9 problems

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms, with $\langle f_i,f_{i'}\rangle=0$ for $i\ne i'$, and let $C_n$ denote the $n$th moment of a standard real…

Let $\mathbb{X}=\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, let $\{f_i\}$ be Hecke–Maass cusp forms normalized by … and define the Gaussian moments by…

Let $\Omega\subset\mathbb{R}^2$ be a chaotic billiard, so its billiard flow is ergodic and has positive topological entropy. Let $\{u_n\}$ be an orthonormal sequence of eigenfuncti…

Hua–Huang–Li conjecture. The powers $\{f_j^{a_j}\}_{j=1}^J$ are statistically independent: for every $\psi\in\mathcal{C}_c^\infty(\mathbb{X})$,

Let $J\geq2$, and let $f_j$, $1\leq j\leq J$, be pairwise orthogonal real-valued Hecke–Maass cusp forms for $\operatorname{SL}_2(\mathbb{Z})$, with spectral parameters $t_{f_j}$. A…

Sarnak's conjecture. There exists $\delta>0$ such that, for every $\ell\in[-\delta,\delta]$, almost surely $\{f=\ell\}\cap\mathbb{R}^3$ contains an unbounded component.

Let $X$ be the universal symmetric space and let $(M_n=\Gamma_n\backslash X)$ be a uniformly discrete expander family that BS-converges toward $X$. Fix an eigenvalue $\lambda_0$ of…

Let $\mathcal{P}$ be the collection of probability measures on the unit ball $B(1)\subseteq\mathbb{R}^{2}$, and let $c_{\max}$ denote the maximal value attained by the Nazarov–Sodi…

Let $\mathcal{P}_{Symm}$ be the family of symmetric probability measures on $\mathcal{S}^{1}$, and let $d_{\max}$ denote the maximal value attained by the Nazarov–Sodin constant…