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Lecture Notes in Mathematics Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesetlschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstn 2146/3140-543210 VORWORT Die mathematische Arbeitstagung in Bonn hatte seit ihrem Anfang im Jahre ]957 bis hin zum 25. Treffen 1984 stets ihren eigenen unver- wechselbaren Charakter. Das Programm wird nicht im voraus festme- legt, sondern Vortragende und Themen ergeben sich spontan aus einer Programmdiskussion aller Teilnehmer unter Leitung von F. Hirzebruch. Dies erm~glicht, auf neueste Entwicklungen in einzelnen mathemati- schen Gebieten unmittelbar einzugehen,und sie auf der Tagung vorzu- stellen. Dar~ber hinaus ist die Arbeitstagung nicht ausschlieSlich einem speziellen mathematischen Thema gewidmet, sondern es finden sich vonder Zahlentheorie [ber die Topologie und Geometrie bis hin zur Analysis ~ber die Jahre hinweg Beitr~ge aus fast allen Gebieten der Mathematik. Gerade diesem informellen und arbeitsmiBigem Charak- ter der Tagung h~tte die Ver~ffentlichung von Ergebnisberichten widersprochen. Da jedoch immer wieder stark danach gefragt wurde, hat man, erst seit dem Jahre 1974, einen Kompromi8 gefunden. Die Sprecher fassen ihre Vortr~ge - meist nur handschriftlich - kurz zusam~,en, und diese werden gesammelt zum Abschlu8 eines jeden Tref- fens an die Teilnehmer verteilt. Anl~81ich der 25. mathematischen Arbeitstagung im Jahre 1984 wurde und wird hiervon in mehrfacher Hinsicht abgewichen: Einige Mathematiker wurden im voraus gebeten, in Uberblicksvortr~qen Ent- wicklungen und Probleme in ihren eigenen Arbeitsgebieten darzustel- len. Diese Beitr~ge finden sich im ersten Teil des vorliegenden Bandes. Den zweiten Teil bilden Ausarbeitungen der meisten der ad- hoc Vortr~ge, deren Sprecher erst w~hrend der Arbeitstagung aus- gewihlt wurden. Vielleicht spiegelt sich trotz dieser ~nderungen im Procedere ein wenig der Charakter der Arbeitstagung wider. Im Anhang sind die Programme der Arbeitstagungen 1957 - 1984 wieder- gegeben. Die M~he der Schreibarbeit fast aller Beitr~ge haben mit viel Geduld und Sorgfalt Frau K. Deutler und Frau C. Pearce auf sich genommen, denen wir daf~r sehr dankbar sind. Bonn, November 1984 Die Herausgeber 25. Mathematische Arbeit~ta~un~ 1984 (15. - 22. Ju~i) J. Tits: Groups and group functors attached to Kac-Moody data M. Atiyah: The eigenvalues of the Dirac operator A. Connes: K-theory, cyclic cohomology and operator algebras G. Segal: Loop groups G. Harder : Special values of Hecke L-functions and abelian integrals H. Wente: A counterexample in 3-space to a conjecture of H. Hopf G. Faltings: Compactification of Ag/Z C.T.C. Wall: Geometric structures and algebraic surfaces J. Harris: Recent work on Hodge structures Y.T. Siu: Some recent results in complex manifold theory related to vanishing theorems for the semipositive case W. Schmid: Recent progress in representation theory W. Ballmann: Manifolds of non-positive curvature B. Mazur - Conjectures of Beilinson on L-functions and K-theory Ch. Soul@: H.-O. Peitgen: Morphology of Julia sets S.S. Chern: Some applications of the method of moving frames S. Lang: Vojta's conjecture on heights and Green's function S. Donaldson: 4-manifolds with indefinite intersection form D. Zagier: Modular points, modular curves, modular surfaces and modular forms G. van der Geer: Schottky's problem R. Bryant: G 2 and Spin(7)-holonomy S. Wolpert: Homology of Teichm~ller spaces J.-P. Serre: l-adic representations M.F. Atiyah: On Manin's manuscript "New dimensions in geometry" Inhaltsverzeichnis ~BERBLI CKS VORTR~GE Connes, A. - Introduction to non commutative differen- 3-16 tial geometry Harder, G., - Special values of Hecke L-functions and 17-49 Schappacher, N. abelian integrals Harris, J. - An introduction to infinitesimal varia- 51-58 tions of Hodge structures Manin, Y. - New dimensions in geometry 59-101 Atiyah, M. - Commentary on the article of Manin 103-109 Peitgen, H.O., - The Mandelbrot set in a model for phase 111-134 Richter, P.H. transitions Schmid, W. - Recent developments in representation 135-153 theory Segal, G.B. - Loop groups 155-168 Siu, Y.-T. - Some recent results in complex manifold 169-192 theory related to vanishing theorems for the semipositive case Tits, J. - Groups and group functors attached to 193-223 Kac-Moody data Zagier, D. - Modular points, modular curves, modular 225-248 surfaces and modular forms Ad-hoc VORTR~GE Atiyah, M. - Eigenvalues of the Dirac operator 251-260 Ballmann, W. - Manifolds of non positive curvature 261-268 Bryant, R. - Metrics with holonomy G 2 or Spin(7) 269-277 Chern, S.S., - On Riemannian metrics adapted to three- 279-308 Hamilton, R.S. dimensional contact manifolds (Appendix by Alan Weinstein) Donaldson, S.K. - 4-manifolds with indefinite intersection 309-320 form Faltings, G. - Arithmetische Kompaktifizierung des 321 - 383 Modulraums der abelschen Variet~ten van der Geer, G. - The Schottky problem 385-406 Lang, S. - Vojta's conjecture 407-419 Wente, H.C. - A counterexample in 3-space to a 421-429 conjecture of H. Hopf Wolpert, S.A. The topology and geometry of the moduli 431-451 space of Riemann surfaces ANHANG Programme der 25 Arbeitstagungen von 453-478 1957-1984 Teilnehmer der Arbeitstagung 1984 479-481 INTRODUCTION TO NON COMMUTATIVE DIFFERENTIAL GEOMETRY A. Connes Institut des Hautes Etudes Scientifiques 35, Route des Chartres F-91440 Bures-Sur-Yvette France This is the introduction to a series of papers in which we shall extend the calculus of differential forms and the de Rham homology of currents beyond their customary framework of manifolds, in order to deal with spaces of a more elaborate nature, such as, a) the space of leaves of a foliation, b) the dual space of a finitely generated non-abelian discrete group (or Lie group), c) the orbit space of the action of a discrete group (or Lie group) on a manifold. What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence. These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b). When the space V is an ordinary manifold, the associated algebra is commutative. It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product. A smooth manifold V can be considered from different points of view such as ~) Measure theory (i.e. V appears as a measure space with a fixed measure class), 8) Topolog~ (i.e. V appears as a locally compact space), T) Differential geometry (i.e. V appears as a smooth manifold). Each of these structures on V is fully specified by the corresponding algebra of functions, namely: ~) The commutative von Neumann algebra L~(V] of classes of essentially bounded measurable functions on V, 8) The C*- algebra C0(V) of continuous functions on V which vanish at infinity, y) The algebra C~(V) of smooth functions with compact support. C It has long been known to operator algebraists that measure theory and topology extend far beyond their usual framework to: A) Th__e theory of weights and yon Neumann algebras. B) C*- algebras, K theory an d index theory. Let us briefly discuss these two fields, A) The theory of weights and yon Neumann algebras. To an ordinary measure space (X,~) correspond the von Neumann algebra L (X,~) and the weight ~ : ~(f) : SX fd~ Vf 6 L~(x,~) + Any pair (M,~) of a commutative yon Neumann algebra M and weight is obtained in this way from a measure space (X,~). Thus the place of ordinary measure theory in the theory of weights on von Neumann algebras is similar to that of commutative algebras among arbitrary ones. This is why A) is often called non-commutative measure theory. Non-cO~autative measure theory has many features which are trivial in the commutative case. For instance to each weight ~ on avon Neumann algebra M corresponds canonically a one-parameter group ot~C Aut M of automorphisms of M, its modular automorphism group. When M is commutative, one has a ~(x)t = x, Vx 6 M, and for any weight ~ on M.
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