Augmented Holomorphic Bundles

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DMS-0072073Steven B. BradlowHolomorphic bundles arise naturally in many different areas of geometry - indeed they lie at the intersection of algebraic, symplectic, and complex differential geometry. At the center of this intersection are sets of partial differential equations, such as the so-called Hermitian-Einstein and Vortex equations, which place constraints on the geometric features of the bundles. Their solutions carry information not only about the geometry of the bundles on which they are defined, but also about their topological and algebraic structure. In recent year it has been discovered how by adding certain extra structure to a holomorphic bundle, interesting new phenomena are revealed and importantapplications can result. The primary goals of this proposalinclude a fuller understanding of these `augmented bundles', the equations defined on them, their moduli spaces, and theirapplications.Holomorphic bundles fall within the class of geometric objectsknown as fiber bundles. In addition to their prominent place in modern geometry, fiber bundles play an important role withinmodern theoretical physics, where the influence of geometry canhardly be overstated. General relativity, electromagnetism and its extensions (known as gauge field theories), and string theorycould not be formulated without sophisticated geometric tools suchas vector and principal bundles.