Problems in Surface Geometry and Topology

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Problems in Surface Geometry and TopologyAbstractHoward MasurThe 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 thesquare with a slit or barrier. The idea is to find asymptotic quadraticestimates. 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 scaleor asymptotic geometry of a group. The principal investigator proposes tostudy the large scale geometry of the mapping class group with a goal ofshowing 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 theplanets. Billiards are another closely related example. There one has apoint mass that moves in straight lines so that when it encountersthe boundary of a region, it rebounds so that the angle of reflectionequals the angle of incidence. Billiards have been studied for over 100years. In order to understand such a dynamical system oneneeds to understand the long term behavior of the orbits. One particularexample of this long term behavior are the study of the periodicorbits. These are the orbits that repeat themselves. The study ofperiodic orbits for billiards in polygons in the plane is the main subjectof this proposal.