CAREER: Geometry of Groups and the Novikov Conjecture

University Of Hawaii, Honolulu

Investigators

Abstract

The Novikov higher signature conjecture, a deep problem in the topology of manifolds, follows from a sufficiently precise understanding of the K-theory of the unitary dual, viewed as a noncommutative space, of the group in question. This project focuses on issues related to this conjecture, in particular (1) approximation properties of group C*-algebras, (2) uniform embeddability of discrete groups in Hilbert space and (3) the use of controlled methods to study the K-theory of group C*-algebras. The investigator will also study parallel problems for C*-algebras associated to metric spaces. In studying the noncommutative dual spaces of discrete groups, with particular emphasis on the important Novikov and Baum-Connes conjectures, this project fits squarely within Alain Connes' program of noncommutative geometry. A recurring theme will be to incorporate a greater variety of ideas from geometric group theory into the study of analytic properties of groups. Indeed, the project will provide a forum for promoting sustained interaction between researchers in noncommutative geometry and geometric group theory.