Reading Group on Moduli space of Abelian Varieties and Curves
Reading Group on Moduli space of Abelian Varieties and Curves Spring 2017
Week 1. The Manifold Structure of Moduli space of Abelian Varieties and Curves by Lei Chen
Week 2. McShane's identity by Paul Apisa
Title: McShane's Identity, Geodesic Laminations, and Mirzakhani's Integration Trick
Abstract: Geodesic laminations were introduced (and "classified") by Thurston as a means of leveraging hands-on hyperbolic geometry to study Teichmuller space. Birman and Series showed that for sufficiently small horocycles, the collection of all geodesic laminations intersect the horocycle in a measure zero Cantor set + finitely may isolated points.
In his thesis, Greg McShane studied the point set topology of this Cantor-ish set and showed that there is a perfect correspondence between point-set topological properties of it and Thurston's classification of geodesic laminations (with some fine print). This allowed McShane to show a surprising, simple, Selberg-zeta-function-esque result that both reflects and comments on the hyperbolic geometry of all cusped finite volume hyperbolic surfaces.
Mirzakhani took this identity as a starting point and, using nothing more than some integration by parts and a Fourier series tricks, calculated the Weil-Petersson volumes of moduli space.
Today's talk will begin with Thurston's definition and work on geodesic laminations, blackbox the Birman-Series result, spend some time with the hands-on hyperbolic geometry in McShane's work, and then illustrate Mirzakhani's argument in the simplest case. There will be lots of pictures and fun exercises in hyperbolic geometry just in case anyone needs practice with their hyperbolic trig.
Week 3. The proof of Torelli theorem by Ping Ngai Chung
Week4: Mirzakhani's thesis: counting curves on a hyperbolic surface by Lei Chen
Week 5. Cohomology of moduli space of abelian variety and moduli space of curves by Lvzhou Chen