minicourse in geometric spectral theory

Geometry and spectrum of random hyperbolic surfaces
Laura Monk (University of Bristol)


March 27, 29, and 31, 2023
12:00 Québec time

The zoom link will be sent to the Spectral geometry in the clouds mailing list. Please contact Alexandre Girouard if you would like to receive it.

The aim of this minicourse is to present recent developments in the study of random hyperbolic surfaces, with a particular emphasis on their spectral geometry.

Part 1: Classical spectral theory of compact hyperbolic surfaces
This first session will be a comprehensive introduction to hyperbolic geometry and the spectral theory of compact hyperbolic surfaces. This will not only allow me to present several important objects and results, but also to motivate the idea of studying random hyperbolic surfaces. Indeed, we will observe that it is quite easy to construct “pathological” hyperbolic surfaces, which seem to be quite peculiar and to stand in the way of proving nice theorems. Our objective moving forward is to discard these pathological examples, by saying they have a very small probability of occuring.
Video recording

Part 2: How to sample and study random hyperbolic surfaces?
Now that we know that using probabilistic methods in hyperbolic geometry is a promising idea, the next question is: how? There are several different models of random hyperbolic surfaces. This minicourse will be centered on the Weil–Petersson probabilistic model exclusively. After presenting the model and some of the tools developed by Mirzakhani that allow to study it, I will explain why the pathological surfaces exhibited in the first session occur only with very low probability
Video recording

Part 3: The spectrum of typical hyperbolic surfaces
This last session will start by a state of the art of the spectral geometry of random hyperbolic surfaces. This will confirm that the Weil–Petersson model does indeed allow to reach a better understanding of the spectral geometry of hyperbolic surfaces, by proving new theorems which are true most of time rather than always. I will then present elements of the proofs of some of those results, in particular of the lower bounds on the spectral gap proven by Mirzakhani in 2013 and Wu–Xue and Lipnowski–Wright in 2020.
Video recording

Bibliography