多圆盘上解析Hilbert模的本质正规性及特异零簇的算子理论分析

Hilbert模的本质正规性是Hilbert模几何分析的中心问题之一，与代数几何、C*-代数扩张理论、BDF-理论、指标理论等数学领域有密切的联系。本项目旨在研究多圆盘上Hardy商模、加权Bergman商模的本质正规性及其相关的算子理论问题，希望能证明特异商模的本质正规性，并进而探索出商模本质正规性的充分必要条件。在此基础上，我们将在特异零簇上寻找相应版本的Berger-Shaw定理和von Neumann不等式。

Essential normality of Hilbert modules is one of the central problems in Geometrical Analysis of Hilbert module. It is deeply linked to branches of mathematics such as Algebraic Geometry, C*-extension Theory, BDF-theory and Index Theory. The present programme focuses on the essential normality of quotient modules of the Hardy module and weighted Bergman modules over the polydisc, as well as related topics in Operator Theory. We are aming at proving the essential normality of distinguished quotient modules, and finding the necessary and sufficient condition for the essential normality of quotient modules. Moreover, we will search for the counterpart of Berger-Shaw Theorem and von Neumann Inequality over the distinguished varieties.