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Finite flat pullback of the diagonal

Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism. Let $\Delta_X$ be the closed subscheme of $X\times X$ ...
user avatar
5 votes
1 answer
507 views

Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?

$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top} \DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such ...
user494312's user avatar
0 votes
0 answers
83 views

Does it help for graph isomorphism to know power of the permutation matrix?

Here all matrices are square $n \times n$ with integer entries. If you prefer, all entries are $0-1$. Observation: the discrete logarithm for permutation matrices is polynomial in $n$, since the ...
joro's user avatar
  • 24.2k
1 vote
0 answers
68 views

A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz

An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite. A space $X$ is said to be ...
Nur Alam's user avatar
  • 475
1 vote
0 answers
94 views

Does the center of any finitely generated associative algebra over a field have finite type?

Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
GiS's user avatar
  • 321
3 votes
1 answer
401 views

Positive definiteness of a matrix-valued function

This question is a repost from math.se, where I didn't receive an answer. Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \begin{cases} \exp(-B |t - s|^\alpha), &...
tsnao's user avatar
  • 490
6 votes
1 answer
314 views

Deformations of the 4-sphere with 8-dimensional isometry groups

I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.
Thomas Schucker's user avatar
14 votes
3 answers
864 views

(An introduction to) deformation theory (written) for differential geometers

Question is as mentioned in the title: Are there any introductory notes on deformation theory that are easier to read for differential geometers? I am learning about differential graded Lie algebras (...
Praphulla Koushik's user avatar
1 vote
1 answer
137 views

Dense properties of weighted Sobolev space define on $\mathbb{R}^n$

Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....
Houa's user avatar
  • 561
1 vote
1 answer
147 views

Trivial convergent sequences in $\beta X$

Let $X$ be a Tychonoff space and denote by $\beta X$ its Stone-Čech compactification. We know, for example, that if $X$ is an $F$-space then $\beta X$ is an $F$-space and, therefore, in $\beta X$, the ...
Carlos Jiménez's user avatar
0 votes
1 answer
266 views

Estimating a sum involving the von Mangoldt function

I'd like to know the estimate of the following sum $$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$ where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
Beta's user avatar
  • 365
5 votes
3 answers
493 views

If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?

Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that $$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$ Do we have $\|f\...
Feng's user avatar
  • 517
1 vote
0 answers
75 views

Integration over a finite-dimensional subspace of Hilbert space

Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
John's user avatar
  • 483
8 votes
1 answer
278 views

What are twisted Verma modules? Basic properties?

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
Pulcinella's user avatar
  • 5,506
6 votes
0 answers
295 views

Looking for analogs of the rational, $p$-adic, and real numbers in homotopy theory via the passage $\mathbb{N}\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{R}$

We can view the construction of the real or $p$-adic numbers as the end result of a series of constructions each of which adds more and more structure to the object at hand: First, we start with the ...
Emily's user avatar
  • 10.3k
6 votes
1 answer
229 views

Is there a general method for computing finitely generated normalizers?

I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ ...
Jim's user avatar
  • 215
0 votes
0 answers
170 views

Geometric interpretation of normalization inside a finite extension of function field

$\DeclareMathOperator\Spec{Spec}$Suppose $X = \Spec A$ is a smooth affine variety over $\mathbb C$ and suppose $L/K$ is a finite extension of its function field. Let $Y = \Spec B$, where $B$ is the ...
Mohan Swaminathan's user avatar
2 votes
0 answers
90 views

References (and a question) on the "fine" topology of powersets

Recently I've been trying to understand powerset topologies better, and came upon the following reference: Frank Wattenberg, Topologies on the set of closed subsets. Pacific J. Math. 68(2): 537-551 (...
Emily's user avatar
  • 10.3k
2 votes
2 answers
151 views

expectation and variance of the norm of a random matrix

Suppose $X \in \mathbb{R}^{n \times d}$ is a random matrix where $n > d$. Given a matrix $A \in \mathbb{R}^{n \times n}$ such that $AX$ is a zero matrix in expectation, i.e., $\mathbb{E}_{X}[AX] = ...
Hao He's user avatar
  • 225
3 votes
0 answers
140 views

Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion

(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
Tom Copeland's user avatar
  • 9,937
5 votes
1 answer
325 views

About an asymptotic behavior in number theory

Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
3 votes
0 answers
170 views

Are "very conservative" connectives already definable?

I'm sadly an outsider to nonclassical propositional logics. All terminology below comes from Humberstone's book The Connectives, specifically section 4.2. A new connective - a bit more precisely, a ...
Noah Schweber's user avatar
2 votes
0 answers
156 views

Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"

I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16. If I understood correctly, in the second ...
Breakfastisready's user avatar
2 votes
0 answers
96 views

Hamiltonian action as a group homomorphism

It is sometimes demanded that a Hamiltonian group action $G \times M \to M$ allow for a Lie algebra homomorphism from $\mathfrak{g}$ to $C^\infty(M)$ with the Poisson bracket. Is there a natural ...
MomentumMap's user avatar
2 votes
0 answers
83 views

Proving that $u_0$ is a canonical solution

I've asked this question on stack exchange before but no one could help me so I wish I can get some help here. Let's first start with the definition of the canonical solution: Consider $\frac{\...
User-123's user avatar
16 votes
1 answer
740 views

The second stable homotopy group

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
Leo's user avatar
  • 541
1 vote
0 answers
155 views

Interpretation of Tate conjecture using motivic homotopy

For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps $$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$ are surjective. To ...
TCiur's user avatar
  • 469
15 votes
1 answer
2k views

How did Fermi calculate this integral?

In his 1926 paper Fermi states without further explanation that it follows from the Thomas-Fermi equation $$\frac{d^2\psi(x)}{dx^2}=\frac{\psi(x)^{3/2}}{\sqrt{x}},\label{1} \tag{1}$$ and boundary ...
Zurab Silagadze's user avatar
5 votes
1 answer
196 views

Step in the Bruhat decomposition for reductive Lie groups

Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on MSE. I'm stuck on a line in the proof of Theorem 7.40 in Knapp's '...
Chertopkhanov on Malek Adel's user avatar
6 votes
1 answer
251 views

A ZFC example of a Menger space which is not Scheepers

$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
Nur Alam's user avatar
  • 475
2 votes
1 answer
77 views

Request for literature recommendations on isotonic mappings

An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a ...
stalinon's user avatar
2 votes
0 answers
50 views

Convergence of minimiser of empirical risk to minimiser of population risk

Let $X_1, \dots, X_n \sim \mu$ be some random elements of a space $\mathcal{X}$. Let $H$ be a Hilbert space of functions $f: S \to \Re$ with norm $\|\cdot\|_H$. Let $\|f^*\|_{L_2(\mu)} < \infty$ ...
user27182's user avatar
  • 315
3 votes
1 answer
231 views

Closed subset of unit ball with peculiar connected components

Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball. Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii? i) $\{0\}$...
user_1789's user avatar
  • 722
1 vote
0 answers
98 views

Schrödinger equation approximation – continuity of eigenvalues with respect to potential

The question has been crossposted from Stackexchange after receiving no answers. Setup: the time-independent Schrödinger equation (eigenvalue problem): $(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$ (On ...
Rohan Didmishe's user avatar
3 votes
1 answer
193 views

Does positive set theory prove the existence of a set of all ordinals and itself?

Does positive set theory $\sf GPK^+_\infty$ prove the existence of a set $K= \{x \mid x \text { is von Neumann ordinal } \lor x=K\}$
Zuhair Al-Johar's user avatar
1 vote
0 answers
64 views

Inside-out dissections of polygons - a generalization

Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...
Nandakumar R's user avatar
  • 5,453
3 votes
0 answers
226 views

Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)

In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as $$D\; X^n = F(\tfrac{...
Tom Copeland's user avatar
  • 9,937
1 vote
0 answers
57 views

Probability that a Lévy process "closely" follows a predefined trajectory

For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\...
Falrach's user avatar
  • 131
3 votes
0 answers
215 views

An attempt to extend polynomial rings

Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, $L_n$ be its fraction field and $W_n=\mathbb{K}[x_1,\...
Zerox's user avatar
  • 1,111
5 votes
1 answer
394 views

What is the correct definition of semisimple linear category?

I have been using the notion of "semisimple linear category" for a while now, but I never bothered to write down a definition. I've always just waved my hands and said "Schur's lemma ...
Milo Moses's user avatar
  • 2,809
11 votes
0 answers
434 views

What sequence maximizes the final distance?

This problem was created by professor Ronaldo Garcia from Universidade Federal de Goiás (UFG) and he showed it to me at an event in my university. This problem has a lot of history and he told me he ...
Arthur Queiroz Moura's user avatar
4 votes
1 answer
133 views

Kernels of actions on truncated polynomial algebra

Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
Ehud Meir's user avatar
  • 4,969
3 votes
1 answer
153 views

Fastest algorithm for calculating optimal tours in weighted $K_5$

Weighted $K_5$ have the unique property that their edge set can be interpreted as the disjoint union of their shortest and their longest Hamilton cycle. That makes $K_5$ attractive for designing new ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
66 views

Computing and isotopy of curves in $\mathbb{R}^3$

Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length. The (stationary) string can be modelled by a unit speed curve: $$[0,1] \...
sitiposit's user avatar
  • 171
0 votes
0 answers
78 views

Describing a time-varying process with a manifold

I am a beginner in topology and I am trying to define a model for some computations. My questions are speculative: I am wondering what is the proper way to add time in a manifold so as to describe a ...
Phys. Student's user avatar
2 votes
1 answer
184 views

Dispersion of a "random" subset of $[-1,1]^2$

Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
Mathews Boban's user avatar
1 vote
1 answer
152 views

Quantitative version of Lebesgue points theorem

Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
tommy1996q's user avatar
1 vote
2 answers
206 views

How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$

Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that \begin{equation}\tag{1}\label{1} \int_a^b \...
Hyperbolic PDE friend's user avatar
4 votes
0 answers
97 views

Systems of parabolic equations -- Petrovskii's condition

Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$. Given a matrix field $A:Q_T\rightarrow\text{M}...
Ayman Moussa's user avatar
  • 2,575
0 votes
1 answer
150 views

Littlewood-Paley characterisation of Hölder regularity

I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
Tham's user avatar
  • 103

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