5 problems

Yau's nodal length conjecture. The nodal length satisfies

Let $(\phi_j,\lambda_j)_{j\ge 1}$ be Laplace eigenfunctions on a smooth compact surface $\mathcal{M}$, with eigenvalues $\lambda_j$ ordered increasingly and counted with multiplici…

A nodal count is the number of nodal domains of an eigenfunction, namely the connected components of the complement of its zero set. Two planar domains are isospectral when their L…

Maximal Nazarov–Sodin constant conjecture. For every $\mu\in\mathcal{P}_{\text{symm}}$ that is a weak-$*$ limit of $\{\mu_n\}$, the maximal value $d_{\max}$ is uniquely attained by…

Let $\Omega$ be a simply connected domain, let $\mathcal D_k$ be a minimal $k$-partition of $\Omega$, let $N(\mathcal D_k)$ be its associated partition graph, and let…