FRG: Rational billards and geometry and dynamics on Teichmuller Space

University Of Chicago, Chicago IL

Investigators

Abstract

Abstract The principal investigators will work on two fundamental projects in the field of geometric analysis. The first project concerns the interrelated analytic study of rational billiards, moduli spaces of abelian differentials, and the dynamics of the Teichmuller geodesic flow. A recent theme has been the interplay between the dynamics on flows on homogeneous spaces on the one hand, and the dynamics of flows on rational billiards and flat surfaces on the other. This synergy has led to recent breakthroughs in both subjects. Broadly speaking, the principal investigators propose to apply the methods and ideas developed in the case of homogeneous spaces to study rational billiards and dynamics on Teichmuller space. The second project is to give a combinatorial approach to the study of Teichmuller Space and its geometry. One of the major goals is to develop a complete analogue between this theory and the classical theory of the geometry of the modular curve. Many natural phenomena are studied via the branch of mathematics known as dynamical systems. One important example is planetary motion. Another centuries old example is the theory of billiards, or the study of collisions of molecules. Triangular billiards occur in the study of elastic collisions of two masses in an interval. The study of a dynamical system can take many forms. One example is to study periodic orbits (or repeating behavior); another is to study choatic behavior. The principal investigators plan to study a variety of dynamical systems that occur in mathematics. One of the prominent topics will be billiards in polygons in the plane, and a generalization known as flat surfaces. These surfaces also fit together into a family known as a moduli space, and the principal investigators plan to study dynamics on moduli spaces. A major component of the project will be educational. The principal investigators plan to give workshops for students, both at the undergraduate and graduate levels, in order to introduce them to these fields of mathematics.