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.