Geometry and Dynamics in Low Dimensional Topology

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A major role of mathematics is to provide a framework or language for scientific research. This mathematical research project centers on the concept of shapes and their evolution, a fundamental object of study in many parts of science. Examples range from the shapes of complicated molecules, such as proteins and enzymes, to the configurations of the solar system. In mathematics, Teichmueller theory is the study of the shapes that a surface can assume. In this project, surfaces, their geodesics (shortest paths), and their evolution will be studied in two different settings. In the first, the surfaces have a geometry in which they are flat and so have zero curvature. In the second, the surfaces have negative curvature. Results of the project will deepen and extend understanding in geometry and topology.This project addresses questions in a wide spectrum of mathematics, including the fields of complex analysis, dynamical systems, geometry, and topology. Important examples of dynamical systems are interval exchange transformations and flows on translation surfaces. Important aspects of these dynamical systems are ergodicity and counting problems. The properties of an individual translation surface are intimately related to the study of the moduli space of all translation surfaces. The project concerns questions in both these subjects. The investigator plans to continue study of the mixing properties of the Weil-Petersson geodesic flow on the moduli space of Riemann surfaces. Knowledge of this flow will greatly expand our understanding of the geometry of Riemann surfaces. A fundamental object in topology and geometry is the mapping class group of a surface; the project aims to further understanding by studying the lattice counting problem for the mapping class group acting on Teichmueller space.