The Novikov and Exactness Conjectures for Discrete Groups

Indiana University, Bloomington IN

Investigators

Abstract

Abstract Guentner The proposed research comprises two distinct projects; the relationship between the Novikov and exactness conjectures for discrete groups, and the use of global analytic techniques to study quantum mechanical systems. The first is motivated by the observation that the classes of groups for which the Novikov and exactness conjectures are known coincide to a large degree; both classes contain amenable groups, hyperbolic groups and Coxeter groups, for example. This project is joint with J. Kaminker. The second is based on the idea that the Berezin-Toeplitz quantization can be analyzed using spectral properties of family of Dirac-type operators. This project is joint with J. Trout. The Novikov conjecture, one of the most important problems in topology, has stimulated a tremendous amount of mathematical research over the last thirty years. The exactness conjecture is purely analytic. That there could exist a connection between these two conjectures from entirely different branches of mathematics is somewhat surprising, but nevertheless is supported by empirical evidence. We plan to develop more fully the relationship between these conjectures. This research has bearing on a number of important outstanding problems including the Baum-Connes Conjecture. At the heart of the connection between mathematics and physics is the theory of quantum mechanics. We propose a new method to analyze quantum mechanical systems based on geometric properties of certain systems of differential equations. This work should lead to a better understanding of a number of quantum mechanical systems from mathematical physics.