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

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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.