Problems in Surface Geometry and Topology

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Problems in Surface Geometry and Topology Abstract Howard Masur The principal investigator proposes to work on problems in the geometry and topology of surfaces. The geometric problems consist of studying the dynamics of flows on flat surfaces with cone angle singularities. A particular example is the study of billiards in polygons with angles that are rational multiples of pi. The particular aspect of this study will be to attempt to find the asymptotic growth rate for the number of periodic orbits. For some billiards such as the square, the growth rate is known. Other more interesting examples are certain right triangles. The principal investigator will study additional examples such as the square with a slit or barrier. The idea is to find asymptotic quadratic estimates. The topological problems concern the mapping class group of a surface. This group is one of the main objects of study in surface topology. One of the recent developments in group theory is to study the large scale or asymptotic geometry of a group. The principal investigator proposes to study the large scale geometry of the mapping class group with a goal of showing that the mapping class group is quasi-isometrically rigid. In the field of dynamical systems one studies the motion of objects. The field had its origins in the study of the motion of the planets. Billiards are another closely related example. There one has a point mass that moves in straight lines so that when it encounters the boundary of a region, it rebounds so that the angle of reflection equals the angle of incidence. Billiards have been studied for over 100 years. In order to understand such a dynamical system one needs to understand the long term behavior of the orbits. One particular example of this long term behavior are the study of the periodic orbits. These are the orbits that repeat themselves. The study of periodic orbits for billiards in polygons in the plane is the main subject of this proposal.