Mathematical Sciences: Riemann-Roch Theory and Harmonic Forms

该奖项支持美国普渡大学（Purdue University）的Tong教授在代数几何方面的研究。汤博士的项目由两部分组成。第一个是应用他和托莱多几年前在纯几何背景下发展起来的方法，来验证数学物理中的贝林森-谢赫特曼猜想。这包括在格罗滕迪克二十多年前提出的基础上，对经典的黎曼-洛克定理进行进一步的改进。汤博士项目的第二部分是研究局部对称空间中的谐波形式及其对偶几何循环。这些例子和由此得出的见解将是攻击霍奇猜想的重要工具。这是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 award supports the research in Algebraic Geometry of Professor Yue L. Tong of Purdue University. Dr. Tong's project has two parts. The first is to apply methods that he and D. Toledo developed some years ago in a purely geometric setting, to the problem of verifying the Beilinson-Shechtman conjecture in mathematical physics. This involves getting further refinements of the classical Riemann-Roch Theorem, beyond those developed by Grothendieck over twenty years ago. The second part of Dr. Tong's project is to study harmonic forms and their dual geometric cycles in locally symmetric spaces. The examples and the insights derived will be important tools in attacking the Hodge Con- jecture. This is research in the field of Algebraic Geometry, 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.