GSTW04 — Random and arithmetic models in spectral theory

There are several widely open problems about the geometry and the spectral theory of the Laplacian, many of them coming from physics, in particular from "quantum chaos'', including the statistics of the number of eigenvalues in appropriately sized intervals, which is related to the influential universality conjectures of Berry and Tabor and of Bohigas, Giannoni and Schmit; quantum unique ergodicity and Berry's random wave model; optimal lower bounds for the first eigenvalue; asymptotic estimates of Cheeger constants. Randomness plays an important role in the study of such problems. One incarnation is random wave theory, which is the study of random linear combinations of eigenfunctions. This and related models are used to understand the fine structure of nodal sets.