Mathematical Sciences: Differential Equations and Differential Geometry

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This research focuses on differential equations and differential geometry. The research involves the study of manifolds without conjugate points, sharp isoperimetric inequalities for manifolds of various dimensions, degenerate systems of partial differential equations (especially equations of prescribed curvature, the geometry of isospectral manifolds, certain classes of minimal submanifolds, and certain applications of geometry to physiology. In general, this research concerns mathematical models of the Riemannian geometry of manifolds. Some of this research has application to physiology; for example, the traditional treatment of gallstones is cholecystectomy (removal of the gallbladder). However, lithotripsy (fragmentation) and dissolution of gallstones has been advanced as an alternative in recent years. The investigators on this award are involved in developing a sophisticated mathematical model which predicts the behavior of fragments in solvent over time. The model predicts the evolution of the stone/solvent interface and is a modification of the curvature flow in differential geometry studied by many researchers.