Extremal hyperbolic manifolds and the Selberg trace formula

Virtual talk.

The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic manifold.  I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the largest possible kissing number or the largest spectral gap of a hyperbolic manifold of bounded volume? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.