4 problems

Poisson statistics conjecture. If $a^2/b^2$ is a “good” irrational number, then the local spectral statistics of the rectangular billiard are well described by Poisson statistics f…

Pseudo-random phase conjecture. As $k\to\fty$, the quantities $phi_m$ modulo $2\textrm{\pi}$ become pseudo-random, possibly after averaging over a small window of $k$.

Let $P\subset\mathbb{R}^2$ be a rational polygon, meaning that all its angles are rational multiples of $\pi$. Consider the Laplace eigenfunctions of the associated quantum billiar…

Let $\mathbb{M}$ be a chaotic surface, such as an ergodic billiard, and consider the nodal lines of its high-energy eigenfunctions. Nodal-line equidistribution conjecture. The noda…