2
votes
0answers
216 views

Rational structures on the flag variety over a finite field

Some Notions A variety over a field is defined to be a scheme of finite type over this field. An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety ...
4
votes
0answers
92 views

If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with ...
0
votes
0answers
11 views

edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...
0
votes
1answer
20 views

Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
1
vote
1answer
53 views

Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...
2
votes
3answers
294 views

Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
0
votes
0answers
20 views

Reformulating Sophie Germain Primes [on hold]

$$\frac{2p+1}{p}=\frac{1}{1}+\frac{1}{p}+\frac{1}{1}$$ Is it enough to first parametrize a proof?
0
votes
0answers
27 views

Tarski's undefinability theorem and IF logic

The Tarski undefinability theorem is valid when there are several constrains on the logic on which it is applied. The most important is that the logic mus to be closed under contradictory negation. ...
1
vote
0answers
19 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
0
votes
1answer
58 views

Topology on the dual of a Frechet space

If $F$ is a Frechet space, is there any locally convex space topology on the dual $F'$, such that for each local diffeomorphism $f$ from an open subset $U$ of $F$ to $F$, the map $U \times F' ...
1
vote
0answers
30 views

Projection onto $\ell^{2,1}$ ball

Does anyone have an idea how to project onto the $\ell^{2,1}$ ball efficiently, i.e. how to solve $$ u = \arg \min_u \|u-f \|^2 \text{such that } \left(\sum_i \big(\sum_j |u_{i,j}|\big)^2 ...
0
votes
0answers
13 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
0
votes
0answers
37 views

Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...
0
votes
0answers
44 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
6
votes
0answers
368 views

Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...
4
votes
1answer
157 views

The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$

Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?: 1)$S^{6}\to S^{7}$ 2)$S^{7}\to S^{8}$
0
votes
0answers
11 views

Existence and Uniqueness of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
3
votes
1answer
71 views

Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define: $G$ has property A iff it is edge-transitive. $G$ has property B iff each edge belongs to the same number of ...
46
votes
13answers
4k views

What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others? What makes four-dimensional topological manifolds special? What makes four-dimensional differentiable manifolds ...
6
votes
5answers
640 views

Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...
7
votes
2answers
2k views

What's the current state of Yang Mills Mass Gap question?

What's the current state of Yang Mills Mass Gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction ...
3
votes
2answers
90 views

A Lie algebra identity

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e_1,\dots,e_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant ...
1
vote
0answers
21 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross each other. I am ...
18
votes
7answers
4k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
-4
votes
0answers
33 views

What are the odds in the card game war that the game will start and finish on the first battle? [on hold]

What are the odds? My 6 year old daughter and I played war. On the first card we both played sevens and the battle bagan. We played three face down and one up and they matched again. Again, we ...
0
votes
1answer
93 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
0
votes
0answers
20 views

Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension? I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is $$ ...
2
votes
4answers
2k views

Irreducibility of determinant of symmetric matrix

It is quite known fact that the determinant of arbitrary symmetric matrix is an irreducible polynomial in algebra $\mathbb C [x_{ij}, 1\leq i,j\leq n]$ ($x_{ij}=x_{ji}$) (see this: ...
2
votes
1answer
315 views

Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
3
votes
1answer
126 views

Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
4
votes
1answer
249 views

Set-theoretic tautologies

Let us consider unquantifed formulas of a set theory (for example, NBG), more precisely, the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set and the class of all ...
11
votes
2answers
384 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
1
vote
1answer
40 views

Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$. Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...
2
votes
1answer
247 views

a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
2
votes
1answer
88 views

Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...
-5
votes
0answers
30 views

Precalculus Help? [on hold]

The lengths of the sides in a right triangle form three consecutive terms of a geometric sequence. Find the common ratio of the sequence. (There are two distinct answers. Enter your answers as a ...
2
votes
0answers
51 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
12
votes
3answers
755 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
13
votes
2answers
521 views

Geometric Quantization

I'm curious about geometric quantization. Of course, I know the procedure: Start with a classical phase space $T^{*}X$, $X$ is the configuration space, then do prequantization by creating a ...
3
votes
1answer
49 views

$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
6
votes
7answers
1k views

A good primer for geometric quantization.

Hello everyone: I'm searching for a good primer on geometric quantization. I found the following: Mathematical foundations of geometric quantization (A. Echeverria-Enriquez, et al.) Symplectic ...
16
votes
1answer
1k views

How big is the sum of smallest multinomial coefficients?

Given positive integers $n$ and $d$, let $S$ indicate the list of all $d$-tuples of non-negative integers $(c_1,\ldots,c_d)$ such that $c_1+\cdots+c_d=n$. Let $v_i$ be the value of the multinomial ...
2
votes
1answer
66 views

References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...
1
vote
2answers
73 views

Estimate maximal coefficient of a polynomial from a circle containing all roots

Suppose I have a polynomial $$ p(x)=\sum_{i=0}^n p_ix^i. $$ For simplicity furthermore assume $p_n=1$. As it is well known we may use Gershgorin circles to give an upper bound for the absolute ...
23
votes
4answers
1k views

What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra. In the former, one assigns to ...
5
votes
1answer
177 views

Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
-2
votes
2answers
83 views

Lack of parabolicity of PDE due to invariancy under diffeomorphisms?

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
6
votes
0answers
65 views

Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
-1
votes
0answers
39 views

Question about lim and sup of a function [on hold]

Let f(t,d) be a continuous function in t. Prove that: $\lim\limits_{m\to \infty}\sup\limits_{t\in[t_1,t_2]}f(t,d_m)=\sup\limits_{t\in[t_1,t_2]}\lim\limits_{m\to \infty}f(t,d_m)$ Does anyone have any ...
1
vote
1answer
196 views

labeling state vectors in representation space of a simple lie algebra

Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...

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