13 problems

Let $H$ be the Schrödinger operator considered in the paper, let $N(\lambda)$ be its eigenvalue counting function, and let $R_{\mathrm h}(\lambda)$ denote the half-line hyperunifor…

Transport-distance finiteness conjecture. If $\Phi$ is $(p+\vartheta)$-uniform for some $\vartheta>0$, then

Let a strictly jammed configuration mean a collectively jammed configuration that disallows all uniform volume-decreasing strains of the system boundary, and let an infinite config…

Let $d\geq 2$, and consider a polynomial perturbation of the lattice in dimension $d$, including perturbations whose displacement tails decay arbitrarily slowly. A translation inva…

Domain-extension conjecture. The same conclusions should hold for more general domains, say Lipschitz domains.

Let $d\ge 2$ and consider the facilitated exclusion process (FEP), with critical density $\rho_c$ and stationary measure $\nu_\rho$. Write $\delta=\rho_c-\rho>0$, and let…

Let $T(x_1,\ldots,x_n):=2[h_1(x_1,\ldots,x_n)-h_0(x_1,\ldots,x_n)]$ be the nonnegative likelihood-ratio test statistic, where $h_0$ and $h_1$ are the hypotheses' maximized log-like…

Let $\mathsf{F}_N$ denote the energy functional of the two-dimensional one-component plasma for $N$ particles, and call its minimizers Fekete points. The lattice property concerns…

Let $g_2({\bf r})$ or, equivalently, $S({\bf k})$, be the pair correlation function or structure factor of a realizable statistically homogeneous many-particle ensemble at number d…

A substitution tiling has a substitution matrix ${\bm M}$ with integer entries and eigenvalues governing its scaling properties. Nonexistence conjecture. There are no limit-periodi…

Gap-dependent bounded holes conjecture. The upper bound on the hole size depends only on the size of the gap for the structure function of the process.

Bounded holes conjecture. The hole sizes for stealthy hyperuniform processes are uniformly bounded.

Let $X$ be a point process, and let $N_\Lambda$ denote the number of points of $X$ in a bounded region $\Lambda$. The process is number rigid if $N_\Lambda$ is determined by the co…