The Magic Wand Theorem of A. Eskin and M. Mirzakhani

TL;DR

This paper explains the groundbreaking work of Eskin and Mirzakhani on the dynamics of SL(2,R) acting on moduli spaces of Abelian differentials, highlighting its significance and applications in billiards in polygons.

Contribution

It presents a popularized overview of Eskin and Mirzakhani's recent results, emphasizing motivations and potential applications without providing detailed proofs.

Findings

SL(2,R) action on moduli space is ergodic and has deep implications

Connections between dynamics on moduli space and billiards in polygons

Potential applications in understanding polygonal billiard trajectories

Abstract

This is the English translation of a short note published by Gazette des Math\'ematiciens. The author was asked to present the recent work of Alex Eskin and of Maryam Mirzakhani, arXiv:1302.3320, and their joint work with Amir Mohammadi, arXiv:1305.3015, on SL(2,R)-action on the moduli space of Abelian differentials. This is a popularization paper with no proofs. Its only purpose is to suggest some motivations, and to outline possible applications considering billiards in polygons as an example.