Gauge Theory on Bundles with Sections

The proposed research lies at the boundary between Gauge Theory and Holomorphic Bundles. The latter are geometric objects with a particularly rich structure, enabling them to be studied from a number of different points of view. They have been extensively studied within the paradigm of Algebraic Geometry, but in recent years Analytic techniques have been shown to be useful. This development was made possible by the discovery that certain key properties of the holomorphic bundles can be understood as, or correspond to, conditions for the solvability of certain partial differential equations. These equations are familiar in Differential Geometry, and, in particular, in Gauge Theory. The projects described in Professor Bradlow's proposal attempt to exploit these correspondences. In one project, the idea is to use well-known relations between equations in Gauge Theory in order to explore undiscovered relations between the corresponding holomorphic objects. In another project, the aim is to use the fact that the set of all holomorphic bundles with a given property (called stability) can be viewed as the set of all solutions to a corresponding equation. This provides a useful tool for studying the sets of holomorphic objects, i.e., for addressing moduli space questions.