Society Meeting in Honour of Maryam Mirzakhani
The meeting commemorates the work of Professor Maryam Mirzakhani (1977 – 2017), Fields Medallist and LMS Honorary Member.
The meeting forms part of the 2017-2018 Warwick EPSRC Symposium workshop on Teichmüller dynamics from 19-23 March.
2.00pm Opening of the meeting
2.15pm Alex Wright (Stanford) Mirzakhani's computation of Weil-Petersson volumes and intersection numbers
Abstract: We will reproduce Mirzakhani's remarkable computation of the Weil-Petersson volume in the simplest example of the moduli space of genus 1, once punctured Riemann surfaces, and discuss how Mirzakhani generalized the techniques to determine the volume of all moduli spaces of Riemann surfaces.Then we will discuss Mirzakhani's proof of Witten's Conjecture concerning intersection numbers of tautological classes on moduli spaces of Riemann surfaces. We will emphasize definitions, examples, and the connection that Mirzakhani established to Weil-Petersson volumes.
3.30pm Tea & coffee break
4.00pm Anton Zorich (Paris) Mirzakhani's count of simple closed geodesics
Abstract: We will try to give an outline of Mirzakhani's count of simple closed geodesics of bounded length on a hyperbolic surface with cusps.
We shall start with basic facts about simple closed geodesics and about moduli spaces of hyperbolic surfaces to make the exposition self-contained. We shall use McShane's identity to give an idea of how Mirzakhani computes integrals over moduli space, in particular, how she counts the average number of simple closed geodesics of bounded length.
We shall proceed with more detailed consideration of simple closed curves on a four-punctured sphere which would lead us to the notion of the space of measured geodesic laminations and of Thurston measure in this particularly simple case. Using these notions we shall state (and even give an idea of the proof) of Mirzakhani's particularly beautiful result: the asymptotic frequencies of simple closed geodesics of different topological types do not depend on the hyperbolic metric and are explicitly computable.
We will finish with an announcement of the result showing that "random" simple closed hyperbolic geodesics have the same asymptotic frequencies as the corresponding "random" simple closed flat geodesics (under appropriate interpretation of "randomness”).
5.00pm Wine reception
6.30pm Dinner at the Mathematics Institute
Regisration: Please register for your place here