PI: Fefferman DMS-9540525 The PI will investigate analytic problems which arise in differential geometry. One project is concerned with semilinear elliptic equations on simply connected domains in Euclidean 4-space. The domain is viewed as the universal cover of a complete conformally flat manifold. The second project will be the study of coupled elliptic-parabolic equations. By coupling an elliptic equation to Hamilton's Ricci flow, one can impose control on the scalar curvature, so that if the scalar curvature is a negative to begin with, then it remains so at least for short time. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.
