Noncommutative Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 3-9, 2000 (Lecture Notes in Mathematics / Fondazione C.I.M.E., Firenze)

Noncommutative Geometry Is One Of The Most Deep And Vital Research Subjects Of Present-day Mathematics. Its Development, Mainly Due To Alain Connes, Is Providing An Increasing Number Of Applications And Deeper Insights For Instance In Foliations, K-theory, Index Theory, Number Theory But Also In Quantum Physics Of Elementary Particles. The Purpose Of The Summer School In Martina Franca Was To Offer A Fresh Invitation To The Subject And Closely Related Topics; The Contributions In This Volume Include The Four Main Lectures, Cover Advanced Developments And Are Delivered By Prominent Specialists. Preface -- A. Connes: Cyclic Cohomology, Noncommutative Geometry And Quantum Group Symmetries -- J. Cuntz: Cyclic Theory And The Bivariant Chern-connes Character -- N. Higson, E. Guentner: Group C*-algebras And K-theory -- E. Guentner, J. Kaminker: Geometric And Analytic Properties Of Groups -- J.e. Roberts: More Lectures On Algebraic Quantum Field Theory. A. Connes ... [et Al.] ; Editors, S. Doplicher, R. Longo. Includes Bibliographical References (p. 339-342). Also Available In An Electronic Version. Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K- Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists. TOC:Preface.- A. Connes: Cyclic Cohomology, NoncommutativeGeometry and Quantum Group symmetries.- J. Cuntz: Cyclic Theory and the Bivariant Chern-Connes Character.- N. Higson, E. Guentner: Group C*-Algebras and K-Theory.- E. Guentner, J. Kaminker: Geometric and Analytic Properties ofGroups.- J. E. Roberts: More Lectures on Algebraic Quantum Field Theory Our purpose in these notes is to give a detailed introduction to basic notions of non-commutative geometry and to illustrate them in a concrete manner in a very specific example.