Atiyah-Singer指标定理和非交换几何是困难而且重要的研究方向。本课题主要是研究这两个方向中的一些问题。我们将计算一族奇数维带边流形上Dirac算子的高阶谱流。定义b-eta上链，把b-eta不变量分解为b-eta上链与幂等矩阵的陈特征的配对加上边界上的量。同时证明b-eta上链的相对指标定理。我们将计算eta上链的绝热极限。 我们将推广Goette的比较公式到Killing向量场有零点情形，一族算子情形，b-eta情形。我们将推广Boutet de monvel代数上带边流形的非交换留数到等变情形，再证明等变带边Kastler-Kalau-Walze类型定理。对于带挠率的Dirac算子和非极小de-Rham Hodge算子，证明一般维数的Kastler-Kalau-Walze类型定理.

Atiyah-Singer index theorems and noncommutative geometry are the difficult and important research directions. In this project, we study some problems in these two research directions。We compute the higher spectral flow for a family Dirac operator on odd dimensional manifolds with boundaty. Define b-eta cochain and decompose a b-eta invariant as a pairing of the b-eta cochain with the Chern character of an idempotent matrix plusing the amount on the boundary. We prove the relative index theorem on b-eta cochain and we compute the adiabatic limit of the b-eta cochain.We generalize the Goette comparision formula to the Killing vector field having zero points case and the family case and the b-eta case. We generalize the noncommutative residue on Boutet de monvel algebra to equivariant case, then prove an equivariant Kastler-Kalau-Walze theorem for manifolds with boundary. We prove general Kastler-Kalau-Walze theorems for Dirac operators with torsion and the nonminimal de-Rham Hodge operator.