Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ Note that for distinct $\frac{a}{q}, \frac{a'}{q'}\in A$, we have $$\left\lVert \frac{a}{q} - \frac{a'}{q'}\right\rVert\ge\frac{1}{qq'}\ge\frac{1}{X^2},\quad\text{where}\quad\lVert\beta\rVert := \min_{n\in\mathbb{Z}} |\beta - n|.$$ It follows that in any interval of length $X^{-2}$ in $[0, 1]$, there are at most $O(1)$ elements of $A$, and this is clearly the best result possible, since $|A|\gg X^2$. Is it possible to get better results for $B$, since we have that $|B|\asymp X^2 / \log X$ by the prime number theorem. In particular, is it possible to get a bound of $o(\log X)$ for the number of elements of $B$ in an interval of length $$|B|^{-1}\ll X^{-2}\log X?$$ Note that the inclusion $B\subseteq A$ gives the trivial bound $O(\log X)$.

This is a cross-post to the question I asked at MSE over almost a month ago.

Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be continuous and $T(\lambda)$ has constant rank $m$ for every $\lambda \in \mathbb C$. Suppose for every $\lambda \in \mathbb C$, there is an open neighborhood $U(\lambda)$ of $\lambda$ and a locally defined continuous function $h_{\lambda}: U(\lambda) \to \mathcal M(n \times m; \mathbb C)$ where the image $h_{\lambda}(\beta)$ is a basis for $T(\beta)$ for every point $\beta \in U(\lambda)$. Could we construct a globally continuous function $\phi: \mathbb C \to \mathcal M(n \times m; \mathbb C)$ by gluing together these locally defined $h_{\lambda}$'s, such that the image $\phi(x)$, of every point $x \in \mathbb C$, is a basis for $T(x)$?

What I have in mind: Suppose we have two continuous families of $g_1: \mathbb C \supseteq U_1 \to \mathcal M(n \times m; \mathbb C)$ and $g_2 : \mathbb C \supseteq U_2 \to \mathcal M(n \times m; \mathbb C)$ where $U_1$ and $U_2$ are two open disks in $\mathbb C$ with $U_1 \cap U_2 \neq \emptyset$. Then $(g_1^1(x), \dots, g_1^m(x))$ is a basis for $T(x)$ for all $x \in U_1$ and $(g_2^1(y), \dots, g_2^m(y))$ is a basis for $T(y)$ for all $y \in U_2$ where $g_i^j: x \mapsto \mathbf c_j( g_i(x))$ where $\mathbf c_j(\cdot)$ denotes the operation of taking the $j^{th}$ column of a matrix. Now for every $y \in U_2 \cap U_1$, ${g}_1 (y) = g_2(y) S(y)$ for some $S(y) \in GL_m(\mathbb C)$. It follows $S(y) = (g_2^T(y) g_2(y))^{-1} g_2^T(y) g_1(y)$. By Cramer's rule, $S(y)$ is continuous with respect to $y$. That is we have a well defined continuous function on $U_1 \cap U_2$, $S \colon U_1 \cap U_2 \to GL_m(\mathbb C)$. Since $U_1 \cap U_2$ is simply connected, there is a continuous retract $r :\mathbb C \to U_1 \cap U_2$. It follows $S \circ r : \mathbb C \to GL_m(\mathbb C)$ is well defined and continuous with $S \circ r|_{U_1 \cap U_2} = S$. So we can define $g$ by \begin{align*} g(x) = \begin{cases} g_1(x), & \text{ if } x \in U_1 \setminus U_2 \\ g_2(x)(S\circ r)(x) , & \text{ if } x \in U_2. \end{cases} \end{align*} This function is clearly continuous by construction and satisfies the prescribed condition.

I am not sure whether above argument is completely correct. Even if it was correct, my problem to proceed is: after we glue some collection of maps, the domain $U$ would become not so "regular". I am not sure the intersection of $U$ and an open disk would still be simple connected.

Can anybody give reference what are the best institutes all over the world for doing von Neumann algebras and where the groups are strong, and also the procedure for applying Ph.D?

A matrix $G=\left[ \begin{array}{cc} A & B \\ C & 0 \\ \end{array} \right]$, where $A$ is a Metzler matrix, $B$ and $C$ is a non-negative matrices, namely the matrix $G$ is a a Metzler matrix as above special form.

My question is: does the statement hold?:

1.There exists at least a positive eigenvalue for the special matrix, namely $G \not=$ Hurwitz.

Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{gl}_N(k)$ be (the $k$-points of) its Lie-algebra.

Is anybody aware of an accessible reference where I could find a classification of the orbits for the adjoint action of $G$ on $\mathfrak{g}$, including orbit sizes?

Other exceptional groups over $k$ are also of interest to me, so if such a reference for them also exists I'd be happy to hear about it.

Thank you!

Let $A$ be a $*$-ring. Let us have some points:

i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.

ii) On the set of projections, we write $p\leq q$ if $pq=p$.

iii) A projection $p$ is called ** strongly finite** if there exist at most finitely many projections $q_1,\cdots,q_n$ with $q_j\leq p$.

iv) For a given $x\in A$, we put $l(x)$ to be the smallest projection with $l(x)x=x$.

Q. Assume that $e$ is an strongly finite projection. Let $p$ be a projection. Is $l(ep)$ an strongly finite projection?

Remark. In this discussion, we assumed $l(x)$ exists for every element $x\in A$. For example $A$ may be assumed a Baer *-ring.

How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.

One way I can think of is to give a map to $Y_{n-1}=\{(y_1,y_2,\ldots , y_{n-1})\in {(\mathbb C^*)}^{n-1}\ |\ y_i\neq y_j\ for\ i\neq j\}$ which is a locally trivial fibration. After replacing $\mathbb C$ by $\mathbb R$ and for $n=2$, I drew a picture and it seems a suitable map should be $(x_1,x_2,\ldots , x_n)\mapsto (\frac{x_i-x_n}{2},\ldots , \frac{x_{n-1}-x_n}{2})$.

But then how does one check that this map is a locally trivial fibration? $Y_n$ is aspherical by taking projection and applying long exact homotopy sequence and induction. Using a result of Fadell-Neuwirth the projections in the case of $Y_n$ are locally trivial fibrations.

Let $K$ be a finite extension of $Q_p$.

Is the centraliser of $Gal(\overline{K}/K)$ in $Gal(\overline{Q_p} / Q_p)$ trivial ? If yes, how can I show it ?

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$ The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ the map $t \mapsto A(t)x$ is continuous.

Consider the initial value problem

$$\varphi'(t)=A(t)\varphi(t)$$

with $\varphi(0)=\varphi_0.$

Assume that there is a dense subspace $X$ in $H$ that is contained in the domain of $A(t)$ for all $t$ and $A(t)X \subset X.$

I ask:

Let $\varphi_0 \in X$. Does this imply that $\varphi(t) \in X$ for all $t>0?$

This sounds very natural but I do not have any tools/ideas to show this at the moment.

If $G=N\rtimes H$ is the semi-direct product of two finite groups, and $|N|,|H|$ are coprime, then if we know the Schur multiplier of $N$ and $H$ then we can find the Schur multiplier for $G$.

Let $N=C_n^n/C_n$, where $C_n$ is the cyclic group on $n$ elements and we are modding out by the diagonal copy of $C_n$, and let $H=S_n$ act on $N$ via permutations (if it is easier, assume that $n$ is prime). In this case $|N|$ and $|H|$ are never coprime ($n$ divides both), but is it still possible to compute the Schur and Bogomolov multipliers, or at least to understand if they are trivial/non-trivial?

If I did not divide by the central $C_n$, the answer would be well known, see the chapter on wreath products of Karpilovsky's book on group representations.

If $W_1,W_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d_1, d_2 \leq d$, respectively, then $d_1 + d_2 > d$ suffices to guarantee $W_1 \cap W_2 \neq \{0\}$. There are similar results for affine subspaces of a spaces of a $k$-vector space, $E_1,E_2 \subset V$.

I'm looking for analogous results for submodules of a free $R$-module $N_1,N_2 \subset M$ of finite ranks $r_1,r_2 \leq r$, and for "affine submodules" (i.e. torsors/cosets/translates of submodules), $A_1,A_2 \subset R$.

The main question I'm interested in is **what can be said about the intersections** $N_1\cap N_2, A_1\cap A_2$ (i.e. are they nonzero or nonempty?) given certain conditions on the subspaces (i.e. they are (translates of) submodules of certain rank). I currently have in mind $\mathbb{Z}$-modules, but I am also interested in modules over other rings (probably all Noetherian). In addition to intersections, I'd be interested to read more general information. The notion of greatest common divisor is evidently at play here.

I would be interested in **good references/resources** or in **direct statements/proofs** of useful results.

E.g. Consider the translates of rank-2 $\mathbb{Z}$-modules $$A_1 = (u_1,v_2,w_1)+\langle (a_1,b_1,c_1),(a_2,b_2,c_2) \rangle$$ $$A_2=(u_2,v_2,w_2)+\langle (a_3,b_3,c_3),(a_4,b_4,c_4) \rangle \subset \mathbb{Z}^3$$ What is $A_1 \cap A_2$?

Lastly I have a **terminology question**, relating to an important distinction between the module case and the vector space case. Is there a name for the property of a submodule being maximal with respect to its rank? ("primitive"?) For example $\langle (2,4) \rangle \subset \mathbb{Z}^2$ is a free rank-1 submodule, but it is properly contained in the free rank-1 submodule $\langle (1,2) \rangle$, which is not properly contained in another rank-1 submodule.

Thank you.

A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.

In the theorem as stated, $\mathbb{F}$ is any field and $T_n(\mathbb{F})$ denotes the algebra of upper triangular $n\times n$ matrices over $\mathbb{F}$.

**Theorem:** Let $A,B\in T_n(\mathbb{F})$ be such that for all $X\in T_n(\mathbb{F})$, $$AX=XA\implies BX=XB$$ Then $B=p(A)$ for some $p\in \mathbb{F}[t]$.

Does anyone know of a source for this result? I have searched Google, MSE, MO, and the like to no avail.

If we replace $T_n(\mathbb{F})$ by $M_n(\mathbb{F})$, the question is answered in this paper. Unfortunately, the argument doesn't seem to translate directly, as I can't find a way to force the $M_i$ maps to be upper-triangular.

Also, I have already asked this question here on MSE. As the question is for an undergraduate research project, it felt appropriate to ask it here as well.

Thanks for any help!

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible representations of $\operatorname{GL}_n(F)$ over a $p$-adic field $F$ as unique quotients of certain induced representations coming from supercuspidals, in an essentially unique way.

It also allows us to obtain these representations as unique subrepresentations in an essentially unique way.

Suppose $\pi$ is a smooth irreducible representation of $\GL_n(F)$, and we know how to obtain $\pi$ as a quotient $Q(\Delta_1, ... , \Delta_r)$ in the B.Z. classification. Then do we also know how to obtain $\pi$ as a subrepresentation $Z(\Delta_1', ... , \Delta_{r'}')$ in B.Z. classification?

Example: suppose $G = \operatorname{GL}_2(F)$, and $\chi = \chi_1$ is a character of $F^{\ast}$, $\chi_2(x) = \chi_1(x)|x|_F$. Then $\Ind_{TU}^G \chi_1 \otimes \chi_2$ has a unique irreducible quotient $\pi$. If we want to get $\pi$ as subrepresentation instead of a quotient, then we swap $\chi_1$ and $\chi_2$ and use $\Ind_{TU}^G \chi_2 \otimes \chi_1$.

The problem is this: given a graph $G$, to find a decomposition of $G$, i.e. a set F of disjoint proper subgraphs of $G$ such that:

$$\text{inertia}(G) = \sum_{H \in F} \operatorname{inertia}(H)$$

where $\operatorname{inertia}(G)=(a,d,c)$, with $a$ equals the number of negative eigenvalues of $A(G)$, the adjacency matrix of $G$, $b$ equals the dimension of null space of $A(G)$, and $c$ equals the number of positive eigenvalues of $A(G)$. I can not find any paper about this problem. Does anyone know something about this problem?

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now the bracket: $$[X,Y]_a= a^{-1}[a(X),a(Y)]$$ A simple calculus shows that it satisfies the Jacobi identities, so $(F,[,]_a)$ is a new Lie algebra. My question is: when does this new Lie bracket come from the vector fields of a manifold $M_a$? For example, if I take $a=g_*$, with $g$ a diffeomorphism of $M$, I could say that $M_{g_*}=M$. Perhaps that a condition of a certain smoothness could be added for the automorphism $a$?

Let $g(x) = e^x + e^{-x}$. For $x_1 < x_2 < \dots < x_n$ and $b_1 < b_2 < \dots < b_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$:

$\det \left (\begin{bmatrix} \frac{1}{g(x_1-b_1)} & \frac{1}{g(x_1-b_2)} & \cdots & \frac{1}{g(x_1-b_n)}\\ \frac{1}{g(x_2-b_1)} & \frac{1}{g(x_2-b_2)} & \cdots & \frac{1}{g(x_2-b_n)}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{1}{g(x_n-b_1)} & \frac{1}{g(x_n-b_2)} & \cdots & \frac{1}{g(x_n-b_n)} \end{bmatrix} \right ) > 0$.

Case $n = 2$ was proven by observing that $g(x)g(y) = g(x+y)+g(x-y)$, and $g(x_2 - b_1)g(x_1-b_2) = g(x_1+x_2 - b_1-b_2)+g(x_2-x_1+b_2-b_1) > g(x_1+x_2 - b_1-b_2)+g(x_2-x_1-b_2+b_1) = g(x_1-b_1)g(x_2-b_2)$

However, things get difficult for $n \geq 3$. Any ideas or tips?

Thanks!

Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).

If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary decomposition $$ H^k(G;M)=\bigoplus_p H^k(G;M)_{(p)} $$ for each $k>0$, where $p$ ranges over the primes dividing $|G|$ and $H^k(G;M)_{(p)}$ denotes the $p$-primary component of $H^k(G;M)$. It follows that if $H^1(G;M)$ is nonzero, then $H^1(P;M)$ is nonzero for some $p$-Sylow subgroup $P$ of $G$. See Section III.10 of K.S. Brown's "Cohomology of Groups".

If $M$ is nonabelian, then $H^1(G;M)$ is still defined, but is not a group in general, so that asking for a decomposition such as above would not make any sense. But one could still ask the following weaker question:

With $G$ and $M$ as above, suppose that $H^1(G;M)$ is non-trivial. Then must $H^1(P;M)$ be non-trivial for some $p$-Sylow subgroup $P$ of $G$?

**Edit:** Tyler Lawson's answer below is very informative, but unfortunately is not complete because the cohomology of a free product is not necessarily isomorphic to the direct product of the cohomologies. So the question still seeks an answer.

**Question:** Do there exist amenable Thompson-like groups?

I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an independent field of research in the litterature, so my question should make sense.

Among the examples of such groups I know, typically either they contain a non-abelian free group or they contain Thompson's group $F$. In the former case, of course the group is not amenable; and in the latter case, the amenability of the entire group would imply the amenability of $F$, so a proof of the amenability should not be available.

Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces.

For example, if $X$ is an affinoid $K$-variety, then there is a tangent sheaf $\mathcal{T}_X$ on $X_{\text{rig}}$, with sections $$\mathcal{T}_X(U) = \text{Der}_K(U)$$ over affinoid subdomains $U$. It gives rise to a sheaf of $\mathcal{O}_X$-algebras $\mathcal{A} = \text{Sym}_{\mathcal{O}_X} \mathcal{T}_X$. There is then a "relative analytification'' space $Y = \text{Spec}^{\text{an}} \mathcal{A}$, as described by Conrad in Section 2.2 here.

As defined, this $Y$ should be the correct candidate for the cotangent space $T^*X$, and so one might try to make sense of a symplectic structure on $Y$. However, I haven't read anything about tangent spaces or symplectic forms in the rigid setting. Is that because these notions are problematic or because they are obvious? Should tangent spaces to points on $X$ be defined in the same way as if $X$ were a scheme? Thanks for any insight or pointers to the literature.

Let $E$ be vector bundle over smooth scheme $X$. Thom space of $E$ is $Th(E)=E/E-i(X)$ where $i\colon X \longrightarrow E$ is zero section. This space is $\mathbb{A}^{1}$ isomorphic to $\mathbb{P}(E \oplus \mathscr{O})/\mathbb{P}(E)$

As same as in algebraic topology, Thom class of vector bundle $E$ (it is denoted $t_{E}$) was defined in motivic homotopy theory and satisfied the property as follow

(Th) Thom class give the isomorphism $\tilde{H}^{*,*}(F_{･} \wedge X_{+}) \simeq \tilde{H}^{*+2\dim E,*+ \dim E}(F_{･} \wedge Th(E)) $

In Reduced power operations in motivic cohomology, V.Voevodsky was defined Thom class. But I don't understand it. So I think definition of Thom class as follow.

Natural monomorphism between vector bundles $f \colon E \longrightarrow E\oplus \mathscr{O}$ give morphism $\mathbb{P}(f)\colon \mathbb{P}(E) \longrightarrow \mathbb{P}(E \oplus \mathscr{O})$ and $\mathbb{P}(f)^{*}(\mathscr{\sigma}_{E\oplus\mathscr{O}})=\sigma_{E}$ where $\sigma_{E}(resp. \sigma_{E\oplus \mathscr{O}}) =\mathscr{O}(-1) \in H^{2,1}(\mathbb{P}(E),\mathbb{Z})(resp.H^{2,1}(\mathbb{P}(E\oplus \mathscr{O}),\mathbb{Z}))$ if $\dim E=d $, then since $\{1,\sigma_{E},\cdots \sigma_{E}^{d-1}\}$ and $\{1,\sigma_{E\oplus \mathscr{O}},\cdots,\sigma_{E\oplus \mathscr{O}}^{d}\}$ are basis of $H^{*,*}(\mathbb{P}(E),\mathbb{Z})$ and $H^{*,*}(\mathbb{P}(E\oplus \mathscr{O}),\mathbb{Z})$, $\mathbb{P}(f)^{*}$ is epimorphism and dimension of its Kernel 1. so Thom class $t_{E}$ of $E$ defined as genereted element in its Kernel. As $Th(E) \simeq \mathbb{P}(E\oplus \mathscr{O})/\mathbb{P}(E)$ it belong to $H^{*,*}(Th(E),\mathbb{Z})$.

Question. Is this definition right? if isn't please tell me true def of thom class.