Brill-Noether Theory for K3 Surfaces; Compactified Picards and Coadjoint Orbits of Loop Algebras

点击翻译按钮获取中文摘要

Markman 9802532 The problems studied in this project arise from the interaction of algebraic geometry, symplectic geometry and integrable systems. Part 1 of the project consists of the study of a birational duality among moduli spaces of sheaves on K3 surfaces arising from the Brill-Noether stratification of these symplectic moduli spaces. The interaction of this duality with a representation of the Heisenberg algebra on the total cohomology of sequences of moduli spaces is investigated. In Part 2 of the project the theory of integrable systems will be used to compare the boundary of the compactified Picard of an algebraic curve with planar singularities to the boundary of coadjoint orbits of loop algebras. 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.