28 problems

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$,…

Feingold–Peres off-diagonal variance conjecture. For ergodic flow, as $E\to\infty$,

Let $\phi_n$ be Laplacian eigenfunctions on an ergodic billiard, and let a quantum limit mean a measure to which $|\phi_n|^2$ converges weakly. Quantum unique ergodicity conjecture…

Let $\Omega$ be a billiard domain and let $\hat{A}$ be multiplication by a test function $A$. Define the local diagonal variance $V_A(E)$ as the mean square of…

Let $\Omega$ be the billiard domain, let $\hat{A}$ be multiplication by a test function $A$, and define its spatial average by … For eigenfunctions $\phi_n$ of the Laplacian with e…

For a finite field $\mathbb{F}_{q}$, let $\mathcal{C}$ be the multiplicative group $\mathbb{F}_{q}^{*}$ or the group of norm-one elements in a quadratic extension, and let…

Luo–Sarnak variance formula. The displayed asymptotic holds for the Hecke eigenfunction basis. The result is rigorous for the stated arithmetic surface and holomorphic Hecke eigenf…

Let $p$ be prime, let $\rho_{1/p}(A)$ be the quantized operator, and let ${\rm T}_A$ be its Hecke torus. For a character $\chi:{\rm T}_A\@ifnextchar^{\t@@}{\t@@^{}}\mathbb C^*$, le…

Let $(M,g)$ be a compact manifold of negative curvature without boundary, and let $(u_n)_{n\geq 1}$ be any orthogonal eigenbasis of the Laplacian. A semiclassical measure is a weak…

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues…

Let $F$ be the number field in the paper, let $\mathbb{X}_F$ be the associated quotient, and let $\bar{\operatorname{G}}(F)\backslash\bar{\operatorname{G}}(\mathbb{A})$ carry the v…

Nonsplit Bianchi quantum unique ergodicity conjecture. If $H\in C_b(\Gamma\backslash\mathbb{H})$ is fixed, then

Nonsplit quantum unique ergodicity conjecture. For every fixed $H\in C_b(\Gamma\backslash\mathbb{H})$,

Let $(M,g)$ be a compact manifold with Anosov geodesic flow, let $\mu$ be a semiclassical measure on $S^*M$, and let $\mu_L$ denote the Liouville measure. Liouville-component conje…

Let $g\to\frac12+it_g$ be a sequence of newforms in $\mathcal{B}_0^{\ast}(q,\chi)$, and let $R$ denote the radius of a hyperbolic ball. The volume-level shrinking-ball equidistribu…

Let $N$ be a compact Riemannian manifold with negative sectional curvature, let $(\Psi_n)_{n=0}^{\infty}$ be an orthonormal basis of Laplace eigenfunctions on $N$, and let $\mu_n$…

Let $M$ be a compact Riemannian manifold with flow domain $\mathcal{D}$, and suppose that $\mathcal{D}$ is the disjoint union of two invariant subsets $U$ and…

QUE conjecture for quasimodes. Any sequence of $o(1/\log r)$-quasimodes satisfies QUE.

Let $U(z)$ be either $y^{k/2}f(z)$ or $u_j(z)$, and let $\psi:\mathbb{Z}\backslash\mathbb{R}\to\mathbb{R}$ be smooth. For fixed $y>0$, Young's horocycle QUE conjecture. … as the we…

Let $u_j$ be even Hecke–Maass forms normalized by the paper's $L^2$ normalization, and let $f$ be an $L^2$-normalized holomorphic Hecke cusp form of weight $k$. If…

Logarithmic quasimode threshold conjecture. Then the sequence $\{\psi_j\}$ satisfies Quantum Unique Ergodicity.

Entropy threshold conjecture. The entropy of $\mu$ satisfies

Let $\phi$ be a Maass-Hecke cusp form with spectral parameter $t_\phi$, Fourier coefficients $\rho_\phi(n)$, and let $\delta_{0,m}$ denote the Kronecker delta. For…

Let $(M^2,g)$ be a real analytic Riemannian surface with ergodic geodesic flow, and let $\gamma_{x,\xi}$ be a geodesic satisfying the QER hypothesis. Let…

Quantum ergodicity conjecture for resonant states. Under these assumptions, there exists a density-one subsequence of resonant states in every strip…