Mathematical Sciences: Differential Geometric Problems Related To Mathematical Physics

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Tom Parker will continue his work in gauge theory. This is currently an extremely active and important part of mathematics. It is important in mathematical physics because of its significance in quantum Yang-Mills theory and in topology because of its implications for diffeomorphism types of four dimensional manifolds. The central objects of study are the moduli spaces of self dual connections on vector bundles over a four dimensional Riemannian manifold. Parker's intention is to investigate the geometry of these spaces as Riemannian manifolds. This will build on the present topological descriptions and will help to provide answers to some problems of physical interest.