Abstract:

Building on her seminal work on moduli space volumes for Riemann surfaces, Mirzakhani also computed expected values of various geometric quantities on moduli space. In particular, she studied the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. I will review several of Mirzakhani’s key results, which together give a detailed picture of the geometry of random high genus surfaces. I will then discuss some extensions of her work in the setting of translation surfaces.

Bio:

Kasra Rafi is a Professor of Mathematics at the University of Toronto and Director of the Centre for Mathematical AI at the Fields Institute. His research focuses on the geometry and dynamics of Teichmüller space and related moduli spaces, geometric group theory, and low-dimensional geometry and topology. He received his PhD from Stony Brook University in 2001. He is a Fellow of the American Mathematical Society and a Fields Institute Fellow.