Mathematical Sciences: Moduli Problems in Algebraic Geometry

9622912李本课题的目的是研究代数几何中模空间的枚举几何。作为第一步，本项目将研究如何将基于正锥构造的过度相交理论应用于研究维数大于预期维数的模空间上的枚举问题。随后，李将运用这一技术研究光滑投影变量的Gromov-Witten不变量，研究Calabi-Yau流形和Fano变量中有理曲线的枚举问题，并推导代数曲面的Donaldson多项式不变量的递推公式。其中一些是用解析法知道的。李认为，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，并将成为未来优秀数学问题的来源。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。

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.