Mathematical Sciences: Applications of Differential Geometryand Global Analysis to Low Dimensional Topology

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The principal investigator will continue his study of the structure of smooth 4-dimensional manifolds using techniques from geometry and analysis. At the heart of this investigation will be the classification of the homotopy types of closed smooth 4- manifolds. Gauge theoretic methods pioneered by Simon Donaldson were the original impetus for much of this work. In its marriage of theoretical physics, geometry, analysis, and topology, it is one of the most dramatic cross-disciplinary developments of recent years.