BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 3, July 2020, Pages 359–408 https://doi.org/10.1090/bull/1687 Article electronically published on February 3, 2020
A TOUR THROUGH MIRZAKHANI’S WORK ON MODULI SPACES OF RIEMANN SURFACES
ALEX WRIGHT
Abstract. We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non- experts.
Contents 1. Introduction 359 2. Preliminaries on Teichm¨uller theory 361 3. The volume of M1,1 366 4. Integrating geometric functions over moduli space 367 5. Generalizing McShane’s identity 369 6. Computation of volumes using McShane identities 370 7. Computation of volumes using symplectic reduction 371 8. Witten’s conjecture 374 9. Counting simple closed geodesics 376 10. Random surfaces of large genus 379 11. Preliminaries on dynamics on moduli spaces 382 12. Earthquake flow 386 13. Horocyclic measures 389 14. Counting with respect to the Teichm¨uller metric 391 15. From orbits of curves to orbits in Teichm¨uller space 393 16. SL(2, R)-invariant measures and orbit closures 395 17. Classification of SL(2, R)-orbit closures 398 18. Effective counting of simple closed curves 400 19. Random walks on the mapping class group 401 Acknowledgments 402 About the author 402 References 403
1. Introduction This survey aims to be a tour through Maryam Mirzakhani’s remarkable work on Riemann surfaces, dynamics, and geometry. The star characters, all across
Received by the editors May 12, 2019. 2010 Mathematics Subject Classification. Primary 32G15.