Mathematical Sciences: Gauge Theory

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In the last decade, the interaction between gauge theory and geometric topology on 4-manifolds has produced some striking results. Among these results is the discovery of nonstandard smooth structures on real 4-space. This project investigates three problems related to 4-manifolds: 1) The Donaldson invariants of manifolds; 2) the behavior at infinity of the 'fake' smooth structures on real 4-space; and 3) the explicit computation of several of Donaldson's invariants. This project lies at the interface of topology and physics. A 4-manifold may be represented by 3 space coordinates and a time coordinate. It has recently been proven that the standard Euclidean smooth structure on 4-space when restricted to certain subsets homeomorphic to 4-space gives a smooth structure which is not diffeomorphic to the standard one. Thus, there exist exotic 4- dimensional manifolds. This research concerns the geometric and differential properties of these exotic 4-manifolds.