9 problems

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 $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $\varphi_{\lambda}\in C^{\infty}(M)$ satisfy … For sufficiently small $r>0$, write $B_g(x,r)$ for the geodesi…

Let $(\Omega,D,N)$ be a triple where $\Omega$ is simply connected, $D$ is a Euclidean line segment, and … Say that $(\Omega,D,N)$ has no hot spots when each first mixed eigenfuncti…

Let $(M,g)$ be a given Riemannian manifold, let $\psi_{\lambda}$ be a real Laplace eigenfunction, and let $1<p<\infty$. For a set $A\subseteq M$, write $\chi_A$ for its characteris…

Let $(M,g)$ be a smooth Riemannian manifold, and let $\psi_{\lambda}$ be a real Laplace eigenfunction satisfying … Write $\operatorname{vol}$ for Riemannian volume. Symmetry conjec…

Let $\Omega\subset\mathbb R^2$ be a bounded domain, and consider the second eigenfunction of the Laplacian on $\Omega$ with Dirichlet boundary condition. Payne's nodal line conject…

Let $F^N_{n,m}$ be the $L^2$-normalized eigenfunctions of the Neumann Laplace operator on the unit disk, with eigenvalues $\lambda^N_{n,m}$. For a subsequence with…

Dirichlet disk growth conjecture. If $1\leq\gamma<3$, then

Let $(M,g)$ be a smooth, compact Riemannian manifold without boundary. Let $\phi$ be an $L^2$-normalised eigenfunction of the Laplacian with eigenvalue $L$, satisfying … on $(M,g)$…