Research in Geometry and Topology

University Of California-Davis, Davis CA

Investigators

Abstract

The goal of this project is to understand structure of 3-dimensional Poincare duality groups, Kleinian groups, flat conformal structures, polygons in symmetric spaces and apply the latter to the representation theory. This project focuses in part on the structure of symmetries of objects naturally arising in geometry, algebra and physics. Such symmetries include both discrete symmetries (like the ones of wallpaper patterns and regular solids) as well as continuous symmetries (like translations and rotations of the space). Presence of ``negative curvature'' (which implies for instance that sum of angles in a triangle can be less than 180 degrees) makes study of such symmetries more interesting and challenging. Polygons provide a link between geometry and Lie theory, which is a field of mathematics which involves both continuous and discrete symmetries. Polygons, which are familiar objects from the high-school geometry, become much more complex when considered in negatively curved spaces. The ``side-lengths'' of polygons in such spaces become vectors rather than numbers. Such polygons provide a poverful tool in the Lie theory.