Mathematical Sciences: Moduli Problems in Algebraic Geometry

9622912 Li本课题的目标是研究代数几何中模空间的计数几何。作为第一步，本项目将研究如何将基于正规锥构造的超交理论应用于研究模空间的计数问题，这些模空间的维度大于预期维度。然后，Li将利用这一技巧研究光滑射影簇的Gromov-Witten不变量，研究Calabi-Yau流形和Fano簇中有理曲线的计数问题，并推导出代数曲面的Donaldson多项式不变量的递推公式。其中一些是用解析法已知的。李相信，沿着这条线取得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，并将成为未来良好数学问题的来源。这是在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25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

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.