Cowen-Douglas算子与Hilbert模的若干分类与结构问题

Cowen-Douglas理论与Hilbert模理论是分别从几何和代数观点研究算子分类与结构问题的前沿领域和重要框架。对于Cowen-Douglas理论，本项目将以分类问题为切入点，利用Specht型算子值不变量研究多元算子组的局部理论；进一步，我们将结合向量丛上的联络理论研究解析子流形上的局部分类与算子组整体分类的关系，并探讨分类问题与几何和理论物理中相关等价问题的联系。对于Hilbert模，本项目将借助函数模型以及拟自由表示等工具，研究两类重要Hilbert模（压缩Hilbert模和投射Hilbert模）的Jordan型分解，Serre问题等具有深刻代数背景的结构理论问题。

Cowen-Douglas theory and Hilbert module implement important modern frameworks to study the structure and classification of operators, with emphasis on differential geometry and commutative algebra respectively. Motivated by the classification problem, we will study localization theory for operator tuples in the Cowen-Douglas class via operator-valued invariants of Specht's type; moreover, we will discuss the relation between local equivalence(on submanifolds) and global equivalence of the operator tuple via connection theory on vector bundles, and exhibit the connection between classification problem and equivalence problems in geometry and physics. Based on the model theory and resolution theory, we will study the Jordan decomposition and Serre's problem for two important kind of Hilbert modules——contractive Hilbert module and projective Hilbert module, which are important problems on structure of Hilbert modules with deep algebraic background.