General Relativity has very important long term goals among which the weak and strong cosmic censorship, final state and stability of Kerr conjectures are the most outstanding. To complete such goals one needs to solve many fundamental difficulties among which the PI proposes to concentrate on the problems of uniqueness and stability of the Kerr family as well as the bounded curvature conjecture. The PI intends to continue his work with A. Ionescu, and more recently S. Alexakis, to remove the restrictive real analyticity assumptions of the well known results of S. Hawking and Carter-Robinson. The PI also plans to continue to work with I. Rodnianski and J. Szeftel and settle the important bounded curvature conjecture. Together with I. Rodnianski they plan to adapt the recent work of Christodoulou, based on what he calls the short pulse method, with which he was able to derive the first result on the formation of trapped surfaces for solutions of the Einstein vacuum equations. The PI and his collaborator believe that the new method can be signifficantly improved and applied to various situations.
If successful the proposal will give a better understanding of some of the main theoretical open problems of General Relativity. The long term hope is that some of the insight gained in the process will influence related areas such Theoretical Physics and Numerical Relativity. Of paramount interest to the gravitational wave experiments is to de- sign numerical algorithms which can simulate, realistically, black hole interactions. The PI also hopes that the interplay between geometric and an- alytic methods, which are being developed, will influence other areas of nonlinear PDE, Differential Geometry and Analysis. The PI plans to attract and train a larger number of PhD students and postdocs, than he has done so far, in General Relativity. The PI believes that the subject is going now through an exciting period when important problems are solved and many new avenues of research are being open.
