申请人将在本项目中研究量子射影平面希尔伯特概形的几何和算术性质。申请人的研究方法基于量子射影平面的非交换几何。项目的最终目的.是证明射影平面上的Vafa-Witten S对偶猜想。我们希望建立量子射影平面上希尔伯特概形的几何拓扑性质于量子射影平面的模空间算术性质之.间的联系。基于这种联系，申请人将尝试给出希尔伯特概形欧拉示性数生成函数模性质的内蕴证明。本项目涉及了模空间代数几何，杨米尔斯.规范场理论，Artin-Schelter正则代数以及无穷维李代数表示中的一些重要问题。

This project is to study the geometry and arithmetic of Hilbert scheme of quantum projective planes based on the method of noncommutative geometry. The goal is to approach Vafa-Witten's S duality conjecture in the case of projective plane. We try to build up a link between the geometry and topology of Hilbert schemes of points on quantum projective planes and arithmetic of moduli space of quantum projective planes..We expect to use this link to prove modularity of generating series of Euler characteristic. The project covers some important problems in geometry of moduli space, gauge theory, Artin-Schelter regular algebra and representation of infinite dimensional Lie algebra.