Algebraic Geometry Motivated by Gauge Theory

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Thaddeus 9808529 This project is aimed at a number of problems, mostly in algebraic geometry but also in symplectic topology, which are expected to have consequences for gauge theory, and specifically for Seiberg-Witten-Floer theory. First, it is proposed to study the quantum cohomology of symmetric products of Riemann surfaces. This problem has great interest in its own right and is tractable using algebro-geometric methods. Second, considering the Floer theory for a mapping torus of a finite-order map leads to a conjecture, of independent interest, about the symplectic topology of a symplectomorphism of finite order. A longer-range goal is a proper understanding of the quantum category associated to Seiberg-Witten theory. Several other topics at the interface of algebraic geometry and gauge theory, ranging from loop groups to secant varieties, will also be investigated. This is research in the field of algebraic geometry. Algebraic geometry 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.