All Questions
152,874
questions
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 $u$ (...
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 ...
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 ...
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 ...
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 ...
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^...
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}{...
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 ...
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 ...
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?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
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_\...
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(...
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 ...
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 ...
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 ...
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&&&&&\\
&\...
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 $\...
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 ...
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?
($\...
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$ ...
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 ...
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 $\...
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 ...
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 ...
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}(\|...
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 $f:\mathbb{N}\to\mathbb{N}$ ...
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 ...
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(...
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 $...
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" ...
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 ...
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%...
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}...
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 ...
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 ...
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 ...
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}
...
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 ...
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 ...
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 ...
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 ...
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,
...
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 ...
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 ...