非交换几何的历史告诉我们，带有适当结构的群胚可以用来解决很多几何、拓扑与分析问题。Fredholm李群胚就是用来研究奇性空间上Fredholm算子的有力工具。利用Fredholm李群胚，可以得到作用群胚、渐近双曲流形、渐近欧氏流形、多圆柱段流形等奇性空间上Fredholm算子的刻画定理；也为奇性空间上由分析问题所产生的算子的Fredholm性质的判定提供依据。本项目拟研究以下三方面的内容：1)利用Fredholm李群胚及其C*-代数和局部化技术来研究平面‘8字形’区域上双位势算子的本质谱问题；2)研究连续族群胚上Fredholm算子的理论；3)对于一般的局部紧群胚，把Fredholm群胚的定义推广为Fredholm模型的概念，使得新的概念包含了群作用下不变算子的情形和算子族的情形，为研究这些情形下的非交换指标定理提供基础。

The history of noncommutative geometry shows that groupoids with appropriate structures can be used to solve many geometric, topological, and analytical problems. Fredholm Lie groupoids are a powerful tool for studying Fredholm operators on singular spaces. By using Fredholm Lie groupoids, we can obtain the characterization theorems for Fredholm operators on singular spaces such as action groupoids, asymptotically hyperbolic manifolds, asymptotically Euclidean manifolds, and manifolds with poly-cylindrical ends, and provide criteria for the determination of Fredholm properties of operators coming from analysis problems on singular spaces. This project intends to investigate the following three aspects: 1) using Fredholm Lie groupoids and their C*-algebras and localization techniques to study the essential spectrum of the double layer potential operator on a plane 'Figue 8' region; 2) establising the theory of Fredholm operators on continuous family groupoids; 3) generalizing the definition of Fredholm groupoids to the concept of Fredholm model for general locally compact groupoids, such that we are able to deal with the case of invariant operators under group action and the case of a family of operators, which provides a basis for the study of the non-commutative index theorem in these cases.