Applications of Non-Commutative Geometry

A recurrent theme in nineteenth and twentieth century mathematics has been the interaction of analysis and topology. 'Analysis' is mathematics based on calculus (i.e. differentiation and integration), which developed from the pioneering work of Newton and Leibniz. 'Topology' is the most basic form of geometry and was founded by such eminent nineteenth century and twentieth century mathematicians as Riemann, Poincare and Lefschetz. Riemann and his co-worker Roch stated and proved one of the true gems of nineteenth century mathematics: the Riemann-Roch theorem. This remarkable theorem asserts that certain numbers assigned by an analytical method to a mathematical structure known as a 'divisor on a Riemann surface' are, in fact, topological. The Riemann-Roch theorem gives a topological formula for these numbers. This Riemann-Roch phenomenon (i.e. certain numbers which are analytically defined are in practice determined by a topological formula) has continued with astonishing vigor throughout twentieth century mathematics. Examples are the Lefschetz fixed-point formula (which led to the A. Weil conjectures) and the Atiyah-Singer index theorem for elliptic operators.

The mathematics of this project continues this trend. Thirty years ago the investigator and A. Connes conjectured that an analytic invariant (the K-theory of the reduced C*-algebra of a locally compact Hausdorff topological group) is, in fact, topological. The conjecture has drawn wide attention and has been the subject of papers, conferences, lectures, Ph.D. theses, and books. The conjecture is unusual in that it cuts across several different areas of mathematics and reveals connections between problems which previously were thought to be completely unrelated. The main objective of the project is to take the examples (e.g. all reductive p-adic groups) where the conjecture is now known to be true and to develop the implications. This gives a new and different approach to well known problems and issues. Equivalently, the project uses the new point of view known as non-commutative geometry to study classical problems in geometry-topology and representation theory. The non-commutative geometry point of view has led to some startling conjectures and results. In the representation theory of reductive p-adic groups a totally unexpected geometric structure has been revealed. This greatly simplifies the representation theory and links Baum-Connes to the Langlands program. For the index of geometrically-arising Fredholm operators, the non-commutative geometry point of view leads to the surprising conclusion that formulas like the Atiyah-Singer index formula apply well beyond elliptic operators. Thus ellipticity is not the essential point needed to obtain a topological formula for the index of such operators.