The polynomial method for spectral gaps on random graphs and surfaces

Speaker：

Joe Thomas (Durham University)

Time：

Mon. & Thur. 14:00-16:00

3.9-3.19, 2026

Venue：

3.9 & 3.16,

C548 Shuangqing Complex Building A

3.12 & 3.19,

B725 Shuangqing Complex Building A

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.