Mathematical Sciences: Classification Problems in Geometric Topology

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The investigator intends to focus on classifying certain kinds of manifolds using a combination of geometric methods and stable homotopy theory. In particular, he would like to find necessary and sufficient conditions for the existence of positive scalar curvature metrics, and give a diffeomorphism classification of Dupin hypersurfaces and complete intersections. Classification of manifolds with nice geometric properties is as old as the Greeks. Here the properties under consideration are more modern, but the basic idea is much the same.