Mathematical Sciences: Moduli Problems in Algebraic Geometry

数学科学：代数几何中的模问题

• 批准号: 9622912
• 负责人: Jun Li
• 金额: $ 6.3万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622912&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Moduli; Problems; Algebraic

9622912 Li The goal of this project is to study enumerative geometry of moduli spaces in algebraic geometry. As the first step, this project will investigate how excessive intersection theory based on normal cone construction can be applied to study enumerative problems on moduli spaces whose dimensions are bigger than their expected dimensions. Afterward, Li will apply this technique to study Gromov-Witten invariants of smooth projective varieties, to study enumerative problems of rational curves in Calabi-Yau manifolds and Fano varieties and to derive recursion formulas for Donaldson polynomial invariants of algebraic surfaces. Some of them are known using analytic method. Li believes that progress along this line will provide new insight to these problems, new technique to attack similar problems and will be a source of good mathematical problems in the future. This is research in the field of algebraic geometry. Algebraic geometry 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.

小行星9622912 本计画的目标是研究代数几何中模空间的计数几何。 作为第一步，本项目将研究如何基于法锥构造的过交理论可以应用于研究维数大于其期望维数的模空间上的计数问题。 之后，李将运用这种技术来研究光滑投影簇的Gromov-Witten不变量，研究Calabi-Yau流形和Fano簇中有理曲线的计数问题，并推导代数曲面的唐纳森多项式不变量的递归公式。 其中一些是已知的使用分析方法。 他认为，沿着这条路线的进展将为这些问题提供新的见解，为解决类似问题提供新的技术，并将成为未来好的数学问题的来源。 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。