Topology Seminar: A tale of 2 metrics

Seminar | October 23 | 4:10-5 p.m. | 3 Evans Hall

 Jing Tao, University of Oklahoma

 Department of Mathematics

Associated to a surface S is the deformation space of all conformation structures on S up to isotopy. This is the Teichmuller space T(S) of S, a topological space naturally homeomorphic to a cell. Via the Uniformization Theorem, T(S) is also the deformation space of hyperbolic structures on S up to isotopy. Teichmuller space can be equipped with several natural metrics. In this talk, I will focus on two such metrics. The first one, introduced by Teichmuller in 1940, allows one to quantify the difference between two conformal structures. The second one, introduced by Thurston in 1986, quantifies the difference between two hyperbolic structures. The goal of my talk will be to give a broad overview of their similarities and differences.