Hilbert Schemes and Moduli of Vector Bundles

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The study of Hilbert schemes and vector bundles is a fundamental problem in algebraic geometry. Their connections with physics and representation theory were pioneered in the work of Atiyah and Penrose. The Hilbert schemes of points on smooth surfaces together with its relation to representation theory of infinite dimensional Lie algebras is a particularly beautiful subject in algebraic geometry, which is related to other areas including Gromov-Witten theory, Donaldson-Thomas theory, the McKay correspondence, the S-duality conjecture, integrable systems, orbifold cohomology, and the n!-conjecture in algebraic combinatorics. In this project, Professor Qin intends to study several problems concerning Hilbert schemes and moduli of vector bundles in the general context of algebraic geometry and its interplay with representation theory and string theory. The main tools are techniques of vertex algebras, quantum cohomology, the virtual localization formula, derived categories of sheaves, and stable bundles on surfaces as well as on Calabi-Yau 3-folds. This project also involves the participation and training of graduate students and postdoctoral associates.Algebraic geometry studies geometric objects described by polynomial equations.It has been at the central stage of recent confluence between mathematics andphysics. 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, and has many of its roots in mathematical physics