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

马克曼9802532本课题研究的问题来自于代数几何、辛几何和可积系统的相互作用。该项目的第一部分包括研究K3曲面上的板模空间之间的双对偶，这些模空间是由这些辛模空间的Brill-Noether分层而产生的。研究了这种对偶性与模空间序列的全上同调上的Heisenberg代数表示的相互作用。在项目的第二部分，我们将利用可积系统理论来比较具有平面奇点的代数曲线的紧化Picard的边界与循环代数的余伴轨道的边界。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

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.