Spectral statistics for random hyperbolic surfaces

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This research project aims to make progress towards the proof of influential conjectures made in the 1980s in the context of quantum chaos, including the Bohigas-Giannoni-Schmit (BGS) conjecture on spectral statistics of chaotic quantum systems and random matrix theory. Following a large body of work in the physics literature over the past three decades, which provided a heuristic explanation of random matrix statistics through correlations of chaotic classical particle trajectories, we will focus on particularly clean mathematical models of quantum chaos -- the Laplacian on hyperbolic surfaces. The study of hyperbolic surfaces is interesting because they provide a rich family of examples with chaotic dynamics (due to the negative curvature) and allow the application of powerful mathematical tools. The novelty of this project is to use recently developed techniques in ergodic theory to address some of the outstanding conjectures on average, that is, not for a single fixed surface (where the challenges are simply too hard) but by taking the mean over the moduli space of surfaces of a given genus. Using the Selberg trace formula, a standard tool in the subject, the spectral statistics will be mapped to geometric correlations of lengths of closed geodesics, and the key challenge in the analysis will be to prove rigorous limit theorems for the distribution of closed geodesics on random surfaces. Here we will exploit recent exciting breakthroughs by geometers including Fields medalist Mirzakhani and others.