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

Abstract Nistor The proposed research extends and strengthens the field of applications of Operator Algebras. It leads to new results on the structure and representations of groupoid algebras associated to singular spaces and non-compact manifolds. It leads also to a new characterization of the Fredholm operators on non-compact and 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 indices of 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. The homology, K-theory, and other invariants of the relevant operator algebras will also be determined. The spectrum and the structure of the distribution kernels of operators on singular spaces will be investigated. The proposed research will have applications to numerical methods for polyhedral domains, which are important in Engineering (an example is the method of layer potentials), and to the analysis on non-compact manifolds with nice ends, which arise in String Theory and General Relativity. This proposal will contribute to the development of the general techniques necessary to approach mathematical and computational problems from Biology, Chemistry, Engineering, and Physics. It will also contribute to applying mathematical results in practice by interactions with researchers from other fields, by organizing conferences, and by advising students, which will lead to the creation of specialists able to use theoretical tools to handle practical problems.