Topology Seminar: Alex Wright

One of the ways we understand Teichmuller space, endowed with the Teichmuller metric, is by studying Teichmuller discs. They exist in great abundance: There is a Teichmuller disc through any point and in any direction. Typically, their projection to moduli space is dense, and yet infinitely often their projection is a closed subvariety of moduli space called a Teichmuller curve. Recently, in joint work with Eskin, McMullen, and Mukamel, we discovered the first non-trivial examples of higher dimensional analogues of Teichmuller discs, namely totally geodesic submanifolds. 

In this talk, we will explain that such higher dimensional totally geodesic submanifolds are much more rigid and rare than Teichmuller discs: Each must cover a closed subvariety of moduli space, and only finitely many such subvarieties exist in each moduli space. This result is an application of joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. One of the goals of the talk will be to explain what this cocycle is and why it lies at the heart of Teichmuller dynamics.