35 problems

Berry's conjecture. The rescaled restriction of $\varphi_i$ to that wavelength-scale ball should behave like $\phi^{\mathbb{R}^n}_1$. This conjecture proposes a universal local mod…

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 $(\psi_n)_{n\ge0}$ be the eigenfunction ensemble on a compact Riemannian manifold $M$, and let $\mu$ be the probability measure used for local spatial averages. Berry's ensembl…

Let a quantum system have a classically chaotic counterpart, and choose a basis in which to express one of its energy eigenstates. Berry's conjecture. The energy eigenstate should…

Let the full modular group act on the upper half-plane, and consider Hecke–Maass cusp forms with Laplacian eigenvalues tending to infinity. Berry–Hejhal–Rackner random wave conject…

Let $r$ be the spectral parameter of a Maass cusp form, with eigenvalue … Let $F$ be any compact regular subregion of the fundamental domain $\mathcal F$, and let $\sigma$ denote t…

Let $M$ be a compact smooth manifold of dimension $d$ without boundary, and let $\rho_j(u)$ denote the Gaussian kinematic densities … where $H_j$ is the $j$th Hermite polynomial. L…

Let $M$ be a compact Riemannian manifold, let $I$ be a bounded frequency interval, and let $\phi_I^M$ be the associated Gaussian random wave on $M$. Let $\phi_I^{\mathbb{B}^n}$ be…

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 $M$ be a compact smooth $d$-dimensional manifold without boundary, let $K_\lambda$ be the covariance kernel of the Gaussian random wave, and let $\Phi_\lambda$ denote the corre…

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…

Let $M$ be a compact Riemannian manifold without boundary, let $\psi_n$ be eigenfunctions with eigenvalues $\lambda_n$, and let $z^j:M\to TM$ be smooth vector fields. Define … Let…

Let $q\geq 3$ be odd, and consider the variances of odd polyspectra associated with random wave models on spherical geometries. Marinucci's conjecture. The variances of odd polyspe…

Let $(M,g)$ be a smooth, compact, Riemannian manifold of dimension $n$ without boundary, let $x\in M$, and write $d_g(x,y)$ for the Riemannian distance and $J_{\frac{n-2}{2}}$ for…

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 $\mathcal M$ be Seba's rectangular billiard with irrational aspect ratio and a Dirac mass in its interior, and consider the wave functions of its associated quantized billiard…

Let $g$ be a real-valued Maaß newform, let $K$ be a fixed compact subset of $\Gamma_0(q)\backslash\mathbb{H}$, and let $n$ be a positive integer. Consider the suitably normalized…

Let $M$ be a compact $d$-dimensional manifold. An invariant random function $F$ is a Wigner wave for $M$ if there exists a sequence of $L^2$-normalized eigenvectors…

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…

Small-ball factorial moment conjecture. For $n>2$ and $\rho\to 0$,

Let $\chi(R,s)$ be the maximal number of lattice points in a cap of radius $s$ of the sphere $R\mathcal{S}^2$. Bourgain–Rudnick's conjecture. For all $\epsilon>0$ and…

Let $M$ be a general manifold, and consider global quantities associated with the zero set of Laplace eigenfunctions on $M$ that can be expressed in terms of an area or co-area int…

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…