A Lecture on Noncommutative Geometry

Alain Connes

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: S1, page 31-64
  • ISSN: 1120-6330

Abstract

top
The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the Noncommutative torus which plays a role in M-theory compactification and finally the Adele class space which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic. On the other hand the stretching of geometric thinking imposed by passing to Noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. Several of these new developments are described below with emphasis on the surprises from the noncommutative world.

How to cite

top

Connes, Alain. "A Lecture on Noncommutative Geometry." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.S1 (2000): 31-64. <http://eudml.org/doc/289692>.

@article{Connes2000,
abstract = {The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the Noncommutative torus which plays a role in M-theory compactification and finally the Adele class space which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic. On the other hand the stretching of geometric thinking imposed by passing to Noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. Several of these new developments are described below with emphasis on the surprises from the noncommutative world.},
author = {Connes, Alain},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
language = {eng},
month = {12},
number = {S1},
pages = {31-64},
publisher = {Accademia Nazionale dei Lincei},
title = {A Lecture on Noncommutative Geometry},
url = {http://eudml.org/doc/289692},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Connes, Alain
TI - A Lecture on Noncommutative Geometry
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/12//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - S1
SP - 31
EP - 64
AB - The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the Noncommutative torus which plays a role in M-theory compactification and finally the Adele class space which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic. On the other hand the stretching of geometric thinking imposed by passing to Noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. Several of these new developments are described below with emphasis on the surprises from the noncommutative world.
LA - eng
UR - http://eudml.org/doc/289692
ER -

References

top
  1. CONNES, A., Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup., 6, n. 4, 1973, 133-252. Zbl0274.46050MR341115
  2. TAKESAKI, M., Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Math., 28, Springer-Verlag, 1970. Zbl0193.42502MR270168
  3. TAKESAKI, M., Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math., 131, 1973, 249-310. Zbl0268.46058MR438149DOI10.1007/BF02392041
  4. KRIEGER, W., On ergodic flows and the isomorphism of factors. Math. Ann., 223, 1976, 19-70. Zbl0332.46045MR415341DOI10.1007/BF01360278
  5. CONNES, A. - TAKESAKI, M., The flow of weights on factors of type III. Tohoku Math. J., 29, 1977, 473-575. Zbl0408.46047MR480760DOI10.2748/tmj/1178240493
  6. CONNES, A., Classification of injective factors. Ann. of Math., 104, n. 2, 1976, 73-115. Zbl0343.46042MR454659DOI10.2307/1971057
  7. CONNES, A., Outer conjugacy classes of automorphisms of factors. Ann. Sci. Ecole Norm. Sup., 8, n. 4, 1975, 383-419. Zbl0342.46052MR394228
  8. CONNES, A., Factors of type I ⁢ I ⁢ I 1 , property L λ 1 and closure of inner automorphisms. J. Operator Theory, 14, 1985, 189-211. Zbl0597.46063MR789385
  9. HAAGERUP, U., Connes' bicentralizer problem and uniqueness of the injective factor of type. I ⁢ I ⁢ I 1 . Acta Math., 158, 1987, 95-148. Zbl0628.46061MR880070DOI10.1007/BF02392257
  10. CONNES, A., Noncommutative Geometry and the Riemann Zeta Function, invited lecture in IMU 2000 volume. To appear. MR1754766
  11. ATIYAH, M. F., Global theory of elliptic operators. Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969). University of Tokyo Press, Tokyo1970, 21-30. MR266247
  12. SINGER, I. M., Future extensions of index theory and elliptic operators. Ann. of Math. Studies, 70, 1971, 171-185. MR343319
  13. BROWN, L.G. - DOUGLAS, R.G. - FILLMORE, P.A., Extensions of C ⋆ -algebras and K-homology. Ann. of Math., 2, 105, 1977, 265-324. Zbl0376.46036MR458196DOI10.2307/1970999
  14. MISHENKO, A. S., C ⋆ -algebras and K theory. Algebraic Topology, Aarhus 1978, Lecture Notes in Math., 763, Springer-Verlag, 1979, 262-274. MR561226
  15. KASPAROV, G. G., 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. Zbl0448.46051MR582160
  16. BAUM, P. - CONNES, A., Geometric K-theory for Lie groups and foliations. Preprint IHES (M /82/), 1982; l'Enseignement Mathématique, t. 46, 2000, 1-35 (to appear). MR1769535
  17. ATIYAH, M. F. - SCHMID, W., A geometric construction of the discrete series for semisimple Lie groups. Inventiones Math., 42, 1977, 1-62. Zbl0373.22001MR463358DOI10.1007/BF01389783
  18. SKANDALIS, G., Approche de la conjecture de Novikov par la cohomologie cyclique. In: Séminaire Bourbaki, 1990-91, Expose 739, 201-202-203, 1992, 299-316. MR1157846
  19. JULG, P., Travaux de N. Higson et G. Kasparov sur la conjecture de Baum-Connes. In: Séminaire Bourbaki, 1997-98, Expose 841, 252, 1998, 151-183. MR1685573
  20. SKANDALIS, G., Progrès recents sur la conjecture de Baum-Connes, contribution de Vincent Lafforgue. In: Séminaire Bourbaki, 1999-2000, Expose 869. 
  21. CONNES, A., Cohomologie cyclique et foncteurs E ⁢ x ⁢ t n . C.R. Acad. Sci. Paris, Ser. I Math, 296, 1983, 953-958. MR777584
  22. CONNES, A., Spectral sequence and homology of currents for operator algebras. Math. Forschungsinst. Oberwolfach Tagungsber., 41/81; Funktionalanalysis und C ⋆ -Algebren, 27-9/3-10, 1981. 
  23. CONNES, A., 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. 
  24. CONNES, A., Noncommutative differential geometry. Inst. Hautes Etudes Sci. Publ. Math., 62, 1985, 257-360. MR823176
  25. Tsygan, B. L., Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Math. Nauk., 38, 1983,217-218. Zbl0518.17002MR695483
  26. CONNES, A. - MOSCOVICI, H., Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology, 29, 1990, 345-388. Zbl0759.58047MR1066176DOI10.1016/0040-9383(90)90003-3
  27. CONNES, A., 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, Harlow1986, 52-144. MR866491
  28. RIEMANN, B., Mathematical Werke. Dover, New York1953. 
  29. WEINBERG, S., Gravitation and Cosmolog. John Wiley and Sons, New York - London1972. 
  30. DIXMIER, J., Existence de traces non normales. C.R. Acad. Sci. Paris, ser. A-B, 262, 1966. Zbl0141.12902MR196508
  31. WODZICKI, M., Noncommutative residue. Part I. Fundamentals K-theory, arithmetic and geometry. Lecture Notes in Math., 1289, Springer-Verlag, Berlin1987. Zbl0649.58033MR923140DOI10.1007/BFb0078372
  32. MILNOR, J. - STASHEFF, D., Characteristic classes. Ann. of Math. Stud., Princeton University Press, Princeton, N.J.1974. MR440554
  33. SULLIVAN, D., Geometric periodicity and the invariants of manifolds. Lecture Notes in Math., 197, Springer-Verlag, 1971. Zbl0224.57002MR285012
  34. LAWSON, B. - MICHELSON, M. L., Spin Geometry. Princeton1989. MR1031992
  35. CONNES, A., Entire cyclic cohomology of Banach algebras and characters of θ summable Fredholm modules. K-theory, 1, 1988, 519-548. Zbl0657.46049MR953915DOI10.1007/BF00533785
  36. JAFFE, A. - LESNIEWSKI, A. - OSTERWALDER, K., Quantum K-theory: I. The Chem character. Comm. Math. Phys., 118, 1988, 1-14. Zbl0656.58048MR954672
  37. CONNES, A. - MOSCOVICI, H., The local index formula in noncommutative geometry. Geom. Funct. Anal., 5, 1995, 174-243. Zbl0960.46048MR1334867DOI10.1007/BF01895667
  38. CONNES, A., Noncommutative geometry. Academic Press, San Diego Cal.1994. MR1303779
  39. CONNES, A. - MOSCOVICI, H., Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem. Comm. Math. Phys., 198, 1998, 199-246. Zbl0940.58005MR1657389DOI10.1007/s002200050477
  40. CONNES, A., C ⋆ -algèbres et géométrie differentielle. C.R. Acad. Sci. Paris, Ser. A-B, 290, 1980, A599-A604. MR572645
  41. CONNES, A. - MOSCOVICI, H., Cyclic Cohomology and Hopf Algebras. Letters Math. Phys., 48, 1, 1999, 97-108. MR1718047DOI10.1023/A:1007527510226
  42. KREIMER, D., On the Hopf algebra structure of perturbative Quantum Field Theory. Adv. Theor. Math. Phys., 2, 2, 1998, 303-334; q-alg/9707029. MR1633004DOI10.4310/ATMP.1998.v2.n2.a4
  43. KREIMER, D., On overlapping divergencies. Comm. Math. Phys., 204, 1999, 669-689; hep-th/9810022. Zbl0977.81091MR1707611DOI10.1007/s002200050661
  44. KREIMER, D., Chen's iterated integral represents the operator product expansion. Adv. Theor. Math. Phys., 3, 3, 1999, to appear; hep-th/9901099. Zbl0971.81093MR1797019DOI10.4310/ATMP.1999.v3.n3.a7
  45. KREIMER, D. - DELBOURGO, R., Using the Hopf algebra structure of Quantum Field Theory in calculations. Phys. Rev., D 60, 1999, 105025-1-105025-14; hep-th/9903249. MR1757650DOI10.1103/PhysRevD.60.105025
  46. CONNES, A. - KREIMER, D., Hopf algebras, Renormalization and Noncommutative Geometry. Comm. Math. Phys., 199, 1998, 203-242; hep-th/9808042. Zbl0932.16038MR1660199DOI10.1007/s002200050499
  47. CONNES, A. - KREIMER, D., Renormalization in quantum field theory and the Riemann-Hilbert problem. J. High Energy Phys., 09, 1999, 024; hep-th/9909126. Zbl0957.81011MR1720691DOI10.1088/1126-6708/1999/09/024
  48. CONNES, A. - KREIMER, D., Renormalization in quantum field theory and the Riemann-Hilbert problem I. the Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys., 210, 1, 2000, 249-273; hep-th/9912092. Zbl1032.81026MR1748177DOI10.1007/s002200050779
  49. CONNES, A. - KREIMER, D., Renormalization in quantum field theory and the Riemann-Hilbert problem II. The β function, diffeomorphisms and the renormalization group, hep-th/0003188. Zbl1042.81059
  50. CONNES, A., Gravity coupled with matter and foundation of noncommutative geometry. Comm. Math. Phys., 182, 1996, 155-176. Zbl0881.58009MR1441908
  51. KALAU, W. - WALZE, M., Gravity, noncommutative geometry and the Wodzicki residue. J. of Geom. and Phys., 16, 1995, 327-344. Zbl0826.58008MR1336738DOI10.1016/0393-0440(94)00032-Y
  52. KASTLER, D., The Dirac operator and gravitation. Comm. Math. Phys., 166, 1995, 633-643. Zbl0823.58046MR1312438
  53. CONNES, A., Noncommutative geometry and reality. Journal of Math. Physics, 36, n.11, 1995, 6194-6231. Zbl0871.58008MR1355905DOI10.1063/1.531241
  54. RIEFFEL, M. A., Morita equivalence for C ⋆ -algebras and W ⋆ -algebras. J. Pure Appl. Algebra, 5, 1974, 51-96. MR367670DOI10.1016/0022-4049(74)90003-6
  55. GROMOV, M., Carnot-Carathéodory spaces seen from within. Preprint IHES/M/94/6. Zbl0864.53025MR1421823
  56. CHAMSEDINE, A. - CONNES, A., Universal formulas for noncommutative geometry actions. Phys. Rev. Letters, 77, 24, 1996, 4868-4871. Zbl1020.46505MR1419931DOI10.1103/PhysRevLett.77.4868
  57. CONNES, A., Noncommutative Geometry: The Spectral Aspect. Les Houches Session LXIV, Elsevier1998, 643-685. MR1616407
  58. PIMSNER, M. - VOICULESCU, D., Exact sequences for K groups and Ext group of certain crossed product C ⋆ -algebras. J. Operator Theory, 4, 1980, 93-118. Zbl0474.46059MR587369
  59. RIEFFEL, M. A., The cancellation theorem for projective modules over irrational rotation C ⋆ -algebras. Proc. London Math. Soc., 47, 1983, 285-302. Zbl0541.46055MR703981DOI10.1112/plms/s3-47.2.285
  60. CONNES, A. - RIEFFEL, M. A., 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. MR878383DOI10.1090/conm/062/878383
  61. CONNES, A. - DOUGLAS, M. R. - Schwartz, A., Noncommutative geometry and Matrix theory: compactification on tori. J. High Energy Physics, 2, 1998. MR1613978DOI10.1088/1126-6708/1998/02/003
  62. CONNES, A., A short survey of noncommutative geometry. J. Math. Physics, 41, 2000. Zbl0974.58008MR1768641DOI10.1063/1.533329

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.