Richard Schwartz plans to continue his research in geometry and discrete groups. One of his main goals is to explore the connections between three dimensional real hyperbolic geometry and four dimensional complex hyperbolic geometry. Schwartz observed that certain complex hyperbolic deformations of the reflection triangle groups lead to complex hyperbolic four-manifolds whose ideal boundaries are real hyperbolic three-manifolds. As the angles in the triangle group change, the ideal boundary appears to undergo Dehn surgery. Schwartz also plans to study quotients of the circle, based on patterns of geodesics in the hyperbolic plane which are invariant under the action of a surface group. The idea is to make topological models for the limit sets which could arise in connection with deformations of surface groups into Lie groups and then use the models to study actual deformations of surface groups into Lie groups.
Broadly speaking, Schwartz' research deals with the geometry of infinite repeating patterns. A crystal lattice is an example of an infinite repeating pattern in Euclidean space. Analogous patterns exist in curved spaces, and often the curvature of the space allows for the existence of more exotic and geometrically rich patterns. Schwartz is interested in the studying these exotic patterns when they are generated by a simple mechanism which has a finite description. For example, one can place several mirrors in a curved space and look at the pattern generated by the images of an object which is reflected endlessly in the mirrors. The central question is: When does the finite mechanism of generation lead to an infinite discrete pattern?
