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Group scheme representation from action of a group scheme on a variety?

Let f be some homogenous polynomial of d. Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p>0$. Now $G$ is a group scheme acting on $X$. ...
grontim's user avatar
  • 167
16 votes
1 answer
451 views

Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$. By Stirling's ...
H A Helfgott's user avatar
  • 19.3k
3 votes
0 answers
212 views

Renyi's theorem on mixing

I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations: A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...
Anon's user avatar
  • 31
4 votes
1 answer
253 views

Invariant lifts of a closed curve on a surface of genus > 1

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question : Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
TheSilverDoe's user avatar
0 votes
1 answer
247 views

Quasiconvexity property of quasinorms

Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
Turbo's user avatar
  • 13.7k
12 votes
1 answer
294 views

Factoring polynomials over the abelian closure of the rationals

What algorithms are known to perform the following task? Input: a univariate polynomial over the rationals $f \in \mathbb{Q}[t]$. Output: the factorization of $f$ into irreducible factors over the ...
Gro-Tsen's user avatar
  • 29.9k
5 votes
1 answer
277 views

Interesting properties in $...\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to ...$

Let $K(G,n)$ be the Eilenberg Maclane space. Consider the map from $$ K(\mathbb{Z}_2,1) \to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to \dots, $$ It ...
annie marie cœur's user avatar
8 votes
1 answer
591 views

Spectral and derived deformations of schemes

I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is. Let $S = (X, ...
Galois groupie's user avatar
106 votes
2 answers
32k views

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
Keshav Srinivasan's user avatar
6 votes
0 answers
876 views

Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle

I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature. I'm hoping there's a nice ...
MichaelGaudreau's user avatar
5 votes
2 answers
378 views

Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
Ali Taghavi's user avatar
4 votes
1 answer
126 views

On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
user521337's user avatar
  • 1,189
2 votes
0 answers
252 views

Cohomology and base change

Let $f$ be some homogenous polynomial of $d$. Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$. Now let $R$ be a $k$-algebra. What ...
grontim's user avatar
  • 167
2 votes
0 answers
530 views

eigenvalues of a square block matrix

How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix} A&B\\-B& 0 \end{bmatrix} where A, B are real matrices, $A =\...
UserName's user avatar
5 votes
1 answer
338 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar
8 votes
0 answers
288 views

Relationships among constructions of fundamental group for schemes

There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
Galois groupie's user avatar
6 votes
0 answers
470 views

The $E_2$-page of the May spectral sequence

I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS. At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
S. carmeli's user avatar
  • 4,064
4 votes
1 answer
285 views

Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators

I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}...
D_S's user avatar
  • 6,100
1 vote
0 answers
50 views

What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?

Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...
John Bentin's user avatar
  • 2,427
8 votes
1 answer
339 views

Characterization of KL divergence for continuous variables?

This is an analog of an older question: What characterizations of relative information are known? With the modification that I’m interested in the case when the distribution is over something that’s ...
zzz's user avatar
  • 868
1 vote
0 answers
41 views

Numerator-cancellable Modules

I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following : Let $R$ be a ...
Rajkarov's user avatar
  • 933
0 votes
1 answer
248 views

Weak convergences in Bochner spaces

I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $\Omega$: Let $\{x_n\}$ be a bounded sequence in $L^2((0,T)\...
MathAnimal's user avatar
12 votes
2 answers
388 views

Has the arcsine law been generalised to higher order divisor functions?

The arcsine law for the distribution of the logarithms of the divisors of an integer $n$ states that $$ \frac{1}{x}\sum_{n\leq x}\frac{1}{d(n)}\sum_{\substack{q|n\\q\leq n^{A}}}1\sim \frac{2}{\pi}\...
Kevin Smith's user avatar
  • 2,470
3 votes
1 answer
117 views

How many $2d$-elements subsets with specific property at most can we choose from $\{1,2,\cdots,n\}$

Are there any known results about the following problem: Given any integer $n\geqslant4$, how many $4$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any ...
user173856's user avatar
  • 1,987
2 votes
1 answer
140 views

Are separability and ccc equivalent for closed subspaces of $\beta N$?

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
user129208's user avatar
1 vote
0 answers
73 views

relative compact on nonlinear term

On the paper: Decay of Solutions to Nonlinear Schrodinger Equations. Let $u$ be a solution of the equation $$Hu+|u|^2u=0,$$ where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
DLIN's user avatar
  • 1,905
2 votes
1 answer
158 views

Graphs with exactly $n$ perfect matchings

Is there for every $n\in\mathbb{N}$ a connected, simple, undirected graph $G_n=(V_n,E_n)$ such that $G_n$ has exactly $n$ perfect matchings?
Dominic van der Zypen's user avatar
3 votes
0 answers
243 views

Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
Ali Taghavi's user avatar
6 votes
0 answers
324 views

Explicit bounds for the Mertens function

It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
Mayank Pandey's user avatar
15 votes
2 answers
3k views

Measures and differential forms on manifolds

Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$. What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...
Galois groupie's user avatar
2 votes
0 answers
62 views

Negative association in a "k out of n" process

Suppose we have $n$ distinct balls labeled $1,2,\dots,n$ in a black box. Now we want to fetch $k$ balls from the box, one by one. Let event $E_i$ be that the label of the $i$-th ball we fetch is no ...
Lwins's user avatar
  • 1,531
24 votes
2 answers
684 views

What's the maximum probability of associativity for triples in a nonassociative loop?

In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...
John Baez's user avatar
  • 21.3k
6 votes
0 answers
168 views

What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?

Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
Elle Najt's user avatar
  • 1,432
2 votes
0 answers
95 views

Polynomials passing through points with tangential conditions

In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
Turbo's user avatar
  • 13.7k
1 vote
1 answer
240 views

Sage: Evaluation precision for elliptic curves over p-adic fields

Consider the elliptic curve $E: y^2 = x^3 + 23x+11$ over p-adic fields. In Sage I use: k = GF(257) E = EllipticCurve(k,[23,11]) kp = Qp(257,5) # 257-adic Field with capped relative ...
user5507059's user avatar
1 vote
1 answer
412 views

Sufficient conditions for a topological space to be regular $T_3$

There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be completely regular $T_{3^1/_2}$. Please, let me know any known condition(s) that ...
Matti Kiiski's user avatar
4 votes
0 answers
131 views

Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
user122374's user avatar
4 votes
1 answer
273 views

Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
user avatar
1 vote
1 answer
275 views

exact sequence of deformations

Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence: $0\longrightarrow H^0(C,T_C)...
Tom's user avatar
  • 71
4 votes
0 answers
58 views

Separation principle in generic models?

Separation principle $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ claims that any two disjoint pointsets of boldface class $\mathbf\Sigma^1_n$ are separated by a $\mathbf\Delta^1_n$ set. ...
Vladimir Kanovei's user avatar
11 votes
1 answer
568 views

“Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$. Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
user avatar
0 votes
1 answer
91 views

Connected infinite graphs in which all matchings are "small"

Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
Dominic van der Zypen's user avatar
21 votes
1 answer
677 views

Diameter of a quotient of the infinite dimensional sphere

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well. Assume that the action $...
Anton Petrunin's user avatar
12 votes
1 answer
549 views

Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
H A Helfgott's user avatar
  • 19.3k
8 votes
5 answers
463 views

Ideals on $\mathbb N$ and large sets that have small intersection

Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable: $$A,...
Tomasz Kania's user avatar
  • 11.3k
3 votes
0 answers
105 views

Induced $(\mathfrak{g},K)$-modules

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
Hebe's user avatar
  • 821
1 vote
1 answer
539 views

Reformulation of Grothendieck vanishing theorem

Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$? ...
Chen's user avatar
  • 1,583
3 votes
0 answers
101 views

Differential of p-divisible groups

Setting: Let $p$ be a prime number and let $S$ be a scheme such that $p$ is locally nilpotent on $\mathcal O_S$ ($p^N=0$). Let $X$ be a $p$-divisible group over $S$. Let $X[p^n] $ be the kernel of ...
slinshady's user avatar
  • 309
5 votes
1 answer
491 views

A name for a group with finite abelization?

Let us recall that a group $G$ is called perfect if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial. Question. Is there any name for a group ...
Taras Banakh's user avatar
  • 40.8k
1 vote
1 answer
265 views

About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$

It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $...
James Cheung's user avatar
  • 1,855

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