Principal Investigator: Kasra Rafi
These projects pursue two major themes. First, we can think of the action of the mapping class group on Teichmueller space as an analogue of the action of a lattice on a Lie group. This analogy is the motivation of some of the proposed problems, namely the rigidity and the counting problems. The other theme is to understand the relation between different metrics on Teichmueller space. There are several distinct metrics of interest on Teichmueller space and many questions are answered for one metric but not for another. We propose the study of the behavior of geodesics in the Lipschitz metric on Teichmueller space, which has the feature that the Lipschitz metric can act as a bridge between the Teichmueller metric and the Lipschitz metric in Outer space, the model space for outer automorphisms of a free group.
The mathematical structures studied in this research program provide coordinates that determine completely a family of two-dimensional geometries. The oldest questions and constructions in these directions are more than 150 years old but remain vital because of their central role in mathematics and their use in analyzing shape-dependent features of physical systems, including brain anatomy identified in MRI images. One of the newer lines of investigation compensates for the longstanding difficulty of imposing a truly satisfactory geometry or metric on Teichmueller spaces by considering several geometries simultaneously, aiming to take advantage of the good features of each. This project aims to pursue this investigation to improve fundamental mathematical understanding of these important spaces.
