Applications of Operator Algebras and Index Theory to Analysis on Singular Spaces

摘要：本文的研究扩展和加强了算子代数的应用领域。得到了与奇异空间和非紧流形相关的群拟代数的结构和表示的新结果。基于群类群c_c -代数的表示，给出了非紧奇异空间上Fredholm算子的一个新的表征。利用非交换几何中的方法（循环同调、陈氏特征、光滑子代数），计算了奇异和非紧空间上算子的指标和相对指标。这些指标定理是非局部的，所以它们会引出谱不变量，推广不变量，这将会被研究。还将确定相关算子代数的同调、k理论和其他不变量。研究了奇异空间上算子分布核的谱和结构。提出的研究将应用于工程中重要的多面体域的数值方法（例如层势方法），以及弦理论和广义相对论中出现的具有良好末端的非紧流形的分析。本建议将有助于发展处理生物、化学、工程和物理中的数学和计算问题所需的一般技术。它还将通过与其他领域的研究人员的互动、组织会议和为学生提供建议，有助于将数学结果应用于实践，这将导致能够使用理论工具来处理实际问题的专家的产生。

AbstractNistorThe proposed research extends and strengthensthe field of applications of Operator Algebras. It leads to newresults on the structure and representations of groupoid algebras associatedto singular spaces and non-compact manifolds. It leads alsoto a new characterization of the Fredholm operators on non-compactand singular spaces based on representations of groupoid C_-algebras.Using also methods from Noncommutative Geometry (cyclic homology,Chern character, smooth subalgebras) the indices and relative indicesof operators on singular and non-compact spaces will be computed.These index theorems are non-local, so they will lead to spectral invariants,generalizing the eta-invariant, that will be investigated. Thehomology, K-theory, and other invariants of the relevant operator algebraswill also be determined. The spectrum and the structure of thedistribution kernels of operators on singular spaces will be investigated.The proposed research will have applications to numericalmethods for polyhedral domains, which are important in Engineering(an example is the method of layer potentials), and to the analysis onnon-compact manifolds with nice ends, which arise in String Theoryand General Relativity. This proposal will contribute to the developmentof the general techniques necessary to approach mathematicaland computational problems from Biology, Chemistry, Engineering,and Physics. It will also contribute to applying mathematical resultsin practice by interactions with researchers from other fields, by organizingconferences, and by advising students, which will lead to thecreation of specialists able to use theoretical tools to handle practicalproblems.