Mathematical Sciences: Higher Index for Coverings of Manifolds with Boundary

本文研究了具有全局边界条件的Atiyah-Patodi-Singer型流形覆盖上的Dirac算子的高指标问题。求解该问题的主要步骤是：(1)研究由边界流形构造的Calderon投影定义的正则边界条件。(2)将非交换谱流边值问题的k理论指标从原始边界条件降为Calderon投影。这些步骤在k理论水平上提供了边界值指标的识别。为了得到更明确的指标表达式，需要计算循环同调中指标的陈氏性质。利用谱流循环特性的公式，这相当于根据边界流形的几何形状确定边界狄拉克和卡尔德隆投影的较高（循环）eta-不变量。高指数问题的成功解决将导致微分几何和拓扑中许多有趣的应用。它也为研究椭圆型偏微分方程的边值问题提供了有价值的理论认识。

Abstract WU This project studies the higher index problem for Dirac operators on coverings of manifolds with boundary, with global boundary condition of Atiyah-Patodi-Singer type. The main steps in solving this problem are: (1) Study a canonical boundary condition defined by the Calderon projector constructed from the bounding manifold. (2) Reduce the K-theoretic index of the boundary value problem to the noncommutative spectral flow from the original bmundary condition to the Calderon projector. These steps provide an identification of the boundary value index at the K-theory level. To get more explicit formulae for the index, one then needs to calculate the Chern character of the index in cyclic homology. Using the formula for the cyclic character for the spectral flows, this amounts to identifying the higher (cyclic) eta-invariant of the boundary Dirac and the Calderon projector in terms of the geometry of the bounding manifold. A successful solution of the higher index problem would lead to many interesting applications in differential geometry and topology. It also provides valuable theoretical understandings to the study of boundary value problems of elliptic partial differential equations arising from various applied scientific fields and engineering.