Vector Bundles and their Interplay with Representation Theory and String Theory

秦教授将致力于代数几何的一般领域及其与表示论和弦理论的相互作用。 主要工具是点在射影曲面上的Hilbert方案，顶点代数技巧，量子上同调，曲面上Gieseker半稳定丛的模空间，以及Calabi-Yau三重的全纯Casson不变量。这些研究将有助于加深人们对Hilbert概型、顶点代数和量子上同调之间关系的认识，有助于理解Calabi-Yau三重上的稳定丛以及Vafa和维滕提出的物理学S-对偶猜想。秦教授的研究领域为物理学理论提供了坚实的数学基础，而物理学理论对解释宇宙至关重要。

Professor Qin will work in the general area of algebraic geometry and its interplay with representation theory and string theory. The main tools are the Hilbert schemes of points on projective surfaces, techniques of vertex algebras, quantum cohomology, the moduli spaces of Gieseker semistable bundles on surfaces, and the holomorphic Casson invariants for Calabi-Yau three folds. The investigation should increase the knowledge about the relations among Hilbert schemes, vertex algebras and quantum cohomology, and shed light on stable bundles over Calabi-Yau three folds and the S-duality conjecture from physics in the form formulated by Vafa and Witten.Algebraic geometry studies geometric objects described bypolynomial equations. It has been at the central stageof recent confluence between mathematics and physics.Many of these interactions have led to profound improvement inthe understanding of both mathematics and physics.Professor Qin's research areas provide solid mathematical foundation to the physics theories which are vital to explain the universe.