24 problems

Polynomiality conjecture. Each coefficient $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ is a polynomial in $\theta$ of degree $l$. Conjecturally, the constant in the displayed…

Weyl-antisymmetry conjecture. This polynomial satisfies

Let $(M,g)$ be a flat manifold and let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$. Generic-potential bounded-eigenfunction problem. For generic $V$, are the eigenfunctions u…

Let $(M,g)$ be a compact Riemannian manifold whose geodesic flow is completely integrable. Let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$, and suppose that all of its orthon…

For each $d\geq 1$, let $Y_0$ be the closure space for Schrödinger waves introduced in the paper, and let $\tilde{Y}_0\subset Y_0$ be its time-independent subspace. Time-independen…

Let $d\geq 1$, let $\mathbb{T}^d$ be the $d$-dimensional torus, and let $V\in L^\infty_x(\mathbb{T}^d)$. Let $\mathcal{H}_V=-\Delta+V$, and let $f$ be any eigenfunction of…

Let $u$ be an eigenfunction of the Laplacian $-\Delta$ on $\mathbb{T}^3$ with eigenvalue $\lambda^2$. Bourgain's eigenfunction conjecture. One should have … The loss…

Let $\mathbb{H}=\{x+iy:x\in\mathbb{R},\ y>0\}$ be the upper half-plane, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, and let $H_k$ be a Hecke basis of holomorphic cusp forms on…

Let $\mathbb{H}=\{x+iy:x\in\mathbb{R},\ y>0\}$ be the upper half-plane, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, and let $H_k$ be a Hecke basis of holomorphic cusp forms on…

Symmetry conjecture. The limit

Full support eigenfunctions conjecture. For any metric graph $(\Gamma,\boldsymbol{\ell})$, and any choice of a complete orthonormal sequence of eigenfunctions, there are infinitely…

Bourgain–Rudnick conjecture. The lower bound for the number of nodal intersections is of order $\lambda$; equivalently,

Random-surface eigenfunction sup-norm conjecture. With probability tending to $1$ as $g\to+\infty$,

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $u$ be a first Dirichlet eigenfunction of $\Omega$. Write $\operatorname{diam}(\Omega)$ for its diameter and…

Interior $L^p$ estimate conjecture. One should have

Let $C\subset\mathbb{R}^2$ be a bounded ellipse whose indicator function is an eigenfunction in the sense of the adaptive anisotropic total variation eigenfunction equation, and le…

Let ${\boldsymbol \tau}$ be a uniform knot sequence such that the dimension of $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ is $2m$. For any $q\geq 0$, let $Q_p^q$ be th…

Let $(M,g)$ be a compact hyperbolic surface, let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, and let $\gamma$ be a closed geodesic or a geodesi…

Let $(M,g)$ be a compact hyperbolic surface, let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, and let $\gamma$ be a periodic geodesic or a geode…

Let $(M^n,g)$ be a manifold with boundary, and let $u_\lambda$ be a Dirichlet eigenfunction associated to the eigenvalue parameter $\lambda$. For each nodal domain $A_i$ of…

Hamilton-Jacobi conjecture. Any such limit $G$ solves the Hamilton-Jacobi equation

Let $X$ be the Riemannian symmetric space under consideration, let $K/M$ be its Furstenberg boundary, and let $q'$ satisfy $1/q+1/q'=1$. Fix $q\in(1,2)$, and let…

Let $d\geq 2$ and let $\Sigma\subset\mathbb T^d$ be a real analytic hypersurface. For an eigenfunction $\varphi_\lambda$ of the Laplacian on $\mathbb T^d$, write…

Let $\mathbb T^2$ be the two-dimensional torus, let $\varphi$ be an eigenfunction with eigenvalue $E=\lambda^2$, and let $C\subset\mathcal N_\varphi$ be a regular arc. For a convex…