8 problems

Kurlberg–Rudnick's conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is the distribution of

Anantharaman–Koch–Nonnenmacher's conjecture. In all dimensions, the optimal lower bound for the entropy of semiclassical measures of quantum cat maps is half the topological entrop…

Fix a prime $p$, choose a standard observable $\xi$, and let $\Psi_\chi\in \mathrm{L}^{2}(\mathbb{F}_{p},\mathbb{C})$ be a Hecke eigenvector in the standard realization of the Weil…

Let $\Gamma_\theta \subset {\rm SL}(2,\mathbb{Z})$ be the theta group of linear automorphisms of the two-torus, and let $U_N(A)$ denote the quantum propagator associated with…

Quadratic-form IID conjecture. For every $\nu\in\mathbb{Z}$, $Y_\nu(\psi)$ has limiting distribution equal to that of $\operatorname{tr}(U_\nu)$, where $U_\nu$ is Haar-random in…

Let $A\in SL_2(\mathbb{Z})$ be hyperbolic, let $U_N(A)$ be its quantization, and let $\\{\psi_i\\}_{i=1}^N$ be a Hecke basis. Hecke fluctuation conjecture. After normalizing the ma…

Let $P(\theta)$ be the windowed sum of diagonal matrix elements of $\operatorname{Op}_N(f)$ over eigenphases, with a window of length $1/L$, and suppose…

Let $A\in SL_2(\mathbb{Z})$ be a hyperbolic matrix, let $U_N(A)$ be its quantization, and let $\\{\psi_j\\}$ be a Hecke basis. For a nonzero Fourier mode $n\in\mathbb{Z}^2$, consid…