This project addresses specific problems in three important areas of the mathematical sciences. The first two such problems concern the mathematical theory of black holes, namely the problem of uniqueness and stability of Kerr black holes and the problem of formation of trapped surfaces and cosmic censorship. The investigators also investigate stability problems in gas dynamics (rarefaction waves) and electrodynamics (crystal optics). All these problems are related by the quest to understand the behavior of important nonlinear equations of mathematical physics in the strong field regime. They also have in common the fact that their resolution requires the development of new mathematical techniques of relevance to other fields of mathematics or science. In particular, the project further develops new geometric techniques which, though originating in works on general relativity, have proved useful in other fields of mathematical physics. Through training of graduate students and postdoctoral researchers, the project contributes to the development of a vibrant scientific community working on mathematical problems connected to these areas. The main focus of the project is on the problem of the stability of Kerr black holes. These are explicit solutions of the Einstein vacuum equations which are at the heart of theoretical understanding of black holes. For the past decade, the Principal Investigator has been involved in the effort to prove the nonlinear stability in the case