Moduli of vector bundles, Donaldson-Thomas theory and Gromov-Witten theory

向量丛的研究是代数几何中的一个基本问题。彭罗斯和阿蒂亚的工作开创了他们与物理学和表象理论的联系。它们在唐纳森理论和塞伯格-威滕理论中发挥了核心作用。从1995年开始，从事弦论研究的物理学家们推测出了许多令人惊讶但深刻的结果，这些结果涉及向量丛、格罗莫夫-威滕理论以及它们与表象论的相互作用。在过去的几年里，向量丛的模空间（包括点和曲线的Hilbert方案）、Donaldson-Thomas理论和Gromov-Witten理论之间的优美关系已经被揭示出来。在这个项目中，秦教授打算在代数几何的一般背景下研究向量丛的模、Donaldson-Thomas理论和Gromov-Witten理论的几个问题及其与表示论和弦理论的相互作用。主要工具是局部化的虚基本类、顶点代数、量子上同调、曲面上的稳定丛和三重。代数几何研究由多项式方程描述的几何对象。它一直处于最近数学和物理学交汇的中心阶段。这些相互作用极大地促进了人们对数学和物理的理解。秦教授的研究有助于加强这些相互作用。

The study of vector bundles is a fundamental problem in algebraic geometry. Their connections with physics and representation theory were pioneered in the work of Penrose and Atiyah. They played central roles in Donaldson theory and Seiberg-Witten theory. From 1995, physicists working in string theory have speculated many surprising but deep results concerning vector bundles, Gromov-Witten theory and their interplay with representation theory. In the past few years, elegant relations among moduli spaces of vector bundles (including Hilbert schemes of points and curves), Donaldson-Thomas theory and Gromov-Witten theory have been revealed.In this project, Professor Qin intends to study several problems concerning moduli of vector bundles, Donaldson-Thomas theory and Gromov-Witten theory in the general context of algebraic geometry and its interplay with representation theory and string theory. The main tools are localized virtual fundamental classes, techniques of vertex algebras, quantum cohomology, and stable bundles on surfaces and three folds.Algebraic geometry studies geometric objects described by polynomial equations. It has been at the central stage of recent confluence between mathematics and physics. Many of these interactions have led to profound improvement in the understanding of both mathematics and physics.Professor Qin's research helps to strength these interactions.