Mathematical Sciences: Moduli of Vector Bundles on Surfaces

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This award supports research in algebraic geometry. There are two objectives of the project. The first is to obtain universal realtions for the values of Donaldson polynomials on classes of algebraic curvs of arbitrary genus and self-intersection. The second is to understand global properties of moduli spaces of bector bundles on surfaces. The research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past ten years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.