# All Questions

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### Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$). Is every involution in $\mathrm{GL}_n(A)$ diagonalisable? This is of ...
255 views

### Hartshorne's proof of Halphen's theorem

Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty. Hartshorne states the theorem as follows: ...
55 views

### Negation-quantifier-negation blocks in nonclassical logic: reference request

I'm looking for references to discussion of a certain question in the literature on non-classical first-order logics. I suspect it must have been investigated thoroughly, but I can't seem to find ...
55 views

### What is a general way to express the volume of some subset of $\mathbb{R}^n$ [closed]

I saw a theorem which states that Let $B \subset \mathbb{R}^n$ be any subset of $\mathbb{R}^n$, then it is true that for any $c > 0$, $$\text{volume}(cB) = c^n\times \text{volume}(B)$$ How is ...
43 views

### Virtually abelian fundamental groups equivalent to nonnegative curvature

This is a follow up question inspired by Fundamental groups of compact manifolds with non-negative Ricci curvature. In dimensions 3 and 2 (and 1) a manifold has a virtually abelian fundamental group ...
94 views

### On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
121 views

### Hodge's conjecture as a quasi-isomorphism between two complexes of sheaves

A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective ...
203 views

### Best explicit bound on $\zeta'(1+it)/\zeta(1+it)$

Assume the Riemann hypothesis. We know that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$ (see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound ...
63 views

### Generalizations of the idea of automorphic

The notion of an automorphic form/representation (and sometimes, of Langlands program in tandem) has been extended in many directions - from arithmetic to geometric to topological - but two versions I ...
76 views

### Can the injective envelope ever be injective for $*$-homomorphisms?

The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
18 views

234 views

### example of "really" non-existent transferred model structure

I am looking for an example where a transferred model structure fails to exist, even if one is willing to work with semi-model category. But let me be more precise: Let's say I have a combinatorial ...
184 views

### Messing around with $e+\pi$

This question originates from the conjecture that $e+\pi$ is transcendental, and that $e$ is conjectured not to be a period. Jianming Wan in his paper Degrees of periods states that the transcendence ...
21 views

### Constant in Brascamp-Lieb inequality being $1$ when reduced to Loomis-Whitney inequality?

The question is basically that, since I heard that the Loomis-Whitney inequality is a special case of the Brascamp-Lieb inequality, I would like to check the constant factor in B-L inequality is ...
77 views

### Subset which maximizes $\frac{\int_E\min(p(x), q(x))}{\int_E\max(p(x), q(x))}$?

Let $p(x), q(x)$ be two p.d.f.s of distributions on $\mathbb{R}$. I am interested in finding the subset $E$ that maximizes the quantity \frac{\int_{E}\min(p(x),q(x))\mathrm{d}x}{\int_{E}\max(p(x),q(...
49 views

### Clarifications involving automorphisms of projective planes and lines?

I have been learning some classical projective geometry recently and I am hoping to gain some clarity regarding various different automorphism groups. There are three different levels of generality ...
515 views

### Conjectures inspired by AI

Today in Nature a paper described how AI guided mathematicians to make highly non-trivial conjectures, which they managed to prove, one in Knot Theory involving a new invariant, the other in ...
Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...