Research in Geometric Analysis

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AbstractAward: DMS-9971574Principal Investigator: F. Reese HarveyThis project has three components. First, push forward singularconnections will be analyzed and their associated characteristiccurrents will be determined. Second, the calculus of kernels asdeveloped by Harvey and Polking will be used to attack newproblems in differential geometry. Third, calibration theory,introduced by Harvey and Lawson, will be applied to geometricproblems.In practical geometric problems it is important to distinguish"local" from"global" information. Here "local" means in thevicinity of a point while by contrast "global" includesinformation such as the distance from, say Washington D.C., toanother place on the surface of the earth. Results of particularimportance for applications are frequently ones where globalinformation is computed from an assembly of local information.For example local information alone, involving the amount oflocal curvature of a surface can be used to distinguish theglobal nature of a ball from the global nature of a innertube. This research will develope new instances of this importantprocedure.