Non-Commutative Geometry and the Spectral Flow Index Theorem

在本研究中，我们在低维拓扑的基础上推广了Atiyah-Singer指标定理。本课题的研究目标如下：1.在非对易几何的框架下建立精细的指数定理。研究了指数定理与ETA不变量、谱流等解析次不变量之间的关系；2.研究了指数定理与包括Maslov类在内的次特征类之间的关系。2001年7月，我们应邀在日本秋田拓扑学研讨会上作了题为“解析K-理论与指数定理”的演讲。我们也在做这个项目，题目是2001年3月在庆应义乌大学的非对易几何和弦理论研讨会上提出的Eta不变量，Godillon-Vey类和指数定理。我们还发展了非交换Hopf不变量的研究，并于2001年3月在名古屋工业大学发表了题为《谱流、Teplitz指数和Hopf不变量》的结果，并于2002年2月在鹿儿岛大学发表了题为《矢量场的指标定理和Hopf不变量》的结果。

In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the bas1S of Low dimensional Topology. The objective of the project are the followings:1. Establish the elaborated Index Theorem in the framework of Non-commutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invariants and the spectral flow;2. Study the relationship between the elaborated Index Theorem and secondary characteristic classes including the Maslov class.Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. In July 2001, we are invited to give a talk with the title 'Analytic K-theory and the index theorem' at Topology Symposium of Japan, Akita. We are also working on the project are presented with the title 'Eta invariants, the Godbillon-Vey classes and the index theorem' in March 2001 at Workshop on Non-commutative Geometry and String Theory, Keio University. We also developed the research on Non-commutative Hopf invariants, and presented some results at Nagoya Technology Institute with the title 'Spectral flow, the Teoplitz index and the Hopf invariant' in March 2001, and with the title 'A index theorem of vector field and the Hopf invariant' at Kagoshima University in February 2002.