6 problems

Let $G$ be a finite group, $V$ a finite-dimensional real $G$-module, and $\rho \colon G \rightarrow \operatorname{GL}(V)$ the representation afforded by $V$. Let…

Generic double multiplicity conjecture. There exists a generic set $\Omega\subseteq\mathbb R^{n,k}$ such that, for every $U\in\Omega$ and every $\tau\in\mathbb R\setminus\{0\}$, th…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let $PM_n$ be the union of the convex hulls of the $k$-th roots of…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let a convex combination of permutation matrices mean a matrix of t…

Simplex and cube extremality conjecture. (i) Among all bounded open convex sets in $\mathbb R^n$ having a fixed positive volume, the $n$-dimensional regular simplex—the interior of…

Let $n$ be a positive integer, and let $S_r^n$ be the recursively defined set of triples of $r$-element subsets of $[n]=\{1,\dots,n\}$. Let $\lambda,\mu,\nu$ be nonincreasing real…