Large Scale Geometry, Index Theory and Representations of Groups

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

DMS-0204558 Jonathan L. Block In recent years there has been an explosion of interest in the applications of large scale or "coarse" methods in geometry, topology, index theory and group theory. For many topological and geometric problems of interest, this large-scale structure is relevant in answering them. The Principal Investigator will study three such instances. First, on a singular space, phenomena are affected not only by the global fundamental group, but also by a local picture of the fundamental group. The fundamental groups of the link of a stratum changes from stratum to stratum and affects the topological behavior of the space, in particular, the classification of stratified spaces must take into account this behavior. The PI proposes to study via operator K-theory how the index theory of the strata of singular spaces (as non-compact manifolds) fit together to give an index theory for the singular space itself. Second, the PI proposes to study more generally quantitative aspects of homology theory, taking into account such issues as the images of the homology of balls of a given size in the homology of a non-compact manifold. Applications to existence of infinitely many null-homotopic geodesics are proposed. Third, the large-scale / small-scale duality between a discrete group and its unitary representation theory has been studied under the rubric of the Baum-Connes conjecture. The PI, motivated by his work in large scale geometry proposes a broader study of this duality. This requires the development of tools from algebraic geometry (especially moduli space theory) to apply to infinite dimensional unitary representations. These tools place families of representations at the fore, which as two effects. First, families of representations seem to be more rigid then single ones. And secondly, one can concentrate ones attention on specific parts of the representation theory. Thus, the PI demonstrates, in several situations, that for lattices in Lie groups, the representation theory in an infinitesimal neighborhood of the trivial representation coincide for the lattice and the ambient group. A second type of question proposed by the PI are called reconstruction problems. When can one reconstruct representations on the ambient group knowing its values on the lattice? The PI views this as a version of the Sampling theorem from signal processing. Large scale geometry is the study of spaces as seen from an increasingly distant perspective. For many problems in geometry and topology, the phenomena at a small or moderate scale only obscure the situation and the large scale perspective shines the essential light on the subject: seeing the forest for the trees. This proposal focuses the attention of large-scale geometry on problems in index theory and representation theory of discrete groups. Students of PI and his collaborators are involved at various stages of the project. The developments and anticipated results are intended as a bridge between algebraic and geometric topology, harmonic analysis and functional analysis. This should provide a clearer understanding of all these subjects and enhance the interaction between the experts in them.