Mathematical Scienaes: Moduli of Stable Bundles, S-Duality Conjecture, and Gromov-Witten Invariants

9622564 在这个项目中，七个问题的一般背景下的代数几何，量子上同调，以及它们与物理学的相互作用，提出要调查。 主要工具是代数曲面上Gieseker半稳定丛模空间上的Grothendieck-Knudsen-Mumford行列式线丛和Gromov-Witten辛不变量。该研究将增加几何学的知识，包括Gieseker半稳定丛模空间的量子上同调环结构，Vafa和维滕提出的物理学S-对偶猜想，以及某些一般型极小曲面的变形类型（Catanese和Reid提出的猜想）。 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，并且反过来在这些领域以及物理学，理论计算机科学和机器人学中找到应用。

9622564 In this project, seven problems in the general context of algebraic geometry, quantum cohomology, and their interplay with physics are proposed to be investigated. The main tools are Grothendieck-Knudsen-Mumford determinant line bundles over moduli space of Gieseker semistable bundles on algebraic surfaces and Gromov-Witten symplectic invariants. The investigation would increase the knowledge about the geometry including the quantum cohomology ring structure of the moduli space of Gieseker semistable bundles, the S-duality conjecture from physics in the form of Vafa and Witten, and the deformation type of certain minimal surfaces of general type (a conjecture due to Catanese and Reid). 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 the 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.