Title: Optimal spectral gap of random hyperbolic surfaces

Speaker: Joe Thomas (Durham University)

Time/Location: Nov 7 (Room 5207), Nov 10 (Room 5206), Nov 12 (Room 5505), Nov 14 (Room 5207), 10:00–12:00 AM

Abstract: We will focus on the size of the spectral gap of the Laplacian for random hyperbolic surfaces. We will begin with a crash course on hyperbolic geometry and surfaces, with the main highlight being the Selberg trace formula, which provides a direct connection between the Laplacian spectrum and the geometry of a surface. Next, we will give an overview of the construction of Weil–Petersson random hyperbolic surfaces. We will introduce the Mirzakhani integration formula, a fundamental tool in computing probabilities in this random surface model. Finally, we will combine these tools with the key ideas of the polynomial method to prove an optimal spectral gap result holding asymptotically almost surely.