关于Hilbert模的曲率研究

曲率是Sz.-Nagy-Foias理论的一个重要分支,这个理论在复分析,内插和调和分析中有许多应用.在过去的二十年,许多学者专注于交换曲率的研究并且得到了许多重要的结果.然而,关于非交换曲率的研究是非常缓慢的.因此,本项目将专注于非交换曲率的研究.具体来讲,我们主要研究以下三个方面的内容:首先,我们研究子空间的基本性质并且给出对应游荡子空间的完全刻画.其次,我们给出子模的一个完全刻画并且证明曲率关于有限秩子模是一个完全不变量.最后,我们得到了黎曼几何中Gauss-Bonnet-Chern定理的非交换版本.

Curvature is an important branch of Sz.-Nagy-Foias theory. This theory has many applications in complex analysis, interpolation, and harmonic analysis. In the last twenty years, many authors focused on the study of commutative curvature and obtained some significant results. However, the research for noncommutative curvature is still slow. Consequently, this project will focus on the research of noncommutative curvature. More precisely, firstly, we study some basic properties of subspaces and provide an explicite description of the corresponding wandering subspaces. Secondly, we obtain a complete description of the submodules and show that the curvature is a complete invariant for the finite rank submodules. Finally, we present a noncommutative version of the Gauss-Bonnet-Chern theorem from Riemannian geometry.