Mathematical Problems in General Relativity

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General Relativity has very important long term goals among whichthe weak and strong cosmic censorship, final state and stability of Kerrconjectures are the most outstanding. To complete such goals oneneeds to solve many fundamental difficulties among which the PI proposesto concentrate on the problems of uniqueness and stability of the Kerrfamily as well as the bounded curvature conjecture. The PI intends tocontinue his work with A. Ionescu, and more recently S. Alexakis, toremove the restrictive real analyticity assumptions of the well knownresults of S. Hawking and Carter-Robinson. The PI also plans to continueto work with I. Rodnianski and J. Szeftel and settle the importantbounded curvature conjecture. Together with I. Rodnianski they planto adapt the recent work of Christodoulou, based on what he callsthe short pulse method, with which he was able to derive the firstresult on the formation of trapped surfaces for solutions of the Einsteinvacuum 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 someof the main theoretical open problems of General Relativity. The longterm 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 holeinteractions. The PI also hopes that the interplay between geometric and an-alytic methods, which are being developed, will influence other areas ofnonlinear PDE, Differential Geometry and Analysis. The PI plans to attractand train a larger number of PhD students and postdocs, than he hasdone so far, in General Relativity. The PI believes that the subject is goingnow through an exciting period when important problems are solved and many new avenues of research are being open.