Mathematical Sciences: Gauge Theory and Holomorphic Bundles with Sections

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The principal investigator will continue his research in Gauge theory and explore a set of equations known as the Vortex equations. These equations are defined for holomorphic bundles over Kahler manifolds and are closely related to the Hermitian-Yang- Mills equations. The principal investigator will study three areas of Gauge theory in this project: Yang-Mills type functionals, the moduli spaces associated with gauge invariant functionals, and properties of holomorphic vector bundles. Gauge theory is a branch of geometry very closely connected with mathematical physics. It is an area in which ideas from physics are serving to develop and enrich geometry and techniques of mathematics and geometry are used to explore ideas in physics. A number of different methods, some very analytic and others algebraic, are successfully being used to create a powerful and beautiful theory in physics. It is one of the exciting examples of cross disciplinary research being done in mathematics today.