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

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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.