Mathematical Sciences: Topology and Geometry Under Curvature Bounds

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9626419 Wei The proposed research lies in the area of Riemannian Geometry. The investigator wishes to study the effect of Ricci curvature and integral curvature bounds on the local geometric structure and topology of the underlying Riemannian manifold. For example, the investigator is interested in determining the structure of the fundamental group for manifolds with a lower Ricci curvature bound. Riemannian geometry is the study of abstract spaces, so called Riemannian manifolds, possessing a distance function. Riemannian manifolds are abstract spaces in that they do not necessarily lie in an ambient space, unlike curves or surfaces in 3-space. The proposed research has to do with the interplay between the topology of a Riemannian manifold - which determines its overall shape without regard to 'curvature' - and the metric or distance related properties.