Math 613— Fall 2001 Topics in Analysis

This course will concentrate on hyperbolic geometry and Kleinian groups, and should be of interest to students in complex analysis and geometric group theory.

We will start with the foundations of hyperbolic geometry, including collaring theorems and the Margulis lemma, the connection with complex analysis (uniformization, and the isomorphism of $SO(3,1)$ with $PSL_2(\Bbb C)$), and subgroups of $PSL_2(\Bbb R)$ and $PSL_2(\Bbb C)$. We will continue to study the works of Gromov and Thurston about hyperbolic groups.

The level of presentation and the precise choice of topics after the beginning will depend very much on the audience.