Mathematical Sciences: Applications of Elliptic PDE's to Geometry and Topology

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The principal investigators will study geometric scale- invariant elliptic partial differential equations. Such equations arise naturally in many fields of mathematics including topology, geometry , and mathematical physics. The three specific areas of research supported by this award are Dirac type operators, Yang- Mills equations, and curvature functionals. 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.