# Lipschitz geometry of the space of hyperbolic surfaces

## Department of Mathematics Colloquium

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Stephanie Geyer, Sergei Tabachnikov, Dmitri Burago, Alberto Bressan, Nigel Higson.

**Speaker:** David Dumas, University of Illinois Chicago

**Abstract:** The space of marked hyperbolic structures on a surface, or Teichmüller space, is a topological ball of a certain dimension. There are many natural metrics on this space that arise from different ways of understanding the geometry of surfaces (e.g. complex analysis, hyperbolic geometry). These metrics typically induce the same topology while having vastly different metric properties.
In this talk I will discuss how one such "metric" on Teichmueller space fits into this picture---an asymmetric distance function introduced by William Thurston in 1986 that is based on the Lipschitz constants of maps between surfaces. While the asymmetry of this metric may seem like a deficiency, Thurston's metric has many favorable properties. I will describe some of these properties as developed in Thurston's work, as well as in more recent investigations, including my joint work with Kasra Rafi, Anna Lenzhen, and Jing Tao. I will also present some visualizations related to Thurston's metric that were created in collaboration with François Guéritaud.

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 10/17/2019

**Time:** 3:30pm - 4:30pm