### The spectral gap of random hyperbolic surfaces

Ergodic Theory and Dynamical Systems Seminar

16th December 2021, 2:00 pm – 3:00 pm

Fry Building, 2.04

The focus of this talk is the first non-zero eigenvalue of the Laplacian on a compact hyperbolic surface, otherwise called spectral gap. Surfaces with a large spectral gap are well-connected, and have good dynamical properties. Finding surfaces with an optimal spectral gap is a hard and important problem, which has remained opened from the 80s until very recently.

In this talk, based on work in progress with Nalini Anantharaman, I will explain how a modern probabilistic approach has allowed major progress in this topic in the last year. Rather than exhibit examples, we now aim at proving that the probability for a random surface to have an optimal spectral gap is close to 1. I will explain how one can sample random surfaces using the Weil--Petersson probabilistic model, and provide a few tools used to tackle spectral gap questions in this setting.

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