# All Questions

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### Iwasawa decomposition of the pseudo-orthogonal group

This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the ...
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### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
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### Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
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### Does normalization of projective varieties preserve very ampleness

Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...
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### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
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### Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...
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### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
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### Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$. We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...
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### Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem \begin{align} \lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\ &r_i\geq ...
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### identifiability of a linear regression [on hold]

If we have a generative model $X_2=X_1a_1+\varepsilon$ where $\varepsilon \sim \mathcal{N}(0,\sigma_2^2)$ do we have $X_1=X_2a_2+\varepsilon '$ where $\varepsilon \sim \mathcal{N}(0,\sigma_1^2)$ ...
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### Asymptotics and combinatorics

Wright's expansion of $$(1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1}$$ is, in the words of the late, great Mark Kac "well known to those that know it well". (See, for example, ...
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### Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...
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### $C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ ...
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### Length inequalities in trees and CAT(0) spaces

I have a family of possibly related questions. Let me start with an elementary one: Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...
A colleague asked me a topology question which comes down to this: Suppose that $M$ is a smooth $n$-manifold, and $C\subset M$ is a closed set such that $H_{n-p}(M-C)\to H_{n-p}(M)$ is not surjective. ...
I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...