Mathematical Sciences: Higher Index for Coverings of Manifolds with Boundary

本文研究了具有Atiyah-Patodi-Singer型整体边界条件的流形覆盖上的Dirac算子的高指标问题。解决这一问题的主要步骤是：(1)研究由边界流形构造的Calderon投影器定义的正则边界条件。(2)将边值问题的K-理论指标从原边界条件化为非对易谱流。这些步骤提供了在K理论水平上的边值指数的识别。为了得到指数的更明确的公式，需要计算指数在循环同调中的Chern特征标。使用谱流的循环特征的公式，这相当于根据有界流形的几何来确定边界狄拉克和卡尔德隆投影体的高阶(循环)不变量。高指数问题的成功解决将在微分几何和拓扑学中产生许多有趣的应用。为研究各种应用科学领域和工程中的椭圆型偏微分方程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.