Mathematical Sciences: Topics in Riemannian and Complex Geometry

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The principal investigators will study several problems in global Riemannian geometry related to the existence, classification, and degeneration of compact Riemannian manifolds which are extremal for an integral norm of the curvature. They will also study applications of this theory to Kahler geometry, differential topology, and mathematical physics. One principal investigator will study the existence of degenerate solutions to the critical metric problem. The second principal investigator will study the holonomy of complete manifolds of positive scalar curvature. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.