Research conducted in the last ten years has revealed unexpected rigidity properties of noncommutative C*-algebras. Whereas the cohomological invariants of a space (commutative C*-algebra) will determine the space at most up to homotopy equivalence, in the class of nuclear simple C*-algebras, the objects are often determined up to isomorphism by their K-theoretical invariants. Elliott's conjecture states that, far from being an accident, this is always the case for the entire class of separable nuclear simple C*-algebras. (Tracial invariants are needed if the real rank is nonzero.) The proposed research aims to uncover and explain rigidity properties of nuclear C*-algebras. The basic idea beyond the classification program is that that the simplicity and the stable or real rank conditions for a nuclear C*-algebra translate to certain internal dynamical properties of the algebra which forces a behavior typical to that of a combinatorial object. The C*-algebra becomes a rigid object built around its K-theory skeleton. The ramifications of the classification theory into the structure theory of C*-algebras will be explored with emphasis on dynamical systems and group C*-algebras. The investigator will analyze the impact of the recent advances around the Baum-Connes conjecture on the classification theory with the long term goal of formulating and exploring a Baum-Connes type conjecture for general nuclear C*-algebras. This is closely tied with the universal coefficient theorem problem in KK-theory and deformation theory of C*-algebras. Geometry was developed in an attempt to describe the ambient physical space. Its history has seen a series of remarkable achievements from the Euclidian geometry to the non-Euclidian geometries which culminated with the Riemannian geometry providing a successful model for large-scale spacetime in general relativity. The noncommutative geometry of Alain Connes is a far reaching generalization of the Riemannian geometry, well adapted for the study of a variety of large and small scale structures. The theory can be viewed as a significant development in the quest of quantizing of mathematics following the successful quantization of physics. As in quantum physics, the coordinates in this theory are no longer ordinary numbers but noncommuting operators acting on infinite dimensional Hilbert spaces. The ordinary spaces are being replaced by algebras of operators. The proposed project aims to contribute to the extensive effort of a community of researchers to extend the mathematics of commutative spaces to operator algebras.
