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

本文的研究拓展和加强了算子代数的应用领域。这一结果在奇异空间和非紧流形上的广群代数的结构和表示上得到了新的结果。利用广群C_-代数的表示方法，给出了非紧奇异空间上Fredholm算子的一个新的刻画（循环同调，Chern特征标，光滑子代数）计算奇异空间和非紧空间上算子的指数和相对指数，这些指数定理是非局部的，因此它们将导致谱不变量，推广η-不变量，这将被调查。Thehomology，K-理论，和其他不变量的相关运营商algebras也将被确定。本文将研究奇异空间上算子的谱和分布核的结构，并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端点的非紧流形的分析。这一建议将有助于从生物学、化学、工程学和物理学的角度来处理几何学和计算问题所必需的一般技术的发展。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，从而有助于应用数学成果实践，这将导致能够使用理论工具来处理实际问题的专家的创建。

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.