Non-commutative geometry seeks to extend geometry and topology from the classical setting of Riemannian manifolds and topological spaces to a new setting of mathematical structures whose coordinate algebras are non-commutative. In this context, approximately twenty years ago, P.Baum and A.Connes conjectured a formula for the K-theory of the (reduced) C* algebra of any locally compact topological group. At the present time no counter-example is known to the conjecture and due to the work of many mathematicians the conjecture has been proved for several very interesting classes of groups (e.g. real Lie groups, p-adic algebraic groups, adelic algebraic groups, discrete hyperbolic groups, amenable groups). Also established is that the conjecture, when valid, has many corollaries (e.g. Mackey analogy, Atiyah-Schmid construction of the discrete series, Novikov higher signature conjecture, stable Gromov-Lawson-Rosenberg conjecture, Kadison-Kaplansky conjecture). This project aims to discover and develop further corollaries of the conjecture in representation theory and in geometry-topology.
Analysis is the branch of mathematics based on calculus. The fundamental ideas of calculus (differentiation and integration) were introduced by Newton and Leibniz and played a central role in the scientific revolution of their era. Topology is the most basic form of geometry and was founded by such eminent nineteenth and twentieth century mathematicians as Riemann, Poincare and Lefschetz. A major theme in modern mathematics has been the interplay between analysis and topology. For example, Maxwell's equations for electricity-magnetism are formulated via analysis, but many of the implications are topological. This project continues the interaction of topology and analysis by using and applying a new synthesis of analysis and topology known as "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. In topology only the shape and configuration of geometrical objects is used. For example, in the city of Koenigsberg there were seven bridges and the residents of the city began to wonder if it was possible to take a walk crossing each bridge exactly once. This is a topological problem. The length, width, and height of each bridge is irrelevant to the problem. Only the positioning of the bridges vis-a-vis each other counts. (An elementary topological argument proves that it is impossible to take a walk crossing each Koenigsberg bridge exactly once.) In Maxwell's mathematical theory of electricity and magnetism, the underlying geometry is topology. Many of the fundamental ideas introduced by Maxwell are topological in nature. His equations --- which precisely describe the physical properties of electric and magnetic fields --- are, however, analytical. 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 mathemati- cal structure know 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.that certain numbers which are analyti- cally 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. Atiyah-Singer is a direct descendant of Riemann-Roch since it does in higher dimensions exactly what Riemann-roch does in dimension two. The mathematics of this project continued the inter-action of analysis and topology. Thirty years ago the PI and Alain Connes conjectured that an analytically defined mathematical quantity (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 establishes connections between problems which previously were thought to be completely unrelated. This project considered examples (e.g. all reductive p-adic groups) where the Baum-Connes conjecture is now known to be true and developed the implications. This gives a quite new and different approach to well-known problems and issues. In more technical language, the aim was to bring to bear on classical problems in geometry-topology and representation theory the new point of view known as non-commutative geometry. 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. Thus a major simplification in representation theory has been achieved, and Baum-Connes has been connected to the Langlands program. This program-conjecture, due to Robert Langlands, is among the most fascinating and mysterious issues to emerge from twentieth century mathematics. One outcome of the project was a step forward on the Langlands program : the proof (using the non-commutative geometry point of view) that the local Langlands conjecture is valid throughout the principal series of any reductive p-adic group. 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 are valid well beyond the elliptic operators studied by Atiyah and Singer. An outcome of the project was an index formula for a geometrically arising class of non-elliptic Fredholm differential operators. This formula indicates that ellipticity is not the really essential hypothesis needed for index theory. In summary, this project continued to develop the interaction of topology and analysis that has its origins in the discoveries made by Maxwell and Riemann-Roch. Impact will be on the Langlands program, representation theory, index theory, and mathematical physics. 1
