Mirzakhani's recursion formula on Weil-Petersson volume and applications

TL;DR

This paper reviews Mirzakhani's proof of a recursion formula for Weil-Petersson volumes of moduli spaces of hyperbolic surfaces and explores its applications to Witten's conjecture and geodesic length spectrum growth.

Contribution

It provides an overview of Mirzakhani's volume recursion proof and highlights its applications in mathematical physics and geometric analysis.

Findings

Validation of Mirzakhani's recursion formula

Application to Witten's conjecture

Insights into geodesic length spectrum growth

Abstract

We give an overview of the proof for Mirzakhani's volume recursion for the Weil-Petersson volumes of the moduli spaces of genus $g$ hyperbolic surfaces with $n$ labeled geodesic boundary components, and her application of this recursion to Witten's conjecture and the study of simple geodesic length spectrum growth rates.