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39 votes
4 answers
3k views

The symmetric group theory of natural numbers

Sometimes it is not easy to formulate a correct question. Here is a better version of this question (I still do not know if it is optimal, but it is better than the previous one). We say that a set $...
user avatar
4 votes
0 answers
1k views

Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
wonderich's user avatar
  • 10.3k
5 votes
1 answer
339 views

Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper

In Kottwitz's 1985 Compositio paper, Isocrystals with additional structure, first page, paragraph 4: Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
user125609's user avatar
0 votes
0 answers
119 views

Maximizing the sum of hook lengths

Given positive integers $a\geq b$, and $n\in\{1,2,\dots,ab\}$ I am looking for a partition of $n$ into at most $b$ parts of size at most $a$ which maximizes the sum of the hook lengths in the ...
Thomas Kalinowski's user avatar
5 votes
0 answers
181 views

Which rings are the endomorphisms ring of some abelian groups?

Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
Sara.T's user avatar
  • 151
3 votes
1 answer
154 views

Is the level set of a product of affine linear functions comprised of convex curves?

Internet searches haven't helped. Can you? Let $\, f = \prod_{i=1}^n (a_i x + b_i y + c_i).$ Is each component of $\, f^{-1}(1)$ a convex curve? I expect so, and can prove it for $n=2,$ but I'm ...
Eric Zaslow's user avatar
9 votes
2 answers
497 views

Does Easton forcing preserve measurable cardinals?

The question is in the title. For Easton's theorem see Wikipedia. Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the ...
Ioannis Souldatos's user avatar
6 votes
1 answer
414 views

Transitive homeomorphisms of Erdős spaces

A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense. Does either of the Erdös spaces $\...
D.S. Lipham's user avatar
  • 3,055
7 votes
0 answers
270 views

What are $(m,n)$-pseudoplanes?

An incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition ...
Alex Kruckman's user avatar
9 votes
3 answers
448 views

Minimal combinatorial data needed to define a polytope [duplicate]

Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
giulio bullsaver's user avatar
11 votes
0 answers
719 views

Fast computation of matrix product $AXA^T$ with fixed $A$?

Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
hao chen's user avatar
8 votes
1 answer
385 views

projective plane cubics with exactly 9 real points

It is not hard to construct such curves explicitly, e.g. my favourite example is a curve $U$ singular at $(3:4:5i)$ and also passing through $(1:0:0)$, $(0:1:0)$, $(0:0:1)$, $(1:1:0)$, $(1:0:1)$, $(0:...
Dima Pasechnik's user avatar
3 votes
1 answer
251 views

Strong polynomial algorithm for linear programming

What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
Hao Yu's user avatar
  • 771
1 vote
2 answers
232 views

Is there a standard name for this type of multidigraph?

A digraph (direct graph) consists of a set $V$ of vertices and a set $E$ of directed edges $v\to v'$. A multidigraph is a digraph in which $E$ is a multiset, so edges may appear multiple times in $E$, ...
Joe Silverman's user avatar
2 votes
0 answers
288 views

Local coefficients system

Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow ...
Mehmet Onat's user avatar
  • 1,161
3 votes
3 answers
1k views

A few general questions about pre-sheaves and sheaves

I am no specialist in sheaf theory, so I would be glad to get some help regarding the following: I have a pre-sheaf $F$ of abelian groups above a topological space $X$, and I have found an open ...
BrianT's user avatar
  • 1,197
5 votes
0 answers
195 views

When a coherent sheaf on DM stack is locally free?

A coherent sheaf $\mathcal{F}$ on a variety $X$ is locally free if every fiber $\mathcal{F}|_x$ is of the same dimension. My question is if such theorem is also true on a Deligne-Mumford stack, or ...
JJH's user avatar
  • 1,447
1 vote
1 answer
675 views

Restriction of vector bundles

I am trying to compute the Chern classes of the restriction of a rank two vector bundle on $\mathbb{P}^3$, denoted by $E$, with fixed Chern classes, $c_1(E) = c_1$ and $c_2(E) = c_2$, to a hyperplane $...
User43029's user avatar
  • 596
6 votes
1 answer
598 views

The group of isometries of Shahshahani metric

Edit: 28 January 2023 I just realized that this metric is frequently used in this paper https://hal.science/hal-01382281/document Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\...
Ali Taghavi's user avatar
3 votes
0 answers
151 views

exact sequence of fundamental groups associated to "almost" smooth families of curves

Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
Jeff Yelton's user avatar
  • 1,308
3 votes
0 answers
116 views

Obstruction to the existence of a complex-valued determinant function

Let $\mathcal A$ be a Type $II_1$ von Neumann algebra, equipped with a finite trace $\tau: \mathcal A \to \mathbb C$. Further, denote by $\mathcal A^\times \subset \mathcal A$ the group of invertible ...
H1ghfiv3's user avatar
  • 1,225
3 votes
3 answers
335 views

Intersection of a line and a conic in a surface

Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?
user avatar
4 votes
2 answers
277 views

Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements?

Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order? From Strongly real elements of odd order in ...
John Murray's user avatar
  • 1,070
8 votes
1 answer
209 views

Categorification of monotone maps via tilting modules?

It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
Mare's user avatar
  • 25.8k
1 vote
1 answer
890 views

Hironaka's theorem and smooth completion

Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X ...
Federico Barbacovi's user avatar
4 votes
0 answers
133 views

Ambidextrous Yoneda structures, admissibles and adjunctions

Let $\cal K$ be a 2-category, supporting two Yoneda structures, induced by a pseudoadjunction $P^\sharp\dashv P : {\cal K}^\text{coop}\to \cal K$; this means that there is such an adjunction, or in ...
fosco's user avatar
  • 13k
7 votes
2 answers
261 views

Functors in Isbell duality exchange $f^*a$ and $f_*a$

As you maybe remember, Isbell duality is an adjunction $$\mathcal O : [A°,Set] \leftrightarrows [A,Set]° : {\cal S}pec$$ as defined here; since every functor $f : A\to B$ defines both a functor $f^...
fosco's user avatar
  • 13k
1 vote
1 answer
68 views

Example of a certain partitioned manifold

I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$. (At first I ...
geometricK's user avatar
  • 1,851
1 vote
0 answers
74 views

Has this type of pathwise (S)DE been studied before?

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before. Let $(G,\ast)$ be an abelian $C^1$ Lie group....
Julian Newman's user avatar
0 votes
2 answers
197 views

Prime factors of a sequence of integers which differ by consecutive prime differences

[Edited following Gerhard's answer. I had forgotten to state a crucial assumption; my apologies for the confusion. I have reworded slightly to try to make things clearer.] Let $p$ be the $k$-th odd ...
user125562's user avatar
4 votes
0 answers
379 views

Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$

Suppose we're given a particular number $n \in \mathbb{N}$. We're also given that $n=pq$ where $p,q$ are unknown primes satisfying $$ p=a^2+b^2 $$ and $$ q=2ab+1 $$ for some $a,b$. Is there an ...
sfmiller940's user avatar
2 votes
0 answers
40 views

Asymptotic flag in terms of geometry of the stratum of abelian differentials?

Let $C$ be a closed Riemann surface of genus $g\geq 1$. Fix a holomorphic 1-form on $C$; it endows $C$ with a flat structure (i.e. a metric of trivial holonomy which has conical singularities at a ...
user avatar
5 votes
0 answers
193 views

Semisimplicity of the p-adic étale Tate module over $F_p(t)$

Let $k$ be a finitely generated field of positive characteristic p. Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
Emiliano Ambrosi's user avatar
10 votes
0 answers
402 views

Using the universal property of K-theory

A paper of Blumberg, Gepner and Tabuada gives a universal property of K-theory: from their abstract "connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with ...
Jakob's user avatar
  • 1,986
1 vote
0 answers
47 views

What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries. The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
Alexander's user avatar
  • 151
4 votes
0 answers
102 views

Compact subspace of sober space

We know from lemma 1.2.5 in part C of Sketches of an Elephant (by Johnstone) that both open and closed subspaces of a sober space are again sober. This raises the following question. Question: Is a ...
Math Student 020's user avatar
5 votes
1 answer
310 views

Analytic families of compact self-adjoint operators: eigenvalue extension

Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...
Brian Lins's user avatar
6 votes
1 answer
731 views

Are there non-projective, but algebraic, hyperkahler varieties?

Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough. ...
Pierce's user avatar
  • 61
2 votes
1 answer
461 views

On the paradox that $n$-coskeletal simplicial sets model all homotopy types

Please help me resolve the following paradox: False claim: Let $X$ be an $n$-coskeletal, $n$-connected simplicial set. Then $X$ is weakly contractible. Actually, I suppose the claim is ...
Tim Campion's user avatar
  • 60.6k
7 votes
1 answer
217 views

Five-dimensional manifolds fibering over a fixed hyperbolic surface

I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
Nicolas Boerger's user avatar
3 votes
1 answer
223 views

Isogeny of Drinfeld module

Let $\phi$ be a Drinfeld module over a function field $K$. It is known that if the rank of $\phi$ is 1, then it is isomorphic over $K$ to one defined over $O_K$. Is it true that if the rank of $\phi$ ...
user125554's user avatar
0 votes
0 answers
392 views

What can be said about the level set of the real part of an analytic function?

I am cross-listing this question from math.stackexchange since I did not find a satisfactory answer there. This is my first time posting a question on MO, so if this is not the appropriate community I ...
Jeremy Upsal's user avatar
0 votes
1 answer
123 views

Sets of integers represented by products of $q(q^n-1)$

Consider $X=\{q(q^n-1)|q \ \text{is some power of a prime number}, n\in \Bbb N^*\}$, $S=\{ s \in \Bbb N| s=\prod_i s_i, s_i \in X\} $, I am interested in which integers are in $S$. For example, $2, 6 \...
sawdada's user avatar
  • 6,148
1 vote
1 answer
674 views

finite dimensional C*-algebras

Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional. Q. Is $A$ finite dimensional?
ABB's user avatar
  • 3,898
3 votes
1 answer
261 views

Reconstructing the Green's function of an initial-value problem of partial differential equation

Consider a partial differential equation that is of the following form: \begin{equation} (-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t) \end{equation} where $g(x)$ is a real function. Suppose that $f(...
Mr. Gentleman's user avatar
8 votes
2 answers
866 views

What is the polynomial functor for the Bag monad

I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name ...
Ben Sprott's user avatar
  • 1,281
3 votes
0 answers
86 views

Archimedean components of base changed automorphic representations

Let $\pi$ be an automorphic (cuspidal) representation of ${\rm GL}(n)_{/\Bbb Q}$. Let $K\supset\Bbb Q$ a numberfield. The existence of the base change lift $\pi_K$ is known in a number of cases. ...
AdLibitum's user avatar
  • 221
6 votes
1 answer
285 views

Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?

In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
Mario Krenn's user avatar
4 votes
1 answer
127 views

Fréchet–Urysohn subspaces in $[0,1]^{[0,1]}$

A topological space $X$ is called a Fréchet–Urysohn space if for each subset $A\subseteq X$ and for each point in its closure, $x\in\overline{A}$, there is a sequence (not just a net, but a sequence) $...
Sergei Akbarov's user avatar
8 votes
2 answers
1k views

The Ricci Form and the First Chern Class

Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...
Zheng Hai Mu's user avatar

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