GEOMETRY AND DYNAMICS ON MODULI SPACE

University Of Chicago, Chicago IL

Investigators

Abstract

The principal investigator (PI) will work on problems in the geometry and dynamics of Teichmuller space, moduli space of Riemann surfaces, directional flows on polygons and translation surfaces, and the mapping class group. These problems involve a fairly wide spectrum of mathematics including complex analysis, dynamical systems, and geometry. The objective of the first part of the project is to study the Weil-Petersson flow on moduli space and the geometry of its geodesics. This metric is fundamental in studying properties of moduli space. There are many important connections between the Weil-Petersson metric and the hyperbolic geometry and dynamics of surfaces. In the second part the goal is to study properties of Teichmuller geodesics and their relation to flows on surfaces, and in the third part rigidity questions about Teichmuller space with the Teichmuller metric. The last part of the project concerns the action of the mapping class group on Teichmuller space and on its boundary at infinity. If the goals of the project are met, it would add to our understanding of these subjects. A major role of mathematics is to provide a framework or language to discuss processes of science. The framework developed in this project centers on the concept of shape. In mathematics, Teichmuller theory and moduli space theory is the study of the shapes that a surface can assume. Geodesics in a space represent an efficient way of moving or evolving in the space and the study of geodesics is central to geometry. Both Weil-Petersson geodesics and Teichmuller geodesics describe natural and very different ways in which the shapes of surfaces evolve efficiently over time. This project concerns the study of this evolution under these two different processes.