I will report on recent progress on influential conjectures from the 1970s and 1980s, namely those of Berry–Tabor and Bohigas–Giannoni–Schmit, which suggest that the spectral statistics of the Laplace–Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the geodesic flow is integrable or “chaotic.” In the case of flat tori, the spectrum of the Laplacian is given by values of quadratic forms at integer points and is thus closely related to fundamental open questions in number theory.
The two most recent results that I will present in this lecture were obtained in collaboration with Laura Monk, and with Wooyeon Kim and Matthew Welsh.
