Topics in Discrete Groups and Geometry

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Richard Schwartz proposes to study several problems in geometry and dynamics. The first problem has to do with the deformations of complex hyperbolic discrete groups. Schwartz proved a general surgery theorem for many of these groups, along the lines of Thurston's hyperbolic Dehn surgery theorem, and he hopes to use this theorem to analyze some concrete examples. The second problem has to do with triangular billiards. Recently Schwartz proved that a triangular shaped billiard table has a periodic billiard path provided that all its angles are less than 100 degrees. This represents the first substantial progress on the 200-year old triangular billiards problem, which asks if every triangular shaped billiard table has a periodic billiard path. The third problem has to do with the iteration of simple geometric constructions, such as barycentric subdivision.The common theme to Schwartz's research is the idea of looking at the consequences of performing a simple operation infinitely often. In nature one sees great complexity in things like trees and coral reefs, and this complexity is often produced by the same process happening multiple times. For instance, some kind of basic branching operation produces the shape of a tree. The problems studied by Schwartz are idealizations of the sort of thing one might see in nature - for instance, a complex hyperbolic discrete group encodes the way light bounces of mirrors in a certain curved 4 dimensional space, and the triangular billiards problem asks what happens if one plays pool on a frictionless pool table with a billiard ball that is the size of a point. The barycentric subdivision problem is somehow an idealization of the idea of chopping up a high dimensional diamond over and over again, and studying the shapes of the facets.