Consider $\mathrm{SO}(5)$, or maybe $\mathrm{SO}(n)$ over your favorite locally compact non-Archimedian field of characteristic $0$. There are two interesting families of compact open subgroups. The first is those coming from the Moy-Prasad filtration. The second are the stabilizers of lattices. My question is as follows: is it the case that a lattice stabilizer is a subgroup of an arbitrary step in the Moy-Prasad filtration?

I know the top of the Moy-Prasad filtration is a maximial parahoric, which is known (by Gan-Yu) to be a subgroup of the stabilizer of a lattice. But I don't know much more than that.

By an algebraic number expressed in radicals, I mean one that is an element of a set $S$ characterized as follows:

- The integers are in $S$.
- For any $a$ and $b$ in $S$, $a+b$ and $a·b$ are in $S$.
- For $a$ and $b$ in $S$ with $b\neq0$, $a/b$ is in $S$.
- For $a$ and $b$ in $S$, not both $0$ and $b$ rational, $a^b$ is in $S$.

For example, given an expression like $\sqrt{\sqrt[3]{2}+\sqrt[5]{3}}+1$, what is an algorithm that can be used to find similar expressions for all of its conjugates? Since its conjugates are by definition the roots of its minimal polynomial, the obvious approach is to find the minimal polynomial, then split it. Is that the way to do it? If so, I have more questions about both parts of the process, which I'll add in edits.

Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?

**Motivation**: I'm trying to find the first passage time distribution from a master equation. I can impose an absorbing boundary at the threshold $n$, and the master equation with the new boundary condition is of the form $\frac{dp}{dt}=A p$ for a tridiagonal $n\times n$ matrix $A$. Then the first passage time distribution can be written as a particular matrix element of $\exp(A\,t)$. It takes forever for Mathematica to compute $\exp(A\,t)$ for large $n$, so I was wondering if there is a way to compute only the desired matrix element and not the whole matrix.

Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{R}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{R}^{n \times n}$ we denote the operator norm of $A$ as an operator on $\mathbb{R}^n$ by $\|A\|_p$, too. Moreover, let $|A|$ denote the matrix whose entries are the absolute values of the entries of $A$.

The number $\| \,|A|\,\|_p$ is sometimes called the **regular norm** of $A$ (in particular in Banach lattice theory, where a complete norm on the space of regular operators is constructed this way).

We clearly have $\|A\|_p \le \| \,|A|\,\|_p$, and equality holds for $p = 1$ and $p = \infty$. For general $p$, Mark Meckes' explained in his answer to this question that the estimate \begin{align*} \| \,|A|\,\|_p \le n^{\frac{2}{p}(1 - \frac{1}{p})} \|A\|_p \tag{E} \end{align*} holds as a consequence of the Riesz-Thorin theorem. This estimate is sharp for $p = 2$, at least if $n$ is a power of $2$; this is mentioned in Mark Meckes' post quoted above, and it can also be found in [Schaefer: Banach Lattices and Positive Operators (1974), Example 1 on page 231]. Obviously, $(\text{E})$ is also sharp for $p = 1$ und $p = \infty$.

My question is:

**Question 1.** Is the estimate $(\text{E})$ sharp for $p \in (1,\infty) \setminus \{2\}$?

In case that the answer to Question 1 is "no", I would like to ask:

**Question 2.** What is the best known constant (in dependence of $n$) in $(\text{E})$? Is the optimal constant known?

Note: A related question concerned with the Schatten $p$-norm of a matrix can be found here.

If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then $$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$

Let's say we are studying rotationally symmetric solutions to parabolic equation: perhaps $(\partial_t - \Delta) u = A(|x|, u)$. We end up studying \begin{equation} (\partial_t - \partial_r^2) g = n r^{-1}\partial_r g + A(r, g). \label{test}\tag{1} \end{equation} For differentiability of $u$ we will have a a Neumann boundary condition for $g$ at $r=0$.

Now, if we have some standard regularity theorem valid in $\mathbb{R}^{n+1}$ we can apply it to $u$ to learn something about $g$. For instance, if $A$ is a $C^{\infty}_{loc}$ function of $g$ and we have an a priori bound $c < g < C$, then we get interior estimates for derivatives of $g$. This is despite the apparently bad coefficient $r^{-1}$ on $\partial_r$ in \eqref{test}.

I'd like to know if that is clear from \eqref{test} directly. In my situation I have a larger system, and when deriving equations for various quantities derived from the system I end up with coefficients like $n + 4$ on $r^{-1}\partial_r g$. Ideally, I would not like to look silly for saying "consider this function rotated in $\mathbb{R}^{n+5}$" if there is a standard way to deal with regularity for \eqref{test}. Furthermore, that trick would not work for $$(\partial_t - \partial_r^2)g = \alpha r^{-1}\partial_r g + A(r, g)\label{alpha}\tag{2}$$ if $\alpha$ is not a natural number.

To have a concrete question,

Suppose $A$ is smooth, $\alpha \geq 0$, and $g$ satisfies \eqref{alpha} with a bound $|g| < C$ for $(r,t) \in [0, 1] \times [0,1]$. Is there a $C'(A, \alpha)$ such that $|\partial_r g| < C'$ for $(r,t) \in [0, 1/2] \times [1/2, 1]$?

Note that for $A = 0$ the equation has a solution $$g = \frac{1}{-\alpha + 1}r^{-\alpha + 1}.$$ If $\alpha < 0$ is not an integer, then this satisfies the Neumann condition but is not smooth. That is why I ask for $\alpha > 0$.

Ideally Id like an answer that comes from a general principle that lets us see the level of regularity we should expect.

where $r_{min}$ is the root of the denominator. $V(r)$ is a sum of $ae^{br}$ terms, and the rest is constant.

I tried some naive solutions, but the problem is that the thing to integrate (let's call it $Y$) approaches infinity when $r$ approaches $r_{min}$. So mathematically, this integral is supposed to converge (or is it?), but numerically it's ill-posed.

In the naive approach, I find $r_{min}$ numerically, but this gives a very high Y at the beginning of the integration (that should be $\pm \infty$ if $r_{min}$ was exact), and then I try to integrate with very small trapezoids, but it seems this approach is fundamentally flawed.

So... I should probably transform this integral, changing variables and stuff, but I don't know where to begin...

Is there a known method to compute this ?

Let $E$ be a $spin^c$ bundle and $L_E$ be a (complex) line bundle defined using transition functions $\nu \circ g_{U,V}$ where $\nu:spin^c(n) \to \mathbb{T}$ is map such that $\ker \nu=spin(n)$ and $g_{U,V}$ are transition functions for $spin^c(n)$-principial bundle $spin(E)$ (see also here). Let $S$ be a spinor bundle i.e. vector bundle constructed from the system of transition functions $c \circ g_{U,V}$ where $c$ is irreducible representation of Clifford algebra.

How to prove that the bundle $L_E$ satsifies the following $S \otimes L_E \cong S^*$ (where $V^*$ denotes the dual bundle).

I tried to show this using tranistion functions: the best situation would be if transition functions from both sides exactly coincide. This is equivalent to $$c(g_{U,V}(x)) \otimes \nu (g_{U,V}(x))=([c(g_{U,V}(x))]^t)^{-1}.$$

Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider the map $\nu:spin^c(n) \to \mathbb{T}$ defined by $\nu(w)=w^!w$ where $(v_1 \cdot ... \cdot v_r)^!=v_r \cdot ... \cdot v_1$. We can form the composition $\nu \circ g_{U,V}:U \cap V \to \mathbb{T}$. As this satisfies cocycle property we can form the *line* bundle $L_E$.

Let us assume that $E$ is $spin^c$ but is *not* spin.

How to prove that the first Chern class of $L_E$ is *odd* (in the sense that $j^*(c_1(L_E)) \neq 0 $ where $j^*$ is mod 2 reduction of coefficients)?

Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$.

Does there necessarily exist an equivalence relation $\thicksim$ on $X$ such that $\pi_1(X/\thicksim)$ is isomorphic to $\pi_1(X)/N$?

Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.

Let $M\in \mathcal{B}(F)$ be a bounded operator. Suppose

that $M\in \mathcal{B}(F)^+$, i.e., $\langle Mx,x\rangle\geq0$ for all $x\in F$, and

that $M$ is an injective operator on $F$.

Consider $$S_M(0,1)=\{x\in F:\;\langle Mx, x\rangle=1\}.$$ Is $S_M(0,1)$ always homeomorphic to the 1-sphere $S(0,1)$?

A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ which contains $x$.

The faces of the positive semidefinite cone $H_+$ in the real vector space of Hermitian matrices are well characterized, and we know that $F(H_+, A) = \left\{ B : \text{ker}(A) \subset \text{ker}(B) \right\}$.

I am interested in the faces of subsets of this cone of the form

$$ C_p = \text{conv} \{ vv^* : \|v\|_p \leq 1 \} $$

where conv denotes the convex hull and $\|\|_p$ is any $\ell_p$ norm. I am specifically looking at $p=1$ and $p=\infty$, but more general results could also be interesting.

In particular, are there some ways to get lower bounds on the dimensions of the corresponding faces?

An interesting corollary of the characterization of the faces of $H_+$ is that we have $\text{dim}\, F(H_+, A) = \text{rank}(A)^2$, which means that there are faces of dimension 1 (the extreme rays generated by rank-one matrices) and faces of dimension 4 (generated by rank-two matrices), but no faces of dimension 2 or 3. Can anything similar be said about the sets $C_p$?

Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing notion $\mathbb{P}_G$: for $A, B \in \mathcal{B}$ let $A \sim B \iff m(A \bigtriangleup B) = 0$, and let $\mathbb{P}_G$ consists of equivalence classes $[A]_\sim$ where $A \in \mathcal{B}$ and $m(A)>0$, with $[A]_\sim \leq [B]_\sim$ iff $m(A\setminus B)=0.$

Now define an equivalence relation $\equiv$ among infinite compact Abelian groups by

$G \equiv H \iff \mathbb{P}_G$ is forcing equivalent to $\mathbb{P}_H$.

Now my question is the following:

**Question 1.** What non-trivial facts one can say about $\equiv$ relation$?$

As a sample of explicit question, one may ask the following:

**Question 2.** Let $\kappa$ be an infinite cardinal and let $\mathfrak{g}_\kappa$ consists of compact abelian groups of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in \mathfrak{g}_\kappa \}?$

In general one can ask similar question for probability spaces. Let $(\Omega, \mathcal{F}, P)$ be an infinite probability space and assign to it, its natural forcing notion $\mathbb{P}_{(\Omega, \mathcal{F}, P)}$. Also define $\equiv_P$ relation between infinite probability spaces as above.

**Question 3.** What non-trivial facts one can say about $\equiv_P$ relation$?$

Similar to question 2, one may ask:

**Question 4.** Let $\kappa$ be an infinite cardinal and let $\mathfrak{p}_\kappa$ consists of probability spaces of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in \mathfrak{p}_\kappa \}?$

**Remark.** The equivalence of forcing notions is defined in the comments by Professor Hamkins.

Let $E$ be a complex Hilbert space. We recall that an operator $T\in\mathcal{L}(E)$ is said to be hyponormal if $[T^*, T]\geq 0$ (i.e. $\langle (T^*T-TT^*)x,x \rangle\geq 0$ for all $x\in E$). Let $E\overline{\otimes}E$ denotes the completion, endowed with a reasonable uniform cross-norm, of the algebraic tensor product $E\otimes E$.

Recall that a pair $A=(A_1,A_2)\in\mathcal{L}(E)^2$ is called hyponormal if $$\varphi(A)=\begin{pmatrix}[A_1^*, A_1] & [A_2^*,A_1]\\ [A_1^*, A_2 ]& [A_2^*, A_2] \end{pmatrix}$$ is positive on $E\oplus E$ (i.e. $\langle \varphi(A)x,x \rangle\geq 0$ for all $x\in E\oplus E$.

For $A=(A_1,A_2)\in\mathcal{L}(E)^2$ and $B=(B_1,B_2)\in\mathcal{L}(E)^2$ we consider $A\otimes B:=(A_1\otimes B_1,A_2\otimes B_2)$. Assume that $A,B\in\mathcal{L}(E)^2$ are hyponormal. It is true that $A\otimes B\in\mathcal{L}(E\overline{\otimes}E)^2$ is hyponormal?

Note that $T\otimes S\in \mathcal{L}(E\overline{\otimes}E)$ is hyponormal if and only if $T\in\mathcal{L}(E)$ is hyponormal and $S\in\mathcal{L}(E)$ is hyponormal.

In fact: Assume that $T$ and $S$ are hyponormal. This implies that $$(T\otimes S)^*(T\otimes S)=(T^*T)\otimes (S^*S)\geq (T\otimes S)(T\otimes S)^*.$$ Conversely we use the following result discovered by STOCHEL

Lemma: Let $A_1, A_2\in \mathcal{L}(E)$ and $B_1, B_2\in \mathcal{L}(E)$ be nonnegative operators. If $A_1\not=0$ and $B_1\not=0 $, then the following conditions are equivalent:

$A_1\otimes B_1\leq A_2\otimes B_2$.

There exists $c > 0$ such that $A_1 \leq cA_2$ and $B_1\leq c^{-1}B_2.$

Clearly $T\otimes S$ is hyponormal if and only if $T^*T\otimes S^*S\geq TT^*\otimes SS^*$. So by applying STOCHEL result, there exists $c>0$ such that $$cT^*T\geq TT^*\;\;\text{and}\;\;c^{-1}S^*S\geq SS^*.$$ A simple computation shows that $c=1$.

** Question:** Primes $p$ of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$. For example:

- $k=2$ then $p=31$.
- $k=3$ then $p=73$.
- $k=4$ then $p=157.$

Conjecture:

(1) None of the $p$'s is congruent to $6\pmod 7$. Example: 31 is not congruent to $6\pmod 7$, 73 is not congruent to $6\pmod 7$. With PFGW we calculated up to $k=125.800$ and all $p$'s aren't congruent to 6 (mod 7), thus we didn't find any counter-example. Can you find a counterexample or give a proof for the conjecture?

Here it is used in a sentence

It is therefore a priori probable that Plato πυθαγοριζει in the passage where he says that between two planes one mean suffices, but to connect two solids, two means are necessary. - Sir Thomas Heath: A History of Greek Mathematics, Volume I: From Thales to Euclid, page 89

I realize this is not an equation question, but this forum might seems an appropriate place to ask a question about words used in discussions of number theory

Let $I$ and $J$ be two ideal of a ring $R$ (commutative with $1$) such that $I\subseteq Ann_R(Ann_R(J))$ and $I$ is a principal ideal. Is there any conditions on $I$ or $J$ or both of them under which we can deduce that $I\subseteq J$?

A correlation matrix in $M_n(\mathbb{C})$ is a Hermitian positive semidefinite matrix whose diagonal elements are all equal to 1. I will call the set of all such matrices $\mathcal{E}_n$. Notice that we have

$$\mathcal{E}_n = \left\{ A : A\geq 0,\, \text{Tr}(A) = n,\, \|A\|_{\max} \leq 1\right\}$$

where $\|A\|_{\max} = \max_{i,j} |A_{ij}| = \max_i A_{ii}$ for positive semidefinite $A$.

Consider now the set $\mathcal{S}_n$, defined as the convex hull of rank-one correlation matrices, which we can express as

$$\mathcal{S}_n = \text{hull} \left\{ x x^* : \|x\|_{\infty} \leq 1,\, \|x\|_{2} = \sqrt{n} \right\}.$$

We then have $\mathcal{E}_n \supset \mathcal{S}_n$ and the inclusion is strict in general [1].

My question is as follows. If there exists a correlation matrix $A \in \mathcal{E}_n$ such that

$$A \in \text{hull} \left\{ x x^* : \|x\|_{\infty} \leq 1 \right\},$$

does it necessarily mean that $A \in \mathcal{S}_n$?

Equivalently: if a positive semidefinite matrix of trace $n$ is a convex combination of rank-one terms $xx^*$ with $\|x\|_{\infty} \leq 1$, is it necessarily a convex combination of rank-one terms with $\|x\|_{\infty} \leq 1$ and $\|x\|_{2} = \sqrt{n}$? The reason why I hope the above is true is that we have $$\left\{ A : A\geq 0,\, \text{Tr}(A) = n \right\} = \text{hull} \left\{ xx^* : \|x\|_{2} = \sqrt{n} \right\},$$ which makes the conjecture at least slightly plausible. Yet another way to express the question is then whether the extreme points of the set $$\text{hull} \left\{ xx^* : \|x\|_{2} = \sqrt{n} \right\} \cap \text{hull} \left\{ x x^* : \|x\|_{\infty} \leq 1 \right\}$$ are all rank one.

I was unfortunately not able to show this, and my numerical hunt for a counterexample was unsuccessful. I would appreciate any ideas for a possible proof / counterexample.

What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to correct me in the probably various places I am wrong)

I am trying to make some basic computations, like for example compute the whole algebra probably, $H_K$ (the left-invariant on $K$ distributions) and the center (which in the rest of literature is what is usually denoted by $H_K$ I think, essentially the bi-$K$-invariant distributions). Now the center can be computed by the Satake isomorphism (and it is isomorphic to $\mathbb{C}[z_1^{\pm},z_2^{\pm}]^{S_2}$). For the rest I really could not come up with an explicit computation.

Now searching around, I found that most sources do not define the Hecke algebra as the locally constant compactly supported distributions, but by a purely algebraic definition with some generators over the Weyl group. I really cannot understand this definition.

Can you provide me with some source that explains this explicit presentation of the Hecke algebra, and why is it the same as Bernstein's one?

Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. Explicitly, the homotopy types are $ét(T) = B\hat{\mathbb{Z}}$ and $ét(\tilde{T}) = *,$ for $*$ the point. In particular, the natural covering map $$\pi:\tilde{T}\to T$$ gives a basepoint (in a suitable homotopy sense) $$ét(\pi)$$ of $ét(T)$. Now group structure on $T$ lets us define a new point $\pi^2 := \pi*\pi,$ which is the composition of $\pi$ with the squaring map $[2]:T\to T$. While both $\pi, \pi^2:\tilde{T}\to T$ realize $\tilde{T}$ as a universal cover of $T$, they are not equal (as can be seen e.g. by looking at the map on tangent spaces at $1$). On the other hand, since $B\hat{\mathbb{Z}}$ is (I think?) connected, the maps $$ét(\pi), ét(\pi^2):ét(\tilde{T})\to ét(T)$$ should be homotopy equivalent in a suitable homotopy category.

**Question** is there a way to see the homotopy equivalence between $ét(\pi)$ and $ét(\pi^2)$ *explicitly*? Here by "explicitly", I mean as an interval in the mapping space between natural topological models, or an interval in some other model category. Edit: I'd also like for the functor from varieties to etale types to take etale maps to fibrations, so that in particular the etale types of $\pi, \pi^2$ are not a priori equal.