Date:

Thu, 08/12/202214:30-15:30

**Title:**Eigenvalue statistics for random hyperbolic surfaces

**Abstract:**I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections between random matrix theory, the zeros of the Riemann zeta function, and spectral statistics on the moduli space of hyperbolic surfaces.

In particular, I will describe some recent progress on an outstanding conjecture in quantum chaos, that the statistics of the energy levels of “generic” chaotic systems with time reversal symmetry are described by those of the Gaussian Orthogonal Ensemble (GOE) in Random Matrix Theory. Conjectural examples are the eigenvalues of the Laplacian on a “generic” hyperbolic surface. This conjecture has proved to be extremely difficult, with no single case being proved. It has long been desired to improve the situation by averaging over a suitable ensemble of chaotic systems. I will describe a version of such ensemble averaging on the moduli space of compact hyperbolic surfaces, equipped with the Weil-Petersson measure, using the pioneering work of Maryam Mirzakhani.

**Recording will be available at:**

__https://huji.cloud.panopto.eu/Panopto/Pages/Sessions/List.aspx?folderID=a6e6beab-eb0e-47c6-9d5d-aeb500acb5ad__