AUGMENTED HOLOMORPHIC BUNDLES

Professor Bradlow studies the geometry of vector bundles. Geometry is a multifaceted, highly active area of research within modern mathematics. While its subject matter can ultimately be related to the shapes of objects, it allows for very general and very abstract notions of shape and object. A vector bundle is a particular type of geometric object, the simplest examples of which are the cylinder and the Mobius band. These, and their higher-dimensional generalizations, play a central role in areas of mathematics such as algebraic and complex differential geometry, and in mathematical gauge theory. They are, moreover, an important part of the mathematical framework in physical theories such as superconductivity and electromagnetism, as well as in string theory and quantum field theory, which seek to explain physical phenomena at the smallest distances and highest energy scales.

In recent years the study of such bundles has been given a new twist: by prescribing certain kinds of extra structure on the bundles, interesting new phenomena have been found and unexpected applications have emerged. There are by now many different examples of such augmented bundles, and a considerable body of knowledge about them has accumulated. This body of knowledge is currently scattered over many places in the literature. During his Center appointment Professor Bradlow and his co-author will work on a monograph that provides a unified, systematic theory of augmented bundles and compiles in one place all examples studied to date.