15 problems

Let $\mu_0$ be a rotationally invariant, compactly supported probability measure satisfying some mild assumptions, and let $P^N_0$ be any sequence of polynomials whose empirical ro…

Let $m,n$ be positive integers, let $\varepsilon>0$, and let $T_{m,n}^{(\mu)}(z)$ be the generalised Laguerre polynomial. At the coalescence values $\mu=-n-j$, roots approaching or…

Let $m,n$ be positive integers, let $T_{m,n}^{(\mu)}(z)$ be the generalised Laguerre polynomial, and write $\lceil\mu\rceil$ for the ceiling of $\mu$. For…

Non-local PDE conjecture. For $0\leq t<1$, the density $p_t$ satisfies

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…

Uniform-disk radial-density conjecture. This formula applies to several such models, including i.i.d. roots, Weyl polynomials, and eigenvalues of a random Ginibre matrix. The Weyl-…

Let $P_n$ be a sequence of degree-$n$ polynomials whose empirical root measures converge weakly to the uniform probability distribution on the unit circle, and impose an additional…

Strong convergence conjecture. The Cauchy transform satisfies, almost everywhere in $\mathbb C$,

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…