Harmonic Forms and Topology of Complete Manifolds

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DMS-0404817P.I.: Jiaping Wang, University of Minnesota-Twin CitiesTitle: Harmonic forms and topology of complete manifoldsABSTRACTThis proposed project is about relating the analysis and topology on complete noncomapct manifolds. In particular, it is proposed to searchfor an extension of the de Rham-Hodge theory, and to relate the spectrum ofthe Hodge Laplacian to the topology of manifolds.The main objective of this project is to describe some large scalestructuresof objects which are high dimensional analogue of the unbounded surfaces.The approach is to inspect the physical steady states of a naturallydefinedenergy on such objects. The results and techniques are expected to beusefulin describing more complicated systems arising from physics, biology andotherbranches of science.