Mathematical Sciences: Non-Abelian Hodge Theory and Applications

这支持了Tony Pantev博士的工作，他是Kleiman教授的博士后助理。将研究三个问题。首先是分析广义函数的单群。第二个问题要求对动力局部系统在曲线上的轨迹进行几何描述。要研究的第三个问题是超卡勒流形的猜想构造。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。* * *

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. ***