Mathematical Sciences: Partial Differential Equations in Riemannian and CR Geometry

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Professor Lee will study nonlinear partial differential equations related to problems in geometry. One of the goals will be to find complete Einstein metrics on the ball with prescribed conformal infinity. A second area of research will be the space of CR structures on a three dimensional manifold. The work on partial differential equations will involve an attempt to extend the idea of the maximum principle to systems. There have been numerous semi- successful attempts at this in the past and any progress made by the principal investigator in this direction should find immediate application. This award will support research in the cross disciplinary area of global analysis which studies geometry through partial differential equations. One of the interesting applications of the research involves the inverse problem of measuring the conductivity properties of a material from electrical measurements made on the boundary of the material. It is possible to reformulate this problem as one in differential geometry and then uses methods from this well developed field to attempt to answer questions concerning the inverse problem.