Augmented Holomorphic Bundles

DMS-0072073史蒂文B。布拉德洛全纯丛自然出现在许多不同领域的几何-事实上，他们躺在交叉的代数，辛，和复杂的微分几何。在这个交叉点的中心是一组偏微分方程，例如所谓的厄米-爱因斯坦方程和涡方程，它们对束的几何特征施加了约束。他们的解决方案携带的信息不仅对几何的丛，他们被定义，但也对他们的拓扑和代数结构。近年来，人们发现了如何通过给全纯丛增加某种额外结构，揭示了一些有趣的新现象，并得到了一些重要的应用。 这个命题的主要目标包括更全面地理解这些“增广丛”，定义在它们上面的方程，它们的模空间，以及它们的应用。全纯丛属于被称为纤维丛的几何对象的范畴。 除了在现代几何中的突出地位外，纤维束在现代理论物理中也起着重要的作用，几何的影响怎么强调都不过分。广义相对论、电磁学及其扩展（称为规范场论）和弦论，如果没有复杂的几何工具，如向量和主丛，就无法形成公式。

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.