A Lecture on Noncommutative Geometry

Connes, Alain

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 11 (2000), p. 31-64 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

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.

Détail

Publié le : 2000-12-01

MR 1839740 | Zbl 1149.58302

Bibliographie

[1] Connes, A., Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup., 6, n. 4, 1973, 133-252. | MR 341115 | Zbl 0274.46050

[2] Takesaki, M., Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Math., 28, Springer-Verlag, 1970. | MR 270168 | Zbl 0193.42502

[3] Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math., 131, 1973, 249-310. | MR 438149 | Zbl 0268.46058

[4] Krieger, W., On ergodic flows and the isomorphism of factors. Math. Ann., 223, 1976, 19-70. | MR 415341 | Zbl 0332.46045

[5] Connes, A. - Takesaki, M., The flow of weights on factors of type III. Tohoku Math. J., 29, 1977, 473-575. | MR 480760 | Zbl 0408.46047

[6] Connes, A., Classification of injective factors. Ann. of Math., 104, n. 2, 1976, 73-115. | MR 454659 | Zbl 0343.46042

[7] Connes, A., Outer conjugacy classes of automorphisms of factors. Ann. Sci. Ecole Norm. Sup., 8, n. 4, 1975, 383-419. | MR 394228 | Zbl 0342.46052

[8] Connes, A., Factors of type $III_{1}$, property $L^{1}_{\lambda}$ and closure of inner automorphisms. J. Operator Theory, 14, 1985, 189-211. | MR 789385 | Zbl 0597.46063

[9] Haagerup, U., Connes' bicentralizer problem and uniqueness of the injective factor of type. $III_{1}$. Acta Math., 158, 1987, 95-148. | MR 880070 | Zbl 0628.46061

[10] Connes, A., Noncommutative Geometry and the Riemann Zeta Function, invited lecture in IMU 2000 volume. To appear. | MR 1754766

[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. | MR 266247

[12] Singer, I. M., Future extensions of index theory and elliptic operators. Ann. of Math. Studies, 70, 1971, 171-185. | MR 343319

[13] Brown, L.G. - Douglas, R.G. - Fillmore, P.A., Extensions of $C^{\star}$-algebras and K-homology. Ann. of Math., 2, 105, 1977, 265-324. | MR 458196 | Zbl 0376.46036

[14] Mishenko, A. S., $C^{\star}$-algebras and K theory. Algebraic Topology, Aarhus 1978, Lecture Notes in Math., 763, Springer-Verlag, 1979, 262-274. | MR 561226

[15] Kasparov, G. G., The operator K-functor and extensions of $C^{\star}$-algebras. Izv. Akad. Nauk SSSR, Ser. Mat., 44, 1980, 571-636; Math. USSR Izv., 16, 1981, 513-572. | MR 582160 | Zbl 0448.46051

[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). | MR 1769535

[17] Atiyah, M. F. - Schmid, W., A geometric construction of the discrete series for semisimple Lie groups. Inventiones Math., 42, 1977, 1-62. | MR 463358 | Zbl 0373.22001

[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. | MR 1157846

[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. | MR 1685573

[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 $Ext^{n}$. C.R. Acad. Sci. Paris, Ser. I Math, 296, 1983, 953-958. | MR 777584

[22] Connes, A., Spectral sequence and homology of currents for operator algebras. Math. Forschungsinst. Oberwolfach Tagungsber., 41/81; Funktionalanalysis und $C^{\star}$-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. | MR 823176

[25] Tsygan, B. L., Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Math. Nauk., 38, 1983,217-218. | MR 695483 | Zbl 0518.17002

[26] Connes, A. - Moscovici, H., Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology, 29, 1990, 345-388. | MR 1066176 | Zbl 0759.58047

[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. | MR 866491

[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. | MR 196508 | Zbl 0141.12902

[31] Wodzicki, M., Noncommutative residue. Part I. Fundamentals K-theory, arithmetic and geometry. Lecture Notes in Math., 1289, Springer-Verlag, Berlin1987. | MR 923140 | Zbl 0649.58033

[32] Milnor, J. - Stasheff, D., Characteristic classes. Ann. of Math. Stud., Princeton University Press, Princeton, N.J. 1974. | MR 440554

[33] Sullivan, D., Geometric periodicity and the invariants of manifolds. Lecture Notes in Math., 197, Springer-Verlag, 1971. | MR 285012 | Zbl 0224.57002

[34] Lawson, B. - Michelson, M. L., Spin Geometry. Princeton 1989. | MR 1031992

[35] Connes, A., Entire cyclic cohomology of Banach algebras and characters of $\theta$ summable Fredholm modules. K-theory, 1, 1988, 519-548. | MR 953915 | Zbl 0657.46049

[36] Jaffe, A. - Lesniewski, A. - Osterwalder, K., Quantum K-theory: I. The Chem character. Comm. Math. Phys., 118, 1988, 1-14. | MR 954672 | Zbl 0656.58048

[37] Connes, A. - Moscovici, H., The local index formula in noncommutative geometry. Geom. Funct. Anal., 5, 1995, 174-243. | MR 1334867 | Zbl 0960.46048

[38] Connes, A., Noncommutative geometry. Academic Press, San Diego Cal.1994. | MR 1303779

[39] Connes, A. - Moscovici, H., Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem. Comm. Math. Phys., 198, 1998, 199-246. | MR 1657389 | Zbl 0940.58005

[40] Connes, A., $C^{\star}$-algèbres et géométrie differentielle. C.R. Acad. Sci. Paris, Ser. A-B, 290, 1980, A599-A604. | MR 572645

[41] Connes, A. - Moscovici, H., Cyclic Cohomology and Hopf Algebras. Letters Math. Phys., 48, 1, 1999, 97-108. | MR 1718047

[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. | MR 1633004

[43] Kreimer, D., On overlapping divergencies. Comm. Math. Phys., 204, 1999, 669-689; hep-th/9810022. | MR 1707611 | Zbl 0977.81091

[44] Kreimer, D., Chen's iterated integral represents the operator product expansion. Adv. Theor. Math. Phys., 3, 3, 1999, to appear; hep-th/9901099. | MR 1797019 | Zbl 0971.81093

[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. | MR 1757650

[46] Connes, A. - Kreimer, D., Hopf algebras, Renormalization and Noncommutative Geometry. Comm. Math. Phys., 199, 1998, 203-242; hep-th/9808042. | MR 1660199 | Zbl 0932.16038

[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. | MR 1720691 | Zbl 0957.81011

[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. | MR 1748177 | Zbl 1032.81026

[49] Connes, A. - Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem II. The $\beta$ function, diffeomorphisms and the renormalization group, hep-th/0003188. | Zbl 1042.81059

[50] Connes, A., Gravity coupled with matter and foundation of noncommutative geometry. Comm. Math. Phys., 182, 1996, 155-176. | MR 1441908 | Zbl 0881.58009

[51] Kalau, W. - Walze, M., Gravity, noncommutative geometry and the Wodzicki residue. J. of Geom. and Phys., 16, 1995, 327-344. | MR 1336738 | Zbl 0826.58008

[52] Kastler, D., The Dirac operator and gravitation. Comm. Math. Phys., 166, 1995, 633-643. | MR 1312438 | Zbl 0823.58046

[53] Connes, A., Noncommutative geometry and reality. Journal of Math. Physics, 36, n.11, 1995, 6194-6231. | MR 1355905 | Zbl 0871.58008

[54] Rieffel, M. A., Morita equivalence for $C^{\star}$-algebras and $W^{\star}$-algebras. J. Pure Appl. Algebra, 5, 1974, 51-96. | MR 367670

[55] Gromov, M., Carnot-Carathéodory spaces seen from within. Preprint IHES/M/94/6. | MR 1421823 | Zbl 0864.53025

[56] Chamsedine, A. - Connes, A., Universal formulas for noncommutative geometry actions. Phys. Rev. Letters, 77, 24, 1996, 4868-4871. | MR 1419931 | Zbl 1020.46505

[57] Connes, A., Noncommutative Geometry: The Spectral Aspect. Les Houches Session LXIV, Elsevier1998, 643-685. | MR 1616407

[58] Pimsner, M. - Voiculescu, D., Exact sequences for K groups and Ext group of certain crossed product $C^{\star}$-algebras. J. Operator Theory, 4, 1980, 93-118. | MR 587369 | Zbl 0474.46059

[59] Rieffel, M. A., The cancellation theorem for projective modules over irrational rotation $C^{\star}$-algebras. Proc. London Math. Soc., 47, 1983, 285-302. | MR 703981 | Zbl 0541.46055

[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. | MR 878383

[61] Connes, A. - Douglas, M. R. - Schwartz, A., Noncommutative geometry and Matrix theory: compactification on tori. J. High Energy Physics, 2, 1998. | MR 1613978

[62] Connes, A., A short survey of noncommutative geometry. J. Math. Physics, 41, 2000. | MR 1768641 | Zbl 0974.58008