Seminar on Noncommutative Geometry

Mini-Workshop 21-23 May, 2003

Wednesday, 21 May (Room P.015A)

11:00-13:00 R. Popescu, Actions of groupoids on C*-algebras and equivariant E-theory
14:00-16:00 A. Wassermann, von Neumann algebras in mathematics and physics

Thursday, 22 May (Room P.015B)

11:00-13:00 E. van Erp, The Atiyah-Singer index formula for subelliptic operators
14:00-16:00 M. Benameur, The index theorem in Haefliger cohomology of foliations

Friday, 23 May (Room P.019)

11:00-13:00 P.-E. Paradan, Transversally elliptic operators and geometric quantization
14:00-15:00 E. Opdam, Elliptic representations of Hecke algebras
15:30-16:30 N.P. Landsman, Functoriality of quantization, KK-theory, and the Guillemin-Sternberg conjecture


Program for Friday, February 14, 2003

11:00-13:00 M. Solleveld, Cyclic cohomology I
14:00-16:00 M. Mueger, K-theory for locally convex algebras and the Chern-Connes character I

Program for Friday, February 28, 2003

11:00-13:00 E. Opdam, Index functions and A. Wassermann's proof of the Baum-Connes conjecture for real reductive groups.
14:00-16:00 M. Solleveld, Cyclic cohomology II

Program for Friday, March 14, 2003

11:00-13:00 R. Bos, Spin^c structures and K-theory III
14:00-16:00 M. Mueger, K-theory for locally convex algebras and the Chern-Connes character II

Program for Friday, April 11, 2003

11:00-13:00 N.P. Landsman, The general Baum-Connes conjecture
14:00-16:00 E. Opdam, Towards K-theory for affine Hecke algebras

Old program (2002)

Program for Friday, December 6th:

11:00-13:00 N.P. Landsman, KK_theory
14:00-16:00 R.D. Bos, Spin^c structures and K-theory II

Program for Friday, November 22nd:

11:00-13:00 E.M. Opdam, Index theory and trace formula II
14:00-16:00 R.D. Bos, Spin^c structures and K-theory

Program for Friday, November 1st:

11:00-13:00 E.M. Opdam, Index theory and trace formula
14:00-16:00 N.P. Landsman, Connes's proof of the Atiyah-Singer index theorem II

Program for Friday, October 18th:

11:00-13:00 N.P. Landsman, Connes's proof of the Atiyah-Singer index theorem I
14:00-16:00 E. Emsiz, K-theory II sheets

Program for Friday, October 6th:

11:00-13:00 E.P. van den Ban, Harish-Chandra's Plancherel formula for semisimple Lie groups III
14:00-16:00 E. Emsiz, K-theory

Program for Friday, September 20th, 2000:

11:00-13:00 N.P. Landsman, The Atiyah-Singer index theorem
14:00-16:00 E.P. van den Ban, Harish-Chandra's Plancherel formula for semisimple Lie groups II

Program for Friday, September 6th, 2002:

11:00-13:00 N.P. Landsman, Overview: from Atiyah-Singer to V. Lafforgue
14:00-16:00 E.P. van den Ban, Harish-Chandra's Plancherel formula for semisimple Lie groups I

Outline

The aim of this seminar is to explore the relationship between index theory and representation theory. We intend to do this in the setting of noncommutative geometry, which provides such powerful tools as (noncommutative) K-theory, K-homology, cyclic cohomology, and E-theory. The relationship in question goes back to the work of Atiyah-Schmid and Parthasarathy on discrete series representations of semisimple Lie groups, and was further developed by Connes-Moscovici and others. It culminates in the so-called Baum-Connes conjecture, which describes the K-theory of a group algebra in terms of topological data.

More precisely, the unitary dual G^ of a topological group G, which plays a basic role in the harmonic analysis of G, was studied for several classes of topological groups G long before the advent of noncommutative geometry. From the 1970's onwards it became apparent that equivariant K-theory and index theory are indispensable tools for the understanding of the basic building blocks of G^, and in fact G^ came to be seen as a natural example of a noncommutative space, technically described by the group C*-algebra C*(G). In 1982, these developments led Baum and Connes to formulate their now famous conjecture relating the noncommutative K-theory of the (reduced) group C*-algebra of G to the equivariant topological K-theory of maximal compact subgroups of G. This relation is essentially given by a generalized index, or, in E-theory, by a deformation.

The Baum-Connes conjecture has been checked for various special classes of topological groups. Unfortunately, many of these proofs rely on detailed explicit knowledge of G^, and, in addition, do not give much insight why the Baum-Connes conjecture is true in those cases. Exciting recent work of Vincent Lafforgue partly remedies at least the former drawback, providing an entirely new way of analyzing G^. The latter problem is to some extent addressed by reformulating the Baum-Connes conjecture in terms of deformation theory.

The first half of the seminar (September-December 2002) will be devoted to introductory lectures on the following topics:

In the second half of the seminar (January-May 2003), guest speakers will give up to date lectures on the Baum-Connes conjecture and related matters. We hope to end with Lafforgue's recent work, and in addition intend to clarify the E-theoretic description of the Baum-Connes conjecture and its relationship with deformation quantization.

Although we intend to cover all pertinent background, we expect the seminar to be appropriate for PhD students and staff, as well as to undergraduates working in one of the relevant areas.

Organizers: Klaas Landsman and Eric Opdam, npl@science.uva.nl, 020-5255101 and opdam@science.uva.nl, 020-5255205

Dates:

(Almost) biweekly on Fridays: September 6, 20, October 4,18, November 1, 22, December 6, 2002 (no meeting on 15/11 in view of the 10 years of Stieltjes celebration, no meeting on 29/11 in view of the Kaashoek Symposium). Dates in 2003 to be announced.

Time:

11:00-13:00 and 14:00-16:00

Location:

Room P. 017, Korteweg-de Vries Instituut, Universiteit van Amsterdam, Gebouw Euclides,
Plantage Muidergracht 24, 1018 TV AMSTERDAM

Literature:

Atiyah, M.F., Singer, I.: The index of elliptic operators I. Ann. Math. 87, 485--530 (1968).

Atiyah, M.F., Singer, I.: The index of elliptic operators III. Ann. Math. 87, 546--604 (1968).

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

Baum, P., Connes, A.: Geometric K-theory for Lie groups and foliations, Ens. Math. 46, 3--42 (2000).

Baum, P., Connes, A., Higson, N.: Classifying space for proper actions and K-theory of group C*-algebras. Contemp. Math. 167, 241--291 (1994).

Blackadar, B.: K-theory for Operator Algebras, 2nd ed. Cambridge: Cambridge University Press, 1999.

Brodzki, J.: An Introduction to K-theory and Cyclic Cohomology. Polish Scientific Publishers, Warsaw, 1998. arXiv:funct-an/9606001.

Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994.

Connes, A., Moscovici, H.: The L^2-index theorem for homogeneous spaces of Lie groups. Ann. of Math. (2) 115, 291--330 (1982).

Connes, A., Skandalis, G.: The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20, 1139--1183 (1984).

J. Cuntz, K-Theorie fuer lokalkonvexe Algebren und der Chern-Connes-Character. Documenta Mathematica 2, 139-182 (1997).

Higson, N., Roe, J.: Analytic K-homology. Oxford: Oxford University Press, 2000.

Lafforgue, V.: Banach KK-theory and the Baum-Connes conjecture. ICM 2002 talk

Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, 1989.

Loday, J.-L.: Cyclic homology. 2nd ed., Springer-Verlag, Berlin, 1998.

Parthasarathy, R.: Dirac Operators and the discrete series. Ann. Math. 96, 1--30 (1972).

Rosenberg, J.: K-theory of group C^*-algebras, foliation C^*-algebras, and crossed products. Index theory of elliptic operators, foliations, and operator algebras 251--301, Contemp. Math., 70, 1988.

Valette, A.: Introduction to the Baum--Connes conjecture. Basel: Birkhauser, 2002.