Clay Mathematics Proceedings Volume 11 Quanta of Maths Conference in honor of Alain Connes Non Commutative Geometry Institut Henri Poincaré, Institut des Hautes Études Scienti(cid:197) ques, Institut de Mathématiques de Jussieu, Paris, France March 29– April 6, 2007 Etienne Blanchard David Ellwood Masoud Khalkhali Matilde Marcolli Henri Moscovici American Mathematical Society Sorin Popa Editors Clay Mathematics Institute Quanta of Maths Clay Mathematics Proceedings Volume 11 Quanta of Maths Conference in honor of Alain Connes Non Commutative Geometry Institut Henri Poincaré, Institut des Hautes Études Scientifiques, Institut de Mathématiques de Jussieu, Paris, France March 29– April 6, 2007 Etienne Blanchard David Ellwood Masoud Khalkhali Matilde Marcolli Henri Moscovici Sorin Popa Editors American Mathematical Society Clay Mathematics Institute Photographs courtesy of Jean-Fran¸cois Dars. 2000 Mathematics Subject Classification. Primary 58B34. Library of Congress Cataloging-in-Publication Data Quanta of maths : non commutative geometry conference in honor of Alain Connes, March 29 to April 6, 2007, Institut Henri Poincar´e, Institut des hautes ´etudes scientifiques, Institut de math´ematiquesdeJussieu,Paris,France/EtienneBlanchard...[etal.],editors. p.cm. —(Claymathematicsproceedings;v.11) Includesbibliographicalreferencesandindex. ISBN978-0-8218-5203-3(alk.paper) 1.Noncommutativedifferentialgeometry—Congresses. I.Connes,Alain. II.Blanchard,Eti- enne, 1968– III. Institut Henri Poincar´e. IV. Institut des hautes ´etudes scientifiques (Paris, France) V.Institutdemath´ematiquesdeJussieu. QA614.Q36 2011 512(cid:2).55—dc22 2010034787 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2010bytheClayMathematicsInstitute. Allrightsreserved. PublishedbytheAmericanMathematicalSociety,Providence,RI, fortheClayMathematicsInstitute,Cambridge,MA. PrintedintheUnitedStatesofAmerica. TheClayMathematicsInstituteretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 151413121110 Contents Preface xi A Geometric Description of Equivariant K-Homology for Proper Actions 1 Paul Baum, Nigel Higson and Thomas Schick The Planar Algebra of Diagonal Subfactors 23 Dietmar Bisch, Paramita Das and Shamindra Kumar Ghosh K -Injectivity for Properly Infinite C∗-Algebras 49 1 E´tienne Blanchard Families of Monads and Instantons from a Noncommutative ADHM Construction 55 Simon Brain and Giovanni Landi Cyclic Theory for Commutative Differential Graded Algebras and S–Cohomology 85 Dan Burghelea Vinberg Algebras, Lie groups and Combinatorics 107 Pierre Cartier Noncommutative Geometry as the Key to Unlock the Secrets of Space-Time 127 Ali H. Chamseddine The Regular C*-Algebra of an Integral Domain 149 Joachim Cuntz and Xin Li Noncommutative Coordinate Algebras 171 Michel Dubois-Violette Random Matrices, Free Probability, Planar Algebras and Subfactors 201 A. Guionnet, V. F. R. Jones, D. Shlyakhtenko The Tangent Groupoid and the Index Theorem 241 Nigel Higson Le Th´eor`eme de P´eriodicit´e en K-Th´eorie Hermitienne 257 Max Karoubi A Short Survey of Cyclic Cohomology 283 Masoud Khalkhali The Core Hopf Algebra 313 Dirk Kreimer Propri´et´e (T) Renforc´ee et Conjecture de Baum-Connes 323 Vincent Lafforgue Combinatorial Hopf algebras 347 J.-L. Loday and M. Ronco Cyclotomy and Analytic Geometry over F 385 1 Yuri I. Manin ix