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Noncommutative Geometry Year 2000

  • Chapter
Visions in Mathematics

Part of the book series: Modern Birkhäuser Classics ((MBC))

Abstract

Our geometric concepts evolved first through the discovery of Non-Euclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. The basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation θ → 1/θ for the noncommutative torus T 2 θ which are unseen in perturbative expansions in θ such as star or Moyal products. We discuss the foundational problem of “what is a manifold in NCG” and explain the fundamental role of Poincare duality in K-homology which is the basic reason for the spectral point of view. This leads us, when specializing to 4-geometries to a universal algebra called the “Instanton algebra”. We describe our joint work with G. Landi which gives noncommutative spheres S 4 θ from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to noncommutative geometries. We give a survey of several recent developments. First our joint work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant (rather than the usual covariant one) geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Second, our joint work with D. Kreimer on renormalization and the Riemann-Hilbert problem. Finally we describe the spectral realization of zeros of zeta and L-functions from the noncommutative space of Adele classes on a global field and its relation with the Arthur-Selberg trace formula in the Langlands program. We end with a tantalizing connection between the renormalization group and the missing Galois theory at Archimedean places.

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References

  1. A. Connes, Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup. 6:4 (1973), 133–252.

    MATH  MathSciNet  Google Scholar 

  2. M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications. Springer Lecture Notes in Math. 28 (1970).

    Google Scholar 

  3. A. Connes, Noncommutative Geometry and the Riemann Zeta Function, Mathematics: Frontiers and Perspectives, IMU 2000 volume, 35–55.

    Google Scholar 

  4. M.F. Atiyah, Global theory of elliptic operators, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), University of Tokyo Press, Tokyo (1970), 21–30.

    Google Scholar 

  5. I.M. Singer, Future extensions of index theory and elliptic operators, Ann. of Math. Studies 70 (1971), 171–185.

    Google Scholar 

  6. L.G. Brown, R.G. Douglas, P.A. Fillmore, Extensions of C*-algebras and K-homology, Ann. of Math. 2:105 (1977), 265–324.

    Article  MathSciNet  Google Scholar 

  7. A.S. Miscenko, C* algebras and K theory, Algebraic Topology, Aarhus 1978, Springer Lecture Notes in Math. 763 (1979), 262–274.

    Article  Google Scholar 

  8. G.G. Kasparov, The operator K-functor and extensions of C* algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571–636; Math. USSR Izv. 16 (1981), 513–572.

    MATH  MathSciNet  Google Scholar 

  9. P. Baum, A. Connes, Geometric K-theory for Lie groups and foliations, Preprint IHES (M/82/), 1982; l’Enseignement Mathematique, t. 46 (2000), 1–35 (to appear).

    MathSciNet  Google Scholar 

  10. M.F. Atiyah, W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Inventiones Math. 42 (1977), 1–62.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Skandalis, Approche de la conjecture de Novikov par la cohomologie cyclique, in “Seminaire Bourbaki, 1990–91”, Expose 739, 201–202–203 (1992), 299–316.

    Google Scholar 

  12. P. Julg, Travaux de N. Higson et G. Kasparov Sur la conjecture de Baum-Connes, in “Seminaire Bourbaki, 1997–98”, Expose 841,252 (1998), 151–183.

    MathSciNet  Google Scholar 

  13. G. Skandalis, Progres recents sur la conjecture de Baum-Connes, contribution de Vincent Lafforgue, in “Seminaire Bourbaki, 1999–2000”, Expose 869.

    Google Scholar 

  14. A. Connes, Cohomologie cyclique et foncteur Ext n, C.R. Acad. Sci. Paris, Ser. I Math. 296 (1983), 963–968.

    MathSciNet  Google Scholar 

  15. A. Connes, Spectral sequence and homology of currents for operator algebras, Math. Forschungsinst. Oberwolfach Tagungsber. 41/81; Funktionalanalysis und C*-Algebren, 27–9/3–10, 1981.

    Google Scholar 

  16. A. Connes, Noncommutative differential geometry. Part I: The Chern character in K-homology, Preprint IHES, M/82/53, 1982; Part II: de Rham homology and noncommutative algebra, Preprint IHES, M/83/19, 1983.

    Google Scholar 

  17. A. Connes, Noncommutative differential geometry, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985), 257–360.

    Article  MathSciNet  Google Scholar 

  18. B.L. Tsygan, Homology of matrix Lie algebras over rings and the Hochschild homology, Uspekhi Math. Nauk. 38 (1983), 217–218.

    MATH  MathSciNet  Google Scholar 

  19. A. Connes, H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345–388.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in “Geometric Methods in Operator Algebras, (Kyoto, 1983) ”, Pitman Res. Notes in Math. 123, Longman, Harlow (1986), 52–144.

    MathSciNet  Google Scholar 

  21. J.L. Loday, Cyclic Homology, Springer, Berlin-Heidelberg-New York, 1998.

    MATH  Google Scholar 

  22. D. Btjrghelea, The cyclic homology of the group rings, Comment. Math. Helv. 60 (1985), 354–365.

    Article  MathSciNet  Google Scholar 

  23. J. Cuntz, D. Qjjillen, Cyclic homology and singularity, J. Amer. Math. Soc. 8 (1995), 373–442.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Cuntz, D. Quillen, Operators on noncommutative differential forms and cyclic homology, J. Differential Geometry, to appear.

    Google Scholar 

  25. J. Cuntz, D. Quillen, On excision in periodic cyclic cohomology, I and II, C.R. Acad. Sci. Paris, Ser. I Math., 317 (1993), 917–922; 318 (1994), 11–12.

    MATH  MathSciNet  Google Scholar 

  26. B. Riemann, Mathematical Werke, Dover, New York, 1953.

    Google Scholar 

  27. S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, New York-London, 1972.

    Google Scholar 

  28. J. Dixmier, Existence de traces non normales, C.R. Acad. Sci. Paris, Ser. A-B, 262 (1966).

    Google Scholar 

  29. M. Wodzicki, Noncommutative residue, Part I. Fundamentals, in “K-theory, Arithmetic and Geometry”, Springer Lecture Notes in Math. 1289 (1987).

    Google Scholar 

  30. J. Milnor, D. Stasheff, Characteristic classes, Ann. of Math. Stud. Princeton University Press, Princeton, N.J. 1974.

    Google Scholar 

  31. D. Sullivan, Geometric periodicity and the invariants of manifolds, Springer Lecture Notes in Math. 197 (1971).

    Google Scholar 

  32. B. Lawson, M.L. Michelson, Spin Geometry, Princeton, 1989.

    Google Scholar 

  33. A. Connes, Entire cyclic cohomology of Banach algebras and characters of θ summable Fredholm modules, K-theory 1 (1988), 519–548.

    Article  MATH  MathSciNet  Google Scholar 

  34. A. Jaffe, A. Lesniewski, K. Osterwalder, Quantum K-theory: I. The Chern character, Commun. Math. Phys. 118 (1988), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  35. PA. Connes, H. moscovici, The local index formula in noncommutative geometry, GAFA 5 (1995), 174–243.

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Connes, Noncommutative Geometry, Academic Press, 1994.

    Google Scholar 

  37. A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198 (1998), 199–246.

    Article  MATH  MathSciNet  Google Scholar 

  38. M. Hilsum, G. Skandalis, Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov, Ann. Sci. Ecole Norm. Sup. (4), 20 (1987), 325–390.

    MATH  MathSciNet  Google Scholar 

  39. Y. Manin, Quantum groups and noncommutative geometry, Centre Recherche Math. Univ. Montréal, 1988.

    Google Scholar 

  40. A. Connes, C* algèbres et géométrie differentielle, C.R. Acad. Sci. Paris, Ser. A-B 290 (1980), 599–604.

    MATH  MathSciNet  Google Scholar 

  41. A. Connes, H. Moscovici, Cyclic cohomology and Hopf algebras, Letters Math. Phys. 48:1 (1999), 97–108.

    Article  MATH  MathSciNet  Google Scholar 

  42. D. Kreimer, On the Hopf algebra structure of perturbative Quantum Field Theory, Adv. Theor. Math. Phys. 2.2, 303 (1998); q-alg/9707029. GAFA2000

    Google Scholar 

  43. D. Kreimer, On overlapping divergencies, Commun. Math. Phys. 204, 669 (1999); hep-th/9810022.

    Article  MATH  MathSciNet  Google Scholar 

  44. A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998), 203–242.

    Article  MATH  MathSciNet  Google Scholar 

  45. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem, J. High Energy Phys. 9, Paper 24 (1999), 8pp; hep-th/9909126.

    Google Scholar 

  46. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210 (2000), 249–273; hep-th/9912092.

    Article  MATH  MathSciNet  Google Scholar 

  47. A. Beatjville, Monodromie des systèmes difiérentiels lineaires à pôles simples sur la sphère de Riemann, in “Séminaire Bourbaki, 45ème année”, 1992–1993, n. 765.

    Google Scholar 

  48. A. Bolibrjjch, Fuchsian systems with reducible monodromy and the Riemann-Hilbert problem, Springer Lecture Notes in Math. 1520 (1992), 139–155.

    Article  Google Scholar 

  49. A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II: The β function, diffeomorphisms and the renormalization group, hep-th/0003188.

    Google Scholar 

  50. A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996), 155–176.

    Article  MATH  MathSciNet  Google Scholar 

  51. W. Kalajj, M. Walze, Gravity, noncommutative geometry and the Wodzi-cki residue, J. of Geom. and Phys. 16 (1995), 327–344.

    Article  Google Scholar 

  52. D. Kastler, The Dirac operator and gravitation, Commun. Math. Phys. 166 (1995), 633–643.

    Article  MATH  MathSciNet  Google Scholar 

  53. A. Connes, Noncommutative geometry and reality, Journal of Math. Physics 36:11 (1995), 6194–6231.

    Article  MATH  MathSciNet  Google Scholar 

  54. T. Schucker, Spin group and almost commutative geometry, hep-th/0007047.

    Google Scholar 

  55. M.F. Atiyah, K-theory and reality. Quart. J. Math. Oxford (2)17 (1966),367–386.

    Article  MathSciNet  Google Scholar 

  56. M.A. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51–96.

    Article  MATH  MathSciNet  Google Scholar 

  57. M. Gromov, Carnot-Caratheodory spaces seen from within, Preprint IHES/M/94/6.

    Google Scholar 

  58. A. Chamseddine, A. Connes, Universal formulas for noncommutative geometry actions, Phys. Rev. Letters 77,24 (1996), 4868–4871.

    Article  MATH  MathSciNet  Google Scholar 

  59. A. Connes, Noncommutative Geometry: The Spectral Aspect, in “Les Houches Session LXIV”, Elsevier (1998), 643–685.

    Google Scholar 

  60. M. Karoubi, Homologie cyclique et K-théorie, Asterisque 149 (1987).

    Google Scholar 

  61. M.A. Rieffel, C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429.

    MATH  MathSciNet  Google Scholar 

  62. M. Pimsner, D. Voiculescu, Exact sequences for K groups and Ext group of certain crossed product C*-algebras, J. Operator Theory 4 (1980), 93–118.

    MATH  MathSciNet  Google Scholar 

  63. M.A. Rieffel, The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc. 47 (1983), 285–302.

    Article  MATH  MathSciNet  Google Scholar 

  64. A. Connes, M. Rieffel, Yang-Mills for noncommutative two tori, in “Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985), Contemp. Math. Oper. Algebra Math. Phys. 62, Amer. Math. Soc., Providence, RI (1987), 237–266.

    Google Scholar 

  65. A. Connes, G. Landi, Noncommutative manifolds, the Instanton algebra and isospectral deformations, Math-QA/0011194.

    Google Scholar 

  66. J.M. Gracia-bondia, J.C. Varilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhauser, 2000.

    Google Scholar 

  67. A. Connes, M. Douglas, A. Schwarz, Noncommutative geometry and Matrix theory: compactification on tori, J. High Energy Physics 2, Paper 3 (1998), 35pp.

    Google Scholar 

  68. A. Connes, A short survey of noncommutative geometry, J. Math. Physics 41 (2000), 3832–3866.

    Article  MATH  MathSciNet  Google Scholar 

  69. S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisès de C*-algebres, Ann. Sci. Ec. Norm. Sup., 4 serié, t. 26 (1993), 425–488.

    MathSciNet  Google Scholar 

  70. G.I. Kac, Extensions of Groups to Ring Groups, Math. USSR Sbornik 5:3 (1968).

    Google Scholar 

  71. S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, 1995.

    Google Scholar 

  72. M. Berry, Riemann’s zeta function: a model of quantum chaos, Springer Lecture Notes in Physics 263 (1986).

    Google Scholar 

  73. A. Connes Trace formula in Noncommutative Geometry and the zeros of the Riemann zeta function, Selecta Mathematica New Ser. 5 (1999), 29–106.

    Article  MATH  MathSciNet  Google Scholar 

  74. N. Nekrasov, A. Schwarz, Instantons in noncommutative ℝ4 and (2,0) superconformal six dimensional theory, hep-th/9802068.

    Google Scholar 

  75. N. Seiberg, E. Witten, String theory and noncommutative geometry, J. High Energy Physics 9 (1999).

    Google Scholar 

  76. J. Arthur, The invariant trace formula II. Global theory, J. of the AMS I (1988), 501, 554.

    Google Scholar 

  77. A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951),1–35.

    Article  MATH  MathSciNet  Google Scholar 

  78. A.R. Bernstein, F. Wattenberg, Non standard measure theory, in “Applications of Model Theory to Algebra Analysis and Probability” (W.A.J. Luxenburg Halt, ed.), Rinehart and Winstin, 1969.

    Google Scholar 

  79. N.N. Bogoliubov, D.V. Shirkov, Introduction to the theory of quantized fields, 3rd ed., Wiley 1980; K. Hepp, Comm. Math. Phys. 2 (1966), 301–326; W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Comm. Math. Phys. 15 (1969), 208–234.

    Google Scholar 

  80. M. Dresden, Renormalization in historical perspective-The first stage, in “Renormalization” (L. Brown, ed.) Springer-Verlag, New York-Berlin-Heidelberg, 1994.

    Google Scholar 

  81. H. Epstein, V. Glaser, The role of locality in perturbation theory, Ann. Inst. H. Poincaré A 19 (1973), 211–295.

    MathSciNet  Google Scholar 

  82. A. Goncharov, Polylogarithms in arithmetic and geometry, Proc. of ICM-94 (Zürich), 1,2, Birkhäuser (1995), 374–387.

    Google Scholar 

  83. D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II, Birkhauser, Boston (1994), 497–512

    Google Scholar 

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Authors and Affiliations

  1. Collège de France, 3, rue Ulm, 75005, Paris, France

    Alain Connes

  2. I.H.E.S., 35, route de Chartres, 91440, Bures-sur-Yvette, France

    Alain Connes

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  1. Alain Connes
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Editors and Affiliations

  1. School of Mathematical Sciences, University of Tel Aviv, Tel Aviv, 69978, Israel

    N. Alon

  2. School of Mathematics Institute for Advanced Study, Princeton University, Princeton, NJ, 08540, USA

    J. Bourgain

  3. Collège de France, 3, rue d’Ulm, 75231, Paris cedex 05, France

    A. Connes

  4. Institut des Hautes Études Scientifiques, 35, Route de Chartres, 91440, Bures-sur-Yvette, France

    M. Gromov

  5. Department of Mathematics, University of Tel Aviv, Tel Aviv, 69978, Israel

    V. Milman

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Connes, A. (2000). Noncommutative Geometry Year 2000. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0425-3_3

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