Online: Geometry of the combinatorial moduli spaces of curves II

Online: https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

Mirzakhani obtained a topological recursion for the Weil-Petersson volumes of the moduli space of bordered Riemann surfaces, based on a recursive partition of unity on the Teichmuller space that extended an identity by McShane. Her idea can be generalised to study length statistics of multicurves on surfaces. I will explain that it fits in a larger theory of "geometric  recursion", allowing to define recursively mapping class group invariant functions on the Teichmuller space, whose integration on the moduli space of curves can be computed by a topological recursion. This can be done both in the hyperbolic setting, or in the combinatorial setting using the results of the first talk. This can be applied to obtain uniform geometric proofs of Witten's conjecture, of Norbury's lattice point counting, and several proofs of topological recursion for Masur-Veech volumes of the moduli space of quadratic differentials.

Based on joint works with Jorgen Andersen, Severin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski, Nicolas Orantin, Campbell Wheeler.