Mathematical Sciences: Higher Index for Coverings of Manifolds with Boundary

数学科学：有边界流形覆盖的更高指数

• 批准号: 9706858
• 负责人: Gabriel Nagy
• 金额: $ 6.64万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706858&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Higher; Index; Coverings

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.

摘要WU该项目研究了Dirac操作员的较高指数问题，该项目在具有边界的歧管的覆盖范围内，具有Atiyah-Patodi-Singer类型的全球边界条件。 解决此问题的主要步骤是：（1）研究由边界歧管构建的Calderon投影仪定义的规范边界条件。 （2）将边界值问题的K理论指数减少到从原始的Bmundary条件到Calderon投影仪的非交通频谱流。 这些步骤在K理论级别提供了边界值索引的标识。为了获得该索引的更明确的公式，然后需要计算循环同源性索引的Chern特征。使用用于光谱流的环状特征的公式，这等于识别边界狄拉克的较高（环）ETA不变和Calderon投影仪，并根据边界歧管的几何形状。 较高指数问题的成功解决方案将导致许多有趣的应用在差异几何和拓扑中。 它还为研究由各种应用科学领域和工程引起的椭圆价偏微分方程的边界价值问题的研究提供了宝贵的理论理解。