Mathematical Sciences: Enumerative Geometry of Moduli Spaces

Bertram教授将研究齐次射影变的上同环的变形以及光滑超曲面。这种变形环（通常称为“量子上同调环”）编码了大量关于有理曲线到变化的映射的模空间的枚举几何的数据。这个项目的主要目标是获得这个环存在的纯代数证明。这是代数几何领域的一项研究，但它与本世纪理论物理学的两项重大进展——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。

Professor Bertram will study deformations of the cohomology rings of homogeneous projective varieties as well as smooth hypersurfaces. This deformation ring (usually called the "quantum cohomology ring") encodes a great deal of data about the enumerative geometry of the moduli space of maps of a rational curve to the variety. The primary goal of this project is to obtain a purely algebraic proof of the existence of this ring. This is a 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.