Geometric and Analytic Aspects of Einstein Metrics

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AbstractAward: DMS 1205947, Principal Investigator: Michael T. AndersonThis project concerns several studies in the geometric, analytic and physical aspects of the Einstein equations and applications to related areas. Of particular interest are global issues for boundary value problems for Einstein metrics, including applications to classical differential geometry such as the isometric embedding problem and the study of minimal and constant mean curvature surfaces in space forms. Research will also be carried out on several topics in general relativity, including the study of quasi-local mass and the Bartnik static extension conjecture. Research related to the renormalization group flow in the AdS/CFT correspondence will also be undertaken in a joint project with physicists working in string theory.The Einstein equations have long been a central focus of study and interest to mathematicians and physicists. They are very important on the mathematical side since they are at the forefront of knowledge and research in the areas of geometric analysis and partial differential equations. On the physical side, they govern our understanding of large-scale physics - the formation of stars, galaxies and the structure of the universe as a whole. As a concrete application, GPS would not be possible without a thorough and full understanding of the Einstein equations. In addition, they lie at the core of string theory - the most actively studied subject in high energy theoretical physics. The project will involve collaboration and interaction between mathematicians and physicists seeking to unravel some of the mysteries of these equations. In addition, the project involves the training of graduate students in these areas important for the future of basic research.