Mathematical Sciences: Geometry of Surfaces, Soliton Equations, and Loop Groups

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The principal investigator will continue his study of minimal surfaces in spaces of constant mean curvature. In particular he will study surfaces of higher genus such as the torus embedded in the sphere. The techniques to be used involve integrable systems in infinite dimensional spaces. This technique is very different from the standard methods used to study these problems thus far. 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.