A tale of two volumes of moduli spaces: Weil-Petersson and Masur-Veech

TL;DR

This survey reviews the computation of Weil-Petersson and Masur-Veech volumes of moduli spaces, highlighting their mathematical significance, methods, and open problems, and exploring parallels between these two types of volumes.

Contribution

It provides a comprehensive overview of the key results, methods, and open problems related to the volumes of moduli spaces of Riemann surfaces, emphasizing their mathematical connections.

Findings

Connections between combinatorial enumeration and volume computations

Recursion relations in volume calculations

Open problems in moduli space volume theory

Abstract

Weil-Petersson and Masur-Veech volumes measure the sizes of moduli spaces of Riemann surfaces equipped with hyperbolic and flat metrics, respectively. Over the past several decades, the computation of these volumes has inspired remarkable developments in combinatorial enumeration, intersection theory, and recursion relations. In this survey, we review key results, methods, open problems, as well as interesting parallels that emerge in the approaches to computing both types of volumes.