Mathematical Sciences: Research Proposal in Algebraic Geometry

9503249 Donagi教授将研究代数几何与可积系统理论相互作用产生的问题。特别是，他将研究镜像对称，曲线在雅可比矩阵中的正规函数，几何朗兰兹猜想以及非阿贝尔函数与阿贝尔化函数的连接的相容性。这是代数几何领域的研究，但它直接关系到本世纪理论物理学的两大进步——量子力学和广义相对论。代数几何本身是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。＊＊＊

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. ***