Quanta of Maths

CMI/AMS publication. 2010. 675 pp., Softcover, List price: $129, AMS members: $103.20. Order code: CMIP/11. Students: $103.20. Order code: CLAY MATH. Available at the AMS bookstore

Editors

Étienne Blanchard (University of Paris 7, France), David Ellwood (Clay Mathematics Institute, Cambridge, MA), Masoud Khalkhali (University of Western Ontario, London, ON, Canada), Matilde Marcolli (California Institute of Technology, Pasadena, CA), Henri Moscovici (Ohio State University, Columbus, OH) and Sorin Popa (University of California, Los Angeles, CA)

Description

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics.

Specific themes covered by the articles are as follows:

  • entropy in operator algebras, regular C*-algebras of integral domains, properly infinite C*-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces;
  • von Neumann algebras, fundamental Group of II1 factors, subfactors and planar algebras;
  • Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory;
  • cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and teh index theorem;
  • noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras;
  • Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities;
  • cyclotomy and analytic geometry over F1, quantum modular forms;
  • differential K-theory, cyclic theory and S-cohomology.

cover of Quanta of Maths

Table of Contents

  • A Geometric Description of Equivariant K-Homology for Proper Actions
    Paul Baum, Nigel Higson and Thomas Schick
  • The Planar Algebra of Diagonal Subfactors
    Dietmar Bisch, Paramita Das and Shamindra Kumar Ghosh
  • K1-Injectivity for Properly Infinite C*-Algebras
    Étienne Blanchard
  • Families of Monads and Instantons from a Noncommutative ADHM Construction
    Simon Brain and Giovanni Landi
  • Cyclic Theory for Commutative Differential Graded Algebras and S--Cohomolog
    Dan Burghelea
  • Vinberg Algebras, Lie groups and Combinatorics
    Pierre Cartier
  • Noncommutative Geometry as the Key to Unlock the Secrets of Space-Time
    Ali H. Chamseddine
  • The Regular C*-Algebra of an Integral Domain
    Joachim Cuntz and Xin Li
  • Noncommutative Coordinate Algebras
    Michel Dubois-Violette
  • Random Matrices, Free Probability, Planar Algebras and Subfactors
    A. Guionnet, V. F. R. Jones, D. Shlyakhtenko
  • The Tangent Groupoid and the Index Theorem
    Nigel Higson
  • Le Théorème de Périodicité en K-Théorie Hermitienne
    Max Karoubi
  • A Short Survey of Cyclic Cohomology
    Masoud Khalkhali
  • The Core Hopf Algebra
    Dirk Kreimer
  • Propriété (T) Renforcée et Conjecture de Baum-Connes
    Vincent Lafforgue
  • Combinatorial Hopf Algebras
    J.-L. Loday and M. Ronco
  • Cyclotomy and Analytic Geometry over F1
    Yuri I. Manin
  • Motivic Renormalization and Singularities
    Matilde Marcolli
  • Free Groups and Reduced 1-Cohomology of Unitary Representations
    Florian Martin and Alain Valette
  • Local index Formula and Twisted Spectral Triples
    Henri Moscovici
  • Observables in Quantum Gravity
    Alejandro Perez and Carlo Rovelli
  • On the Fundamental Group of II1 Factors and Equivalence Relations Arising from Group Actions
    Sorin Popa and Stefaan Vaes
  • Leibniz Seminorms for "Matrix Algebras Converge to the Sphere"
    Marc A. Rieffel
  • Structured Vector Bundles Define Differential K-Theory
    James Simons and Dennis Sullivan
  • Entropy in Operator Algebras
    Erling Størmer
  • Non-Compact Spectral Triples with Finite Volume
    Raimar Wulkenhaar
  • A Characterization of the Image of the Baum-Connes Map
    Guoliang Yu
  • Quantum Modular Forms
    Don Zagier