Nigel/Roe A deepening understanding of the role played by large-scale geometry in topology has made it clear that large-scale geometric features of groups determine small-scale features of their unitary duals. The effect is easily observed in abelian groups, thanks to Fourier theory and Pontrjagin duality, but the situation is more involved for nonabelian groups, whose unitary representation theory is too complicated to admit a direct descriptive account. However the perspective on dual spaces provided by Alain Connes noncommutative geometry makes it possible to formulate instances of this large-scale to small-scale phenomenon for nonabelian groups. Moreover the tools of algebraic topology, carried over to the noncommutative realm, make it possible to elevate the phenomenon to a conjectural reciprocity (formulated by Baum and Connes) between the global, homotopy theoretic structures of groups and their reduced duals. The purpose of the research outlined in this proposal is to obtain a more accurate and deeper understanding of the Baum-Connes conjecture in operator K-theory and of the large-to-small scale phenomenon which underlies it. The proposers will investigate issues related to group boundaries, Sobolev theory on the reduced dual of a group, and Hilbert space embeddings of groups. The recent discovery of counterexamples to variants of the Baum-Connes conjecture will be analyzed in depth. Although the tools used to investigate it are rather elaborate, the idea behind large scale-geometry is very simple: ignore the local, small-scale features of a geometric space and concentrate on its large-scale, or long term, structure. By doing so, trends or qualities may become apparent which are obscured by small-scale irregularities. The investigators and others have developed tools to distinguish between different sorts of multi-dimensional, large scale behavior in geometry. Somewhat surprisingly, aside from their intrinsic interest, these tools have found application in ordinary, small-scale geometry and elsewhere. The present proposal focuses on geometric aspects of group theory which are illuminated by large-scale geometry. The proposers are actively involved in training the next generation of mathematical scientists. They lead Penn States' Geometric Functional Analysis group. They run an active, twice-weekly research seminar and between them they have eight doctoral students under their direct supervision (a number of other students attend the seminar regularly). They currently serve as mentors to one VIGRE supported postdoctoral fellow, and will be recruiting a second fellow to be supported by NSF Focussed Research Grant funds this year. The Geometric Functional Analysis group frequently hosts sabbatical visitors as well as visiting graduate students. Besides the seminar, the group runs a continuing program of mini-workshops on research subjects of current interest. The research described in this proposal will be supported by, and carried out as part of, the activities of the Geometric Functional Analysis group.
This grant supported pure mathematical research on applications of noncommutative geometry to problems in group representation theory and related areas. A major emphasis was placed on graduate student training through mentoring in research, summer research support and research assistantships. The general aim of noncommutative geometry is to express geometric concepts such as distance and curvature using the vocabulary of Hilbert space theory, with the aim of broadening the reach of those geometric ideas. The mathematical foundations of quantum mechanics are also written in the language of Hilbert space, and indeed quantum mechanics is a major source of inspiration for noncommutative geometry. The expectation is that, just as some physical problems are best treated using quantum mechanics rather than classical mechanics, so there will be mathematical problems that are best treated using noncommutative geometry rather than the classical geometry of Gauss, Riemann and others. Major portions of group representation theory were developed in response to questions posed by quantum mechanics (representation theory mathematically encodes the symmetries of quantum systems), and so it is natural to hope that noncommutative geometry will provide appropriate tools to study the theory. One expression of this is the program begun in the 1980's by the mathematicians Paul Baum and Alain Connes, and the projects carried out with the support of this grant took the Baum-Connes theory as a starting point. While the Baum-Connes theory has a very broad scope, the projects undertaken with support from this grant were narrower in focus, but aimed for more detailed and more precise results than the general theory provides. A major accomplishment was a new parametrization of the collection of irreducible representations of certain groups. The general form of this work was first suggested in the 1970's by George Mackey, but it was only with the new perspectives offered by noncommutative geometry and the Baum-Connes theory that a re-examination of Mackey's ideas seemed appropriate. The project was completed in part with the assistance of graduate students, who contributed complete analyses of important cases.
