This course will discuss a basic issue in modern differential geometry, namely under what conditions is a given space of metrics compact, and if not compact, how can such metrics degenerate. This topic has many applications, including to the structure of moduli spaces of "special" metrics, (Einstein, self-dual, special holonomy, etc), and the behavior at the boundary of such moduli spaces. Important applications also arise for instance in Ricci flow, (work of Hamilton and Perelman), and in general relativity. Thus, all three basic types of PDE's, elliptic, parabolic and hyperbolic, tie into this area.

There is no formal text that will be used for the course. Some useful general references are:

Grades for the class will be based on class participation and possibly also on a oral presentation or lecture to the class.