Moduli of Differentials: From Euclidean Geometry to Some Fields Medal Work
An Abelian differential on a Riemann surface defines a Euclidean metric with conical singularities such that the underlying surface can be realized as a plane polygon whose edges are identified pairwise by translation. Varying the shape of such polygons induces an action on the moduli space of Abelian differentials, called Teichmueller dynamics. A number of questions about surface geometry reduce to understanding the moduli space of Abelian differentials and the orbit closures in Teichmueller dynamics, whose study has fascinating connections to many other areas such as arithmetic geometry, measure theory, and combinatorics. In this talk I will give an elementary introduction to this topic, with a focus on recent developments from the viewpoint of algebraic geometry.
(Tea served in First Floor Commons Room at 3:30pm.)