Mathematical Sciences: K-Theory of C*-Algebras, Group Representations, and Coarse Geometry

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9500977 Higson This project deals with the Baum-Connes conjecture. The emphasis of the project is not so much on proving new cases of the conjecture as on trying to understand more clearly the connections between the conjecture and the presumably related topics in geometry and representation theory. The long term goal is to use points of view suggested by the conjecture to give new insights into these subjects. Taking advantage of the extremely broad scope of the conjecture (which reaches from the Novikov conjecture to p-adic group representation theory) one might hope to forge new links between apparently dissimilar areas. The work proposed represents some very modest first steps in this direction. The general area of mathematics of this project has its basis in the theory of algebras of operators on Hilbert space. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These seemingly abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA. ***