Evolution problem in General Relativity

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Evolution Problem in General Relativity Abstract of Proposed ResearchSergiu KlainermanThe research focuses on the mathematical analysis of the evolution of the solutions of the Einstein-field equations. Attention will be concentrated on a number of conjectures about these solutions. One concerns the relationship between the curvature tensor and the causal geometry of space-time. Two concern the phenomenon of "cosmic censorship", other concerns the asymptotic behavior or "final state" and one about the stability of the Kerr solution. The resolution of these problems will require the study of the relationship between geometry and curvature of Lorentzian manifolds as well as analysis of the Einstein equations. Einstein's equations are the field equations of general relativity and their analysis is central to the understanding of many issues in cosmology including the theory of black holes and gravitational waves. There are only a few exact solutions known of these equations and this research will pursue a more mathematical analysis that will prove qualitative results about the solutions. This analysis should also help the development of algorithms for the numerical computation of solutions of these equations.