Mathematical Sciences: Riemann-Roch Theory and Harmonic Forms

该奖项支持在代数几何的研究， 余良教授。普渡大学的唐。唐博士的项目 有两部分。第一种是应用他和D. 托莱多几年前在一个纯粹的几何环境中发展起来， Beilinson-Shechtman猜想的验证问题 数学物理这涉及到进一步的改进 经典的Riemann-Roch定理，超越了 格罗滕迪克二十多年前。唐博士的第二部分 项目是研究调和形式及其对偶几何 局部对称空间中的循环这些例子和见解 将是攻击霍奇骗局的重要工具。 jecture。 这是代数几何领域的研究， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。如今，该领域 使用的方法不仅从代数，但从分析和 拓扑学，反过来说， 以及物理学、理论计算机科学和机器人学。

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.