Geometric analysis problems related to surfaces in mathematical physics

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Professor Mu-Tao Wang proposes to study the geometry of surfaces in mathematical physics and their related PDE's. He plans to continue his research on the mean curvature flow of Lagrangian submanifolds of Kahler-Einstein manifolds. Immediate goals are to understand the behavior of the flow under natural convexity conditions and to apply the results to isotopy problems. Professor Wang shall further investigate the notion of quasilocal mass he recently discovered with Professor Shing-Tung Yau. Immediate goals include studying how the quasi-local mass changes in the spatial, null, and timelike directions and evaluating the quasi-local energy-momentum vector on solutions of the Einstein field equation.Professor Mu-Tao Wang purposes to study the geometry and analysis associated with special curved geometric shapes, including special Lagrangians which are higher dimensional analogues of soap films, and surfaces in space-time that are boundaries of spatial regions.The study of special Lagrangians will provide new perspectives and insights of the geometry of Calabi-Yau manifolds, the space of string theory. Professor Wang's newly discovered quasi-local energy measures the total energy contained in a bounded region of the universe even when the gravitational effect on the boundary is very strong.This notion is essential in many unsolved problems involving singularities in general relativity. The proposed research will further our understanding of energy in the universe and related problems such as black hole formations and collisions.