Mathematical Sciences: Applications of Analysis to Problems in Geometry

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One investigator will study problems in the areas of rigidity theorems for Riemannian manifolds, the structure of manifolds without conjugate points, and sharp isoperimetric inequalities. A second investigator will study related problems in the areas of degenerate systems of partial differential equations, isospectral deformations of Riemannian structures, and minimal isometric embeddings of spherical space forms in spheres. The problems to be investigated in this project arise from an example due to Milnor of two isospectral manifolds which are not isometric. While the original example was sixteen dimensional, more recently three dimensional examples have been constructed. Analogously, one might ask if two round drums which have the same rough shape and which sound the same when struck, have the same exact shape. In fact, they must. These two investigators will study related deep questions for higher dimensional hypersurfaces.