Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by

$$exp_p(v)=\gamma_v(1)$$

If we pick any orthonormal basis,$(e_1,...e_n)$ of $T_pM$ then the $x_i$’s, with $x_i = pr_i◦exp^{-1}$ and $pr_i$ the projection onto $Re_i$ , are called normal coordinates at $p$.

My question is , if our manifold $M$ is flat than does this normal coordinates coincide to what we call Cartesian coordinates ?

Ravenel and Wilson showed that $K(\mathbb Z / p^j,q)$ is $K(n)$-acyclic for any $q \geq n+1$, and that $K(\mathbb Z, q)$ is $K(n)$-acyclic for $q \geq n+2$. It follows that $K(A,q)$ is $K(n)$-acyclic for $q \geq n+2$ when $A$ is finitely-generated.

From here, a Serre spectral sequence argument reveals that the map $\tau_{\leq m} X \to \tau_{\leq n+1} X$ (where $\tau$ is Postnikov truncation) is a $K(n)$-local equivalence for all $m \geq n+1$ when $X$ has finitely-generated homotopy groups. (For $\pi$-finite spaces, $\tau_{\leq m} X \to \tau_{\leq n} X$ is in fact a $K(n)$-local equivalence, as observed by Carmeli,Schlank, and Yanovski).

It's tempting to conclude that $X \to \tau_{\leq n+1} X$ is a $K(n)$-local equivalence for any $X$ with finitely-generated homotopy groups, but this can't possibly be true. If it were true, then in particular $X \to \tau_{\leq n+1} X$ would be an equivalence for all simply-connected finite spaces $X$. Then we could conclude $K(n)_\ast(S^2) = K(n)_{\ast+n+1}(\Sigma^{n+1} S^2) = K(n)_{\ast+n+1}(pt)$, which is false.

Indeed, according to Bauer, the convergence of the spectral sequence for the Postnikov tower of $X$ only holds when $X$ is $n$-truncated. This leads to my

**Question:** If $X$ is a space with infinitely many nontrivial homotopy groups, is there any meaningful relationship between $K(n)_\ast(X)$ and $K(n)_\ast(\tau_{\leq n} X)$ or $K(n)_\ast(\tau_{\leq n+1} X)$? (Beyond the mere existence of a map -- for all I know, this map is zero!) How about if $X$ is finite? Or perhaps, what if $X$ is $(n-1)$-connected?

does a group on 2 to the t elements necessarily have a subgroup of index two? It seems close to the Sylow Theorems but not quite. Maybe there is a simple counterexample.

Throughout the question, $X$ denotes the underlying space of an irreducible scheme. In particular, $X$ is non-empty.

Let $n$ be a non-negative integer. Let us say that $X$ satisfies $P(n)$ if $X$ satisfies the sentence "there is a closed point in the closure of any point...in the closure of any point" with "in the closure of any point" repeated $n$ times (if $n=0$, this means that there is a closed point in $X$).

Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.

QuestionDoes there exist a **non-asymptotic** bound of the form $\text{Proba}(KL(P \|\hat{P}_n) \le \epsilon) \le (n + 1)^ke^{-\epsilon n}$ ?

I'm reading the article by Geoffrey Mess The Torelli groups for genus 2 and 3 surfaces (pp. 785 - 786), and I'm trying to understand the part that concerns genus 3. We've got a map $UT(S) \to T_3/\mathcal{I}_3$, where $UT(S)$ is a unit tangent bundle over a curve of genus 2, $T_3$ is Teichmuller space and $\mathcal{I}_3$ is Torelli group, this map induces the following one $\pi_1(UT(S)) \to \pi_1(T_3/\mathcal{I}_3) \cong \mathcal{I}_3$. It's interesting to understand the images of the generators of $\pi_1(UT(S))$ in $\mathcal{I}_3$. I'm trying to do that by studying deck transformations of Teichmuller space and understanding which elements of Torreli group acts the same.

$UT(S) \to T_3/\mathcal{I}_3$ is built in the following way. We just take a point in Siegel space $\mathcal{Z}_3$ that corresponds to a Jacobian of $C \cup E$, where $C$ is a genus 2 curve and $E$ is a torus, then we can move a point where these two curves intersect over the whole genus 2 curve not changing the Jacobian, and then by adding annulli instead of those points we get genus three surfaces that can be considered as points in Torelli space.

Using Fenchel–Nielsen coordinates I guess that I can show that the generator $s$ that corresponds to going around the fiber (that is a circle) maps to Dehn twist of a separating curve. However, I can't understand how to find the images of other generators. When I try to study the process of dragging the torus around some loop corresponding to some (surface) generator, I get a problem with understanding how the coordinates change and what path we are going in Teichmuller space, so it seems that this is not the right way to look at this problem.

The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=\sum v_i\in H^*(M;\mathbb{Z}_2)$ is the unique cohomology class such that $$\langle v\cup x,z\rangle=\langle Sq(x),z\rangle$$ for all $x\in H^*(M;\mathbb{Z}_2)$. Thus, for $k\ge0$, $v_k\cup x=Sq^k(x)$ for all $x\in H^{n-k}(M;\mathbb{Z}_2)$, and $$w_k(M)=\sum_{i+j=k}Sq^i(v_j).$$ Here the Poincare duality guarantees the existence and uniqueness of $v$.

My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula? In this case, there is a fundamental class $z\in H_n(M,\partial M;\mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms $$D:H^p(M,\partial M;\mathbb{Z}_2)\to H_{n-p}(M;\mathbb{Z}_2)$$ and $$D:H^p(M;\mathbb{Z}_2)\to H_{n-p}(M,\partial M;\mathbb{Z}_2).$$

Thank you!

Let $X$ be the underlying space of a scheme.

- If $X$ is irreducible of finite Krull dimension, is it necessarily quasi-compact?
- Is it necessarily Noetherian?
- What if we assume not only that Krull dimension is finite but also that it is 1?

Let $(X_n,Y_n)_{n \in \mathbb{N}}$ be a sequence of independent random variable such that $X_0,...,X_n,...,Y_0,...,Y_n,...$ are identically distributed. We suppose there exists a sequence $(x_n)_{n \in \mathbb{N}}$ of real numbers such that : $P(\limsup_n|\frac{1}{n}\sum_{k=1}^nX_k-x_n|<+\infty)>0$

How can we prove that $\limsup_n \frac{1}{n}|X_n-Y_n|<+\infty$ $a.s$.

By Kolmogorov 0-1 law, I proved that there exists $c \in \mathbb{R^+}$ such that $\limsup_n|\frac{1}{n}\sum_{k=1}^nX_k-x_n|=c$ $a.s,$ but I don't know if this is helpful or not.

Let's suppose we have two graded coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there is an isomorphism $$\mathrm{Gr}(C_1\prod C_2)\cong \mathrm{Gr}(C_1)\prod\mathrm{Gr}(C_2)$$ where the filtration in $C_1\sqcup C_2$ is $F_k(C_1\prod C_2)=\displaystyle\sum_{p+q=k}{F_pC_1\prod F_q C_2}$. I saw the dual result for k-algebras in page 190 of the book Cogroups and Co-rings in Categories of Associative Rings, but without proof. Any suggestion, please?.

Let $\{v_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $V$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $V_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m}$

Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $U$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $U_m = \overline{\operatorname{span}} \{u_n\}_{n \geq m}$

Under the hypothesis that $\forall m \in \mathbb{N}: V_m + U_m$ is closed, is it true that:

$$ \bigcap_{m=1}^\infty \left( V_m + U_m \right) = \{0\} $$

In this *Inventiones Mathematicae* paper, Fischer-Colbrie proved the following result (Proposition 1):

**Proposition:** Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$ has finite index, there exists a compact set $C \subseteq M$ such that $M\setminus C$ is stable and there exists a positive function $u$ on $M$ such that $L u = 0$ on $M\setminus C$, where $L$ is the stability operator coming from the second variation of the area functional.

My question is if this statement is true in any dimension (assuming codimension $1$). I'm reading the proof and it seems to me that the argument is independent from the dimension, but maybe I'm wrong.

Any help will be very appreciated!

Can someone give a reasonably explicit example of an irreducible one-dimensional scheme with no closed points?

Having run into several references, at various places and occasions, to "Serre’s Course at Collège de France, 19xy-19xy+1" for various values of xy, I would genuinely like to know where these lectures are written down and archived.

Has there been an effort to preserve them in any form?

There are overviews of them in Serre’s collected work, but they have no references and don’t give full proofs.

Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ has no a holomorphic extension to any open neighborhood of $\bar D$?

Let $X=(X_1, X_2, \dots, X_n)$ be a smooth vector field on $\mathbb{R}^n$. The operator $L=(\sum_{i=1}^{m}X_i^2)^p$, where $p$ is an integer, is a degenerated operator. If $X$ satisfies the Hörmander condition, for the case of $p=1$, we have the subelliptic estimates

$$|u|_{s+a}\leq |Lu|_{s} + |u|_{L^2}$$

for $u \in L^2(\Omega)\cap C^{\infty}_0$; here $|\cdot|_{s}$ denotes the Sobolev norm.

Now I want to know what if $p\geq 2$? Moreover, what about general case? i.e. Rockland operator (i.e. higher order operator). I want to get the estimate like the case $p=1$.

Given a (ternary) quadratic form over $\mathbb{Z}$ how can I find all quadratic forms (up to equivalence over $\mathbb{Z}$) in the same genus?

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to the interior of the unit circle. These fragments date from 1839, much later then his better-known article from 1822 on conformal mappings. The relevant pages from Gauss's nachlass are in volume 10-1, p. 311-320. Schlesinger comments on these fragments of Gauss on p.192 of his essay, where he mentions that his results match the formula found much later by Hermann Schwarz.

These fragments are very noteworthy, first of all because they seem to anticipate the Riemann's mapping theorem (why did he attempt to find such a solution at all?), and secondly because they introduce powerful mathematical tools for solving the problem of explicit construction of conformal mapppings to the unit circle (this problem seems to be even more complex than the so-called Schwarz-Christoffel maps for mappings of polygonal interiors by elliptic integrals) .

I've already added a very partial answer at HSM stack exchange, of which i'm not very satisfied. According to several articles i found, this is a very dificult problem and surprisingly i didn't find any comment on Gauss's solution to it in the literature (except Schlesinger's comment). So i'll be glad if anyone will explain what is going on there in Gauss's writings from a modern viewpoint.

A left shelf $(S, \rhd)$ is a magma with the left self-distributive law: $$ \forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z). $$ Shelves are generalization of racks and quandles from the knot theory.

I am looking for examples of shelves with the following additional axiom: $$ \forall x, y \in S: x \neq y \implies (x \rhd y = y \iff y \rhd x \neq x). \tag{$\star$} $$ In particular, what kind of knots correspond to quandles with this additional property?

**Update.**
One possible example might be $(\mathbb Z, \rhd)$ with the operation $\rhd$ defined as follows:
$$
\forall x, y \in \mathbb Z: x \rhd y = \cases{
y + 1 & when $x < y$, \\
y & otherwise.}
$$
However, I am not sure if such $(\mathbb Z, \rhd)$ is a quandle that satisfies ($\star$).

Let $\{c_i\}_{i=1}^n$ be a sequence of real numbers such that $c_i \geq 0$ for each $i$ and $\sum_{i=1}^n c_i = 1$. Let $\omega_i \in [\delta, \Delta]$ for each $i$, where $\delta$ and $\Delta$ are strictly positive reals. Define $$f(t) = \sum_{i=1}^n c_i \sin(\omega_i t)$$ for $t \in [0, \infty)$, and let $t'$ be the smallest non-zero value of $t$ for which $f(t) = 0$.

- How can we show that there exists a constant $T > 0$ such that $t' \leq T$ for all possible choices of $c_i$ and $\omega_i$?
- Can we give upper/lower bounds on $T$ in terms of any of the other variables?
- In particular, how can we determine what, if any, is the dependence of $T$ on $n$?

For the first question above, someone has suggested using compactness (of *what*, I'm not certain), along with the claim that $$\lim_{\overline T \rightarrow \infty} \frac{1}{\overline T}\int_0^{\overline T} f(t) = 0,$$ but I'm not sure yet what to do with these ideas. It's possible that I misheard the person, but I'm not able to reach this person for further comment.