Description:

In this minicourse we will introduce the "polynomial method" due to Chen, Garza-Vargas, Tropp and van Handel (Annals of Math., 2024) in the specific context of obtaining spectral gaps for random regular graphs. We will then look at the setting of Weil-Petersson random hyperbolic surfaces and show how the polynomial method can be fused with the Selberg trace formula to obtain near optimal spectral gaps for the Laplacian operator. This is based on joint work with Will Hide and Davide Macera (arXiv:2508.14874). Along the way we will discuss some exciting open questions.

Prerequisite:

Basic knowledge in hyperbolic geometry

Target Audience:

Graduate students and advanced undergraduate students

Teaching Language:

English

About the speaker:

Joe Thomas obtained his Ph.D. degree in 2021 under the supervision of Tuomas Sahlsten and Etienne Le Masson at the University of Manchester. From 2021 to 2024, he was a postdoctoral research associate at Durham University, funded by the ERC starting grant of Michael Magee. Since October 2024, he has been a Leverhulme Early Career Fellow at Durham University.

His research focuses on geometry, spectral theory, combinatorics, random matrix theory, and their relationship to physics. His outstanding research has been published in top-tier international journals, including the Duke Mathematical Journal, Geometric and Functional Analysis, and Communications in Mathematical Physics.