The present proposal is concerned with the global analysis of solutions to the Einstein equations of general relativity. This subject lies at the intersection of the analysis of partial differential equations, differential geometry and theoretical physics and is a field of intense current activity, with several important advances having been achieved in the last decade only.The main objective of the proposal is to establish a research group based at Imperial College to develop novel mathematical techniques that would allow one to move considerably beyond the current limits of the field. These techniques will be devised and mature in the context of two fundamental problems, which we intend to solve.1) Instability of AdS: The stability of Minkowski space and the stability of de Sitter space are celebrated theorems in mathematical general relativity. In contrast, the dynamics near Anti de Sitter (AdS) space, the maximally symmetric solution with negative cosmological constant, is mathematically entirely unexplored. Heuristic andnumerical arguments suggest instability of this spacetime. Instability problems are typically much more intricate than stability problems and require very different techniques. A rigorous proof of instability would resolve a major conjecture in general relativity and have important implications for theoretical physics.2) The Black Hole Stability Problem: A central problem of general relativity is to prove the full non-linear stability of the 2-parameter Kerr family of black holes. Very recently, the dynamics of linear waves on such stationary black holes has been satisfactorily understood. The proposal suggests to suitably enhance the techniques developed for linear scalar waves to be applicable in the non-linear, tensorial setting of of the Einstein equations. Key will be to establish important estimates on the curvature in a class of (non-stationary) spacetimes which are assumed to converge to a fixed member of the Kerr family.