Mathematical Sciences: Enumerative Geometry of Moduli Spaces

数学科学：模空间的枚举几何

• 批准号: 9500865
• 负责人: Aaron Bertram
• 金额: $ 5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500865&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Enumerative; Geometry; Moduli

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.

伯特伦教授将研究变形的上同调环的齐次投影品种以及顺利超曲面。 这个变形环（通常称为“量子上同调环”）编码了大量关于有理曲线到簇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。 这是一项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世纪中却有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。 如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。