Large Genus Asymptotics in Flat Surfaces and Hyperbolic Geodesics

In this talk we will describe the behaviors of flat surfaces and geodesics on hyperbolic surfaces, as their genera tend to infinity. We first discuss enumerative results that count the number of such surfaces or geodesics (which can be viewed as volumes of particular moduli spaces) in the large genus limit, and then we will explain how a randomly sampled such object looks. Prior works of Mirzakhani (for hyperbolic geodesics) and Delecroix-Goujard-Zograf-Zorich (for flat surfaces) express these topological counts in terms of more algebraic quantities, namely, certain intersection numbers. Thus, the large genus limits of these counts will rely on a new asymptotic result on the behaviors of these intersection numbers at high genus, which might be of independent interest.