Randomness in the spectrum of the Laplacian: from flat tori to hyperbolic surfaces of high genus

I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, 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”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh.