Hilbert模的局部化，表示及相关几何分析

Douglas, Paulsen等数学家从交换代数的观点出发，创立了Hilbert模这一现代多元算子理论的基本框架。 局部化和表示作为源于代数的基本方法，对于研究Hilbert模的性质和分类具有重要的意义。本项目拟基于Cowen-Douglas理论，结合复几何的方法系统研究定义在高维复空间区域上的一类重要的Hilbert模——拟自由Hilbert模的局部化及其几何不变量，并将其应用于解析Hilbert模的相关分类问题。在此基础上，本项目拟通过由拟自由Hilbert模给出的表示, 研究更一般的Hilbert模的酉等价和相似分类问题，并揭示表示理论与纤维维数，算子corona问题，换位提升理论等向量值解析函数空间上重要课题的联系。

The Hilbert module，initiated by Douglas and Paulsen from the view of commutative algebra, is a fundamental framework for the study of modern multi-variable operator theory. Localization and resolution as basic tools in algebra are important in the study of properties and classifications of Hilbert modules. Based on the complex geometry approach of Cowen- Douglas theory, we study localization and geometric invariants of an important class of Hilbert modules called quasi-free Hilbert modules and apply our theory to related problems in the classification of analytic Hilbert modules. For more general Hilbert modules, we will study the unitary equivalence and similarity problems via resolutions by quasi-free Hilbert modules, and exhibit connections between the theory of resolutions and important topics in vector-valued function spaces such as fiber dimension, operator corona problem, and commutant lifting theory.