Mathematical Sciences: Research Proposal in Algebraic Geometry

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9503249 Donagi Professor Donagi will study problems that derive from the interaction of algebraic geometry with the theory of integrable systems. In particular, he will study mirror symmetry, normal functions of curves in their Jacobians, the geometpic Langlands conjecture and the compatibility of connections of non-abelian theta functions with an abelianized version. This is research in the field of algebraic geometry, yet it directly connects to two of the great advances in theoretical physics in this century--quantum mechanics and general relativity. Algebraic geometry itself is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics. ***