Quantum Unique Ergodicity and the number of nodal domains of eigenfunctions

It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having a bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I explain how one can prove that this is indeed true for the surfaces where the Laplacian is quantum uniquely ergodic, under certain symmetry assumptions. For instance, we prove that the number of nodal domains of Maass-Hecke eigenforms on a compact arithmetic triangles tends to $+\infty$ as the eigenvalue grows. This talk is based on joint works with S. Zelditch and S. Jang.

Talk sponsored by the Mrs. Hepsa Ely Silliman Memorial Fund.