Hilbert Schemes and Moduli of Vector Bundles

Hilbert格式和向量丛的研究是代数几何中的一个基本问题。他们与物理学和表象理论的联系是在阿蒂亚和彭罗斯的工作中开创的。 光滑曲面上点的Hilbert概型及其与无限维李代数表示理论的关系是代数几何中一个特别美丽的主题，它与其他领域有关，包括Gromov-Witten理论，Donaldson-Thomas理论，McKay对应，S-对偶猜想，可积系统，orbifold上同调和n！代数组合学中的猜想。在这个项目中，秦教授打算在代数几何的一般背景下研究关于希尔伯特方案和向量丛模的几个问题，及其与表示论和弦理论的相互作用。主要的工具是顶点代数，量子上同调，虚局部化公式，导出的范畴层，稳定的丛表面以及卡-丘3倍的技术。该项目还包括研究生和博士后的参与和培训.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.这些相互作用中的许多都导致了对数学和物理的理解的深刻改善。秦教授的研究有助于加强这些相互作用，并在数学物理中有许多根源

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