Mathematical Sciences: Geometry of Moduli Spaces of Vector Bundles

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9503635 Wentworth This project lies in the area of smooth four-manifolds, more specifically, Donaldson theory. The investigator wants to find necessary and sufficient conditions for the existence of symplectic structures on smooth four-manifolds; to find the Donaldson series for the complex projective plane; to understand how the Donaldson series is affected under certain birational transformations. The modern study of smooth four-dimensional spaces was initiated, in large measure, by the work of Simon Donaldson in the early eighties; Donaldson used ideas from theoretical physics - Yang-Mills equations and gauge theories - in an ingenious way to help classify smooth four-manifolds. It is worth mentioning that Nathan Seiberg and Edward Witten have recently come up with an easier and more direct theory of smooth four-manifolds, again motivated by ideas from theoretical physics, namely monopoles and string theory.