounding a graph invariant
We are given a graph $G=(V,E)$, which has clique number $k$. The graph invariant in question is given by
$$q_{\mathrm{a}}(G)=\min_T \min_{A\subset T} |T|-|A|$$
where $T$ is a transversal of the ...
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{\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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
$$
and is known to be asymmetry: $D(q\|p)\neq D(p\|q)$.
If we fix $p$ and try to ...
1
vote
1
answer
315
views
How does $RCA_0$ achieve weak completeness?
Few days ago I asked about $WKL_0$ and the role of binary trees to provide for completeness for first order theories, and the question was nicely answered by Joel David Hamkins: Does $WKL_0$ plus CON(...
4
votes
0
answers
2k
views
What is the definition of "geometric analysis"? [closed]
Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone ...
10
votes
0
answers
216
views
Cospectral mate of rhombic dodecahedron
I am wondering if the following pair of cospectral graphs was previously known.
The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'):
As far as I know, it was previously ...
1
vote
1
answer
86
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
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 $|F'|\geq 2|F|$.
...
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 $|Y| \leq \aleph_1$?
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, ...
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'...
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 ...
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 ...
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 ...
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 ...