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11 votes
1 answer
479 views

Can an enriched functor be expressed as a colimit of representable functors?

Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...
Rich's user avatar
  • 111
2 votes
1 answer
158 views

Uniform power-saving estimate for an exponential sum

Let $N$ be a large natural number. Define an expoential sum $$ I_m=\sum_{x,y=1}^N e^{2\pi i\frac{x^2-y^2}{N}m}, \,\, m=1,2,...,N-1. $$ The trivial bound for $I_m$ is $N^2$, as there are $N^2$ terms. ...
Tony B's user avatar
  • 443
0 votes
0 answers
151 views

Physical significance of some killing vector fields on a $G_2$ manifold

I have recently calculated three linearly independent Killing vector fields on a G^2 manifold with a specific metric, which can be found in this paper: https://arxiv.org/pdf/hep-th/0011256.pdf The ...
user avatar
14 votes
1 answer
390 views

Hilbert series of graded Cohen-Macaulay domains, 28 years later?

I am reading through Richard Stanley's 1990 paper "On the Hilbert Function of a Graded Cohen-Macaulay Domain" to present in a seminar. I am trying to provide a reasonable conclusion for this talk, and ...
Eric Nathan Stucky's user avatar
3 votes
1 answer
164 views

Unbalanced version of incidences between points and unit circles

Let $P$ be a set of $n$ points and let $C$ be a set of $n$ unit circles, both in $\mathbb{R}^2.$ The maximum number of incidences between P and C is $O(n^{\frac{4}{3}}).$ Is there any bound known for ...
Kim's user avatar
  • 389
6 votes
2 answers
716 views

Books building up to the Gross-Zagier formula

I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
TeaFor2's user avatar
  • 169
0 votes
1 answer
158 views

Diameter of pseudoholomorphic curves

Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk. My question: can one prove an a-priori bound on the diameter of $u$ (...
user142700's user avatar
4 votes
1 answer
232 views

Szemerédi–Trotter type problem

Given $n$ points and an integer $k ≥ 2.$ What is the maximum number of unit circles which pass through at least $k$ of the points? I think the answer is $O(n^{4/3}/k),$ but I'm not really sure. Any ...
Kim's user avatar
  • 389
0 votes
1 answer
328 views

Is marginal density function of a Lipschitz continuous joint density function still Lipschitz continuous?

Let $f(x,y)$ be a Lipschitz continuous density function on $\mathbb{R}^2$. And let $f(x) = \int\limits_\mathbb{R} f(x,y)dy$ be marginal density function. Is $f(x)$ Lipschitz continuous? More ...
Zhuoyi Yang's user avatar
12 votes
1 answer
277 views

Rigidity of doubled convex caps

Suppose that we have a convex cap, i.e., a convex surface in $R^3$ homeomorphic to a disk whose boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the ...
Mohammad Ghomi's user avatar
8 votes
1 answer
324 views

Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$?

Consider the $\ell^2$ complex Hilbert space. Let $m\in \mathbb{N}^*$ be a fixed number, and set $$ S=\left\{ x=(x_n)_n\subset \ell^2\ :\ \sum_{n=1}^m \frac{|x_n|^2}{n^2}=1\right\}.$$ I want to ...
Schüler's user avatar
  • 724
0 votes
1 answer
95 views

Fix a continuous function $f:X\times X^k\to Y$ multilinear in $X^k$, for $X,Y$ Banach. Is $f:X\to\mathscr{L}(X,\ldots,X;Y)$ continuous?

Fix two infinite-dimensional Banach spaces $X,Y$. We define the space $$ \mathscr{L}(X,\ldots,X;Y)=\mathscr{L}(\underbrace{X,\ldots,X}_{k};Y) $$ to be the set of continuous multilinear operators $T:X^...
Dominic Wynter's user avatar
4 votes
1 answer
235 views

Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero

I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...
Sylvester W. Zhang's user avatar
2 votes
1 answer
1k views

An upper bound for a vector with given norm 1 and norm 2

Suppose $X = (x_1, \ldots , x_n)$ is given and we know that $x_i$'s are nonnegative, $\sum_{i=1}^n x_i = n$ and $\sum_{i=1}^n x_i^2 = m $. Just by this information, is it possible to find a vector ...
user115608's user avatar
3 votes
1 answer
220 views

A seemingly Groebnerizable problem

Suppose I take a polynomial $p \in \mathbb{C}[x_1, \dotsc, x_n]$ and I consider the zero set $\mathcal{Q}$ of the set of polynomials composed of $p$ and all of the "pure" partials - that is, our set ...
Igor Rivin's user avatar
  • 95.5k
2 votes
1 answer
230 views

Efficient algorithm for solving a convex quadratic program [duplicate]

Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently? $$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
O. Richard's user avatar
1 vote
0 answers
135 views

intersect a subvariety with a Schubert variety

Let $Y$ be an irreducible subvariety inside $Gr(r,n)$ (Grassmannian of $r$-plane inside $\mathbb{C}^n$) and $X_\lambda$ be a Schubert variety corresponding to $\lambda$. Assume that $codim(Y)+codim(X_\...
Ben's user avatar
  • 849
15 votes
2 answers
1k views

The Three Gap Theorem: why should it be true?

Fix $\alpha\in\mathbb R$ and $N\in \mathbb N$, consider the set $S(\alpha,N)$ of $\{k\alpha\},k=1,\dots,N$, where $\{x\}$ denotes the fractional part of $x$. Let $a_1,\ldots a_N$ be the elements of $S(...
Tian An's user avatar
  • 3,689
4 votes
1 answer
131 views

Generalizations of Dehn-Nielsen-Baer for topological branched cover?

For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of ...
Maxime Scott's user avatar
3 votes
0 answers
156 views

Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)

Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows. $$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
Puraṭci Vinnani's user avatar
1 vote
2 answers
396 views

Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]

I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$ is it sufficient to prove the norm of ...
SC_thesard's user avatar
4 votes
1 answer
1k views

Why is $O(n;k)$ not connected, and has four connected components? [closed]

Why is $O(n;k)$ not connected and has four connected components when $nk\ge 1$? Here $O(n;k) =\{A\in GL(n+k,\mathbb{R}) \mid A^{T}GA=G\}$ where $G=\begin{pmatrix} 1&&&&&\\ &\...
user123325's user avatar
5 votes
0 answers
95 views

Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients: An oriented, (assumed here to be smooth) manifold $M^n$ A finite group $G$ (and a field, chosen to be $\...
Manuel Bärenz's user avatar
4 votes
1 answer
1k views

Who was the first to capitalize Real?

For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are ...
JBantje's user avatar
  • 247
25 votes
2 answers
2k views

Is the intersection of two subgroups, defined below, always trivial?

Suppose, $G = \mathbb{Z} \ast H$, where $H$ is an arbitrary group. Suppose, $g \in G$ and $g \notin \langle\langle H \rangle \rangle $. Is $\langle\langle g \rangle \rangle \cap H$ always trivial? ($\...
Chain Markov's user avatar
  • 2,618
3 votes
1 answer
293 views

Lifting a morphism

Suppose we have a morphism $\phi : S_{1} \rightarrow S_{2}$, between quasi-projective varieties of dimension $2$ over $\mathbb{C}$ with at worst quotient singularities. Suppose furthermore that $\phi$ ...
Nick L's user avatar
  • 6,923
8 votes
1 answer
754 views

Who was in the Fields committee for ICM 1962 (the first appointed by IMU)?

Traditionally, at the presentation of the Fields medals at the ICM opening ceremony, the composition of the Fields medal committee is disclosed. This information can be found in the first volume of ...
Olaf Teschke's user avatar
  • 1,114
3 votes
1 answer
770 views

Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?

The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...
Yuji Tachikawa's user avatar
12 votes
0 answers
344 views

Is the quotient of two linear group schemes linear?

Let $S$ be an affine scheme. Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel. Assuming this, suppose $H\to S$ is a central closed ...
Uriya First's user avatar
  • 2,846
3 votes
2 answers
256 views

Square-integrability of non-holomorphic Poincare series

In what follows, $\mathfrak{H}$ denotes the upper half plane, and $\Gamma=SL(2,\mathbb{Z})$. On the modular curve $\Gamma\backslash\mathfrak{H}$, consider the non-holomorphic Poincare series, defined ...
Rick Masters's user avatar
10 votes
2 answers
5k views

Nuclear norm as minimum of Frobenius norm product

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
Hans's user avatar
  • 2,169
2 votes
0 answers
128 views

Can this construction generate bounded aperiodic functions?

This question is based on this old MathOverflow question: How this set of functions is ordered? In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
Harry Altman's user avatar
  • 2,535
18 votes
3 answers
7k views

What is so special about set theory anyway? [closed]

(Later edit - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers ...
მამუკა ჯიბლაძე's user avatar
3 votes
1 answer
945 views

Are there any numerical packages solving Volterra integral equations?

I am looking for numerical packages (ideally python) to solve second kind Volterra integral equations, such as $$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$ or Volterra-Fredholm integral equations $$u(x,t)=g(...
Sarah's user avatar
  • 31
7 votes
1 answer
361 views

Lickorish-Wallace theorem for torsion spin$^c$ 3-manifolds?

The Lickorish-Wallace theorem tells us that any closed 3-manifold $Y$ is an integer link surgery on $S^3$, which yields an oriented cobordism between $S^3$ and $Y$. Filling out the $S^3$ by a 4-ball $...
cjackal's user avatar
  • 355
26 votes
4 answers
3k views

What "metatheory" did early set theory/logic researchers use to prove semantic results?

Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic. The modern approach seems to be, usually, to interpret a "model" ...
Mike Battaglia's user avatar
9 votes
0 answers
126 views

Inverse Galois problem for $2$-groups with an involution as complex conjugation

It is known that the inverse Galois problem for solvable groups was solved by Shafarevich. My question is the following: given $G$ a finite $2$-group and $s$ an element of order $2$ in $G$. Can we ...
orangeskid's user avatar
7 votes
0 answers
404 views

Failure of integral comparison between crystalline and de Rham cohomology over a highly ramified base

Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$. By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%...
SashaP's user avatar
  • 7,027
1 vote
0 answers
142 views

Translates of a line bundle on a complex $n$-torus

Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}...
Mohan Swaminathan's user avatar
5 votes
0 answers
120 views

Is there a non-smoothable punctured manifold?

Does there exist a connected topological manifold $M$ such that $M-\{pt\}$ is non-smoothable? My understanding is that Quinn showed that these are always smoothable in dimension 4 (in fact in ...
Cihan's user avatar
  • 1,586
4 votes
0 answers
220 views

Set of subsequences with the same ultrafilter limit of the original sequence

Let $\mathscr{U}$ be a free ultrafilter on the positive integers $\mathbf{N}$ and fix $U \in \mathscr{U}$ such that $U$ is not cofinite (thanks J.D.Hamkins for the correction.) Consider the natural ...
Paolo Leonetti's user avatar
2 votes
1 answer
503 views

A question about Second-Order ZF and the Axiom of Choice

This question follows Noah Schweber's excellent answer to a corresponding question regarding second-order $ZFC$ and the continuum hypothesis: https://mathoverflow.net/a/78083/24611 Simply put, it ...
Mike Battaglia's user avatar
4 votes
1 answer
345 views

Poincare duality for mixed motives

Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality, \begin{equation} ...
Wenzhe's user avatar
  • 2,961
2 votes
1 answer
286 views

Embedding into $C\times [0,1]$

Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$. Is something like this known? $X$ is totally disconnected means that every ...
D.S. Lipham's user avatar
  • 3,055
7 votes
0 answers
200 views

Equivalent strictly convex norms in spaces of small density

Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm? Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
Tomasz Kania's user avatar
  • 11.3k
3 votes
0 answers
125 views

Does every non-locally compact metric space admit a violation of Lebesgue's theorem?

From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...
Aryeh Kontorovich's user avatar
12 votes
1 answer
521 views

Number theory question from Homotopy groups of spheres

Let $n$ be some integer. Is it true that there exists odd prime $p$ such that $4n = (p-1) \cdot k$, where $k$ is an integer coprime with $p$? This question asked Roman Mikhailov. This is ...
Alexey Milovanov's user avatar
5 votes
0 answers
483 views

Longest simple path through hypercube corners

This is a variation on a previously answered question, Longest path through hypercube corners. Here I am seeking the longest simple (non-self-intersecting) path through the unit hypercube's vertices, ...
Joseph O'Rourke's user avatar
0 votes
2 answers
373 views

$\mathbb{Z}_p[\zeta]$ is Local Ring

Let consider the ring $\mathbb{Z}_p$ and $\zeta$ be a $p$-th root of unity. Especially $\zeta \not \in \mathbb{Z}_p$. Denote with $\Phi _p(x)$ the cyclotomical polynomial in $p$. Since $p$ is a prime ...
user267839's user avatar
  • 5,948
15 votes
1 answer
1k views

Historically, which came first: the Lie algebras or their classification?

The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$. I am trying to find out ...
Matt's user avatar
  • 198

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