非交换留数和等变指标定理的研究

项目名称： 非交换留数和等变指标定理的研究

项目编号： No.11501414

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王剑

作者单位： 天津职业技术师范大学

项目金额： 18万元

中文摘要： Atiyah-Singer指标定理和非交换几何与几何、拓扑、分析、数论、物理都有密切联系,研究非交换几何框架下的非交换留数和等变指标定理对带边流形的重力作用及一些复杂空间上指标定理的证明有重要意义.本课题拟研究非交换几何中带边流形的等变非交换留数和一族Toep-litz算子等变指标理论的相关问题.在非交换几何框架下,证明高维带边流形的Kastler-Kalau- Walze定理;探讨带边情形下的等变非交换留数理论,给出等变Boutet de Monvel代数上的非交换留数,并证明等变的Kastler-Kalau-Walze定理;考虑一族Dirac算子的等变陈-Connes特征,定义等变高阶谱流,建立相应的一族Toeplitz算子的等变指标定理,并推广到非交换几何框架.

中文关键词： 共形不变量；非交换留数；Kastler-Kalau-Walze定理；等变高阶谱流；非交换几何

英文摘要： Atiyah-Singer index theorems and noncommutative geometry have close relations with geometry, topology, analysis, number theory and physics, the study of noncommutative residue in the framework of noncommutative geometry and equivariant index theory enrich the gravitational action for manifolds with boundary and the proof of index theory in some complex spaces. This project is concerned with the equivariant noncommutative residue for manifolds with boundary of noncommutative geometry and related problems for the equivariant family index formula of Toeplitz operators. It includes the following three parts: the proof of a general Kastler- Kalau-Walze type theorem for any dimensional manifolds with boundary in the framework of noncommutative geometry; the establish of the equivariant noncommutative residue for manifolds with boundary on Boutet de Monvel's algebra; the characterize of a family equivariant Chern-Connes character for Dirac operators, the proof of a equivariant family index formula for Toeplitz operators based on defining the equivariant higher spectral flow, and the generalization of the case in the framework of noncommutative geometry.

英文关键词： conformal invariant;noncommutative residue;Kastler-Kalau-Walze theorem;equivariant higher spectral flow;noncommutative geometry