All Questions

Filter by
Sorted by
Tagged with
9 votes
4 answers
410 views

Minimum number of common edges of triangulations

Let $S$ and $T$ be two triangulations. We define $c(S,T)$ as the number of edges shared by $S$ and $T$. With this, we can define $f(n) = \min_{P} \min_{S,T} c(S,T)$. Here the first minimum goes over ...
Till's user avatar
  • 469
9 votes
2 answers
502 views

Characterization of the family of simple groups PSL(2,q) by tensor multiplicity

Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$ Let the ...
Sebastien Palcoux's user avatar
9 votes
2 answers
510 views

Quotients of schemes by connected groups

Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$. By the Keel-Mori theorem, the quotient $X/G$ is ...
ofiz's user avatar
  • 607
9 votes
1 answer
844 views

Why is the Fast Fourier Transform efficient?

Is there a conceptual way to understand where the Fast Fourier Transform is avoiding redundant computation and thus achieving $O(n\log n)$ instead of $O(n^2)$. Consider a standard example of the FFT ...
user16557's user avatar
  • 1,513
9 votes
1 answer
549 views

Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper? Specifically, fix a ...
Noah Schweber's user avatar
9 votes
1 answer
2k views

Reference request for a proof of Ramanujan's tau conjecture

In the Wikipedia article it states that Ramanujan's tau conjecture was shown to be a consequence of Riemann's hypothesis for varieties over finite fields by the efforts of Michio Kuga, Mikio Sato, ...
teil's user avatar
  • 4,261
9 votes
2 answers
448 views

Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?

(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem) Let $f:\mathbb{R}\to [0,\infty)$ be such that (a) $\int_{\mathbb{R}} f(x) dx = 1$, (b) $\...
H A Helfgott's user avatar
  • 19.3k
9 votes
0 answers
294 views

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 ...
monkeymaths's user avatar
  • 1,169
9 votes
1 answer
1k views

When are certain group C*-algebras exact?

This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic. Anyway. There has been a lot of attention given to showing that for certain ...
Yemon Choi's user avatar
  • 25.5k
9 votes
1 answer
1k views

How do $\infty$-categories allow us to do descent on the derived level?

I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the ...
Kim's user avatar
  • 4,034
9 votes
1 answer
525 views

Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
Minimus Heximus's user avatar
9 votes
2 answers
1k views

Is every open convex subset of a Riemannian manifold necessarily contractible?

Question: Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible? Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in $...
Mostafa's user avatar
  • 4,454
9 votes
1 answer
1k views

On Fibonacci numbers that are also highly composite

It is not known if there are infinitely many prime Fibonacci numbers. But can one assert that there is no Fibonacci number >2 that is also highly composite (https://en.wikipedia.org/wiki/...
Nandakumar R's user avatar
  • 5,473
9 votes
2 answers
623 views

Is $\mathbb{Q}$ the orbit of a rational function under iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices. In the ...
Ivan Meir's user avatar
  • 4,782
9 votes
3 answers
788 views

Epimorphisms of relations

Let $\bf Rel$ be the category whose objects are sets and whose morphisms are relations. What is an epimorphism in this category? I have a sufficient condition, which is: $R$ is epic if the associated ...
seldon's user avatar
  • 1,033
9 votes
0 answers
234 views

Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)

(Previously asked at MSE.) Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
Noah Schweber's user avatar
9 votes
1 answer
922 views

Sort-of converse of Kolmogorov zero-one theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
BCLC's user avatar
  • 237
9 votes
3 answers
2k views

Definition of étale (etc) for non-representable morphisms of algebraic stacks?

I've stumbled upon the statement that the morphism $\pi$ from a root stack of the form $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ (i.e. the "generic" version, not the one concentrated along a divisor) to its ...
Qfwfq's user avatar
  • 22.7k
9 votes
2 answers
686 views

Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?

In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". Can someone explain what are the major ...
Sandeep Thilakan's user avatar
9 votes
5 answers
3k views

Negative values of Riemann zeta function on the critical line.

From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that: (1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on ...
Eren Mehmet Kiral's user avatar
9 votes
3 answers
538 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
David Sun's user avatar
  • 309
8 votes
1 answer
369 views

Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
Harrison Chen's user avatar
8 votes
1 answer
580 views

The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...
Tomasz Köner's user avatar
8 votes
1 answer
588 views

Semisimplicity of the category of coherent sheaves?

The category of coherent sheaves on a locally Noetherian scheme is abelian. Are there some geometric conditions on the scheme that imply that the category of coherent sheaves is semisimple? Edited ...
user avatar
8 votes
1 answer
204 views

Computational complexity and commuting functions

EDIT: in this question, I was proposing a conjecture, Prop. 1. Fedor Pakhomov showed a counter-example. In this new question I propose a slightly weaker conjecture that holds even for that example and ...
Doriano Brogioli's user avatar
8 votes
1 answer
1k views

Surjectivity of a map on inverse limits

(The following is crossposted from Math.SE, where the question did not receive any answers.) I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
Pavel Čoupek's user avatar
8 votes
0 answers
497 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
Ali Taghavi's user avatar
8 votes
0 answers
229 views

Intuition for the volume form - combinatorial definition?

I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
Sprotte's user avatar
  • 1,065
8 votes
1 answer
576 views

minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]

Let's consider the space $L^2[a,b]$ of functions on the interval and the norm: $$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$ Now what if we consider only polynomials with integer coefficients: $f(x) \...
john mangual's user avatar
  • 22.6k
8 votes
3 answers
1k views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
Mark Bell's user avatar
  • 3,125
8 votes
2 answers
1k views

Tweetable way to see that Willmore energy is Möbius invariant?

Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional $$\mathcal{W} = \int_M H^2 dA$$ ...
TerronaBell's user avatar
  • 3,039
8 votes
2 answers
5k views

What are the possible numbers of regions that 4 planes can divide space?

What are the possible numbers of regions that 4 planes can create? We know that the minimum number is 5 and the maximum number is 15. (http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html) Is it ...
user9107's user avatar
  • 103
8 votes
3 answers
482 views

Where does $2\sqrt{d-1}$ come from in Ramanujan graphs?

Ramanujan graphs are the best spectral expanders: $\lambda_2 \le 2\sqrt{d-1}$. I'm looking for some intuition for this value $2\sqrt{d-1}$. Friedman showed that every random $d$-regular graph ...
Xiaoyu He's user avatar
  • 1,151
8 votes
1 answer
719 views

Why relative consistency results by forcing arguments are provable in finitistic metatheory

It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....
Ruizhi Yang's user avatar
8 votes
2 answers
537 views

Non-trivial examples of Stably diffeomorphic 4-manifolds

I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not ...
Luigi M's user avatar
  • 503
8 votes
2 answers
444 views

Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist a ...
Asaf Shachar's user avatar
  • 6,611
8 votes
0 answers
2k views

What is the best lower bound for the domination number in regular graphs of girth 5?

The following theorem is a classical result (see [Alon and Spencer, The probabilistic method, 2nd ed., Theorem 1.2.2]): Theorem: Let $G$ be a graph on $n$ vertices with minimum degree $d$. Then $G$ ...
Florent Foucaud's user avatar
8 votes
1 answer
454 views

Cartan formula for Steenrod squares on the cochain level

Steenrod originally defined his squares using explicit cochain-level formulas for simplicial mod-2 cochains. To this end, he introduced higher cup products, which control the failure of the usual cup ...
Anton Kapustin's user avatar
8 votes
3 answers
3k views

Cone of curves and Mori theorem for algebraic surfaces

In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" ...
Csar Lozano Huerta's user avatar
8 votes
1 answer
655 views

Extending the tangent bundle of a submanifold

Let $X$ be a complex manifold, and $Y\subset X$ a compact submanifold. Is it true that the tangent bundle $TY$ may be extended (as a holomorphic vector bundle) to some open neighbourhood of $Y$ in $...
Alex Gavrilov's user avatar
8 votes
1 answer
1k views

From Shortest Paths to Manifold Structure

I'm relatively green in the differential geometry area, so my apologies if what I'm asking is ill-posed and/or not research-level. I have a situation where I know the shortest path between any two ...
Aeryk's user avatar
  • 2,205
8 votes
1 answer
2k views

Integer solution to special system of linear equations

This problem appear in my research, but I am unable to solve it. There should be an easy argument, but I have not yet found it. Informal version An integer $k\geq 2$ is fixed. We are given a matrix (...
Per Alexandersson's user avatar
8 votes
2 answers
392 views

Approximation of the identity by simple functions

Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$. Can we concluded $X$ may be written by a ...
ABB's user avatar
  • 3,898
8 votes
2 answers
734 views

Knot complement diffeomorphism groups and embedding spaces

I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of ...
Craig Westerland's user avatar
8 votes
1 answer
652 views

Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
Saal Hardali's user avatar
  • 7,549
8 votes
1 answer
160 views

How many maximal length Bruhat paths from $u$ to $w$ can there be?

I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
Matt Samuel's user avatar
  • 2,008
8 votes
3 answers
1k views

Examples of manifolds that do not admit scalar flat metrics

The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories: (A) Every (smooth) function is a scalar curvature. (B) The manifold is strongly ...
Ryan Unger's user avatar
8 votes
3 answers
726 views

Natural statements independent from true $\Pi^0_2$ sentences

I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
Kaveh's user avatar
  • 5,362
8 votes
2 answers
683 views

Adem relations of Steenrod square without modding out the coboundaries

In the paper Products of Cocycles and Extensions of Mappings, Steenrod introduced the cup-$i$ product and Steenrod square $Sq^k$: $$ Sq^k(x_n) \equiv x_n \smile_{n-k} x_n,\ \ \ x_n \in C^n(M^d;\...
Xiao-Gang Wen's user avatar
8 votes
1 answer
418 views

Cheeger Numbers for 3-regular Graphs

A student wanted a challenging Graph Theory programming project and I had him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, ...
user avatar

15 30 50 per page
1
190 191
192
193 194
346