Mathematical Sciences: Non-Abelian Hodge Theory and Applications

这支持了克雷曼教授的博士后助理托尼·潘特夫博士的工作。将研究三个问题。第一个是分析广义theta函数的单行群。第二个问题要求对曲线上的动力局部系统的轨迹进行几何描述。第三个要研究的问题是Hyperkahler流形的猜想构造。这是代数几何领域的研究，但它与本世纪理论物理的两大进步--量子力学和广义相对论--直接相关。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。***

This supports the work of Dr. Tony Pantev a post-doctoral associate to Professor Kleiman. Three problems will be studied. The first is to analyze the monodromy group of generalized theta functions. The second problem asks for a geometric description of the locus of motivic local systems on curves. The third problem to be studied is a conjectural construction of Hyperkahler manifolds. 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. ***