Applications of Non-Commutative Geometry

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AbstractBaumNon-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