Course outline: This course develops the theory of hyperbolic manifolds from the ground up. We start by discussing the various models of hyperbolic space, before proceeding to constructing hyperbolic manifolds in various dimensions. We then contrast the flexibility of hyperbolic structures in dimension 2 with the increasing rigidity in higher dimensions. We will then explore some structural results, including the thick-thin decomposition and Thurston's hyperbolic Dehn surgery theorem.

References:

• An introduction to geometric topology, by Bruno Martelli
• Three-dimensional geometry and topology, Volume 1, by William Thurston
• Hyperbolic knot theory, by Jessica Purcell.

Prerequisites: Math 8061-62.

I will keep a running list of homework problems. Some of them will be mentioned in class; others will be drawn from the above references. I expect you to write up one problem per class day, on average. You can choose which problems to do. In addition, please come by my office to explain a solution at least once every two weeks.

At the end of the semester, everyone will give a 30-40 minute presentation on the topic of your choice, related to the class.