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

9622564在这个项目中，在代数几何、量子上同调及其与物理学的相互作用的一般背景下，提出了七个问题进行研究。主要的工具是代数曲面上Gieseker半稳定束模空间上的Grothendieck-Knudsen-Mumford行列式线束和Gromov-Witten辛不变量。该研究将增加对几何的认识，包括Gieseker半稳定束模空间的量子上同环结构，Vafa和Witten形式的物理学s -对偶猜想，以及一般类型的某些最小曲面的变形类型（Catanese和Reid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。在原点中，它处理的图形可以在平面上用最简单的方程来定义，即多项式。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。

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.