9 problems

Associate to $Sp_n(\lambda)$ the finite measure … where $\delta(z-a)$ is the Dirac measure supported at $a$. The measure $\mu_n$ is the root-counting measure of the spectral polyno…

Let $(\xi_j)_{j\geq 1}$ be independent complex-valued random variables with distribution $\mu$, let $P_n(z)=\prod_{j=1}^n(z-\xi_j)$, and let $\mu\lsem P_n^{(k)}\rsem$ denote the em…

Let $P$ be a polynomial and let $\x03bc_{\x03b1,P}$ be the asymptotic root-counting measure from Theorem. Forest conjecture. For $0<\x03b1<\deg P$, the support of…

Let $\x03be$ be a probability measure compactly supported in $\x1d53d$, let $P_n=\x220f_{i=1}^n(x-\x03be_i)$ have roots independently sampled according to $\x03be$, and let…

Consider a fixed order-$k$ linear ordinary differential operator family with monic polynomial leading coefficient $Q_k(z)$ of degree $k$. An operator is exactly solvable if its coe…

Let ${\mathcal P} \subset {\mathbb R}^d$ be a Fano polytope, meaning an integral convex polytope of dimension $d$ whose only interior integer point is the origin. Call…

Let ${\mathcal P} \subset {\mathbb R}^N$ be an integral convex polytope of dimension $d$, and let $i({\mathcal P},n)$ denote its Ehrhart polynomial. For a root $\alpha$ of…

Consider a generalized Heun equation … where $\deg Q_{k+1}(z)=k+1$ and $\deg Q_i(z)\leq i$ for $i=2,3,\ldots,k$. For each sufficiently large positive integer $n$, let $\mu_n$ be th…

Let $Q(z)$ have roots $a_1,a_2,a_3$, and let $\mu$ be the limiting root-counting measure from the Shapiro–Tater conjecture. For $i\in\{1,2,3\}$, let $\gamma_i$ be the curve consist…