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Surveys in Noncommutative Geometry Surveys in Noncommutative Geometry Clay Mathematics Proceedings Volume 6 Surveys in Noncommutative Geometry Proceedings from the Clay Mathematics Institute Instructional Symposium, held in conjunction with the AMS-IMS-SIAM Joint Summer Research Conference on Noncommutative Geometry June 18–29, 2000 Mount Holyoke College South Hadley, MA Nigel Higson John Roe Editors American Mathematical Society Clay Mathematics Institute The 2000 AMS-IMS-SIAM Joint Summer Research Conference on Noncommutative Geometry was held at Mount Holyoke College, South Hadley, MA, June 18–29, 2000, with support from the National Science Foundation, grant DMS 9973450. 2000 Mathematics Subject Classification. Primary 46L80, 46L89, 58B34. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference and Clay Mathematics Institute Instruc- tional Symposium on Noncommutative Geometry (2000 : Mount Holyoke College) Surveys in noncommutative geometry : proceedings from the Clay Mathematics Institute Instructional Symposium, held in conjunction with the AMS-IMS-SIAM Joint Summer Research Conference on Noncommutative Geometry, June 18–29, 2000, Mount Holyoke College, South Hadley, MA / Nigel Higson, John Roe, editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 6) Includes bibliographical references. ISBN-13: 978-0-8218-3846-4 (alk. paper) ISBN-10: 0-8218-3846-6 (alk. paper) 1. Geometry, Algebraic—Congresses. 2. Noncommutative algebra—Congresses. 3. Noncom- mutative function spaces—Congresses. I. Higson, Nigel, 1963– II. Roe, John, 1959– III. Title. QA564.A525 2000 516.35—dc22 2006049865 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribu- tion, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam- bridge, MA 02138, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2006 by the Clay Mathematics Institute. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. The Clay Mathematics Institute retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 111009080706 Contents Preface vii A Minicourse on Applications of Non-Commutative Geometry to Topology 1 Jonathan Rosenberg On Novikov-Type Conjectures 43 Stanley S. Chang and Shmuel Weinberger The Residue Index Theorem of Connes and Moscovici 71 Nigel Higson The Riemann Hypothesis: Arithmetic and Geometry 127 Jeffrey C. Lagarias Noncommutative Geometry and Number Theory 143 Paula Tretkoff v Preface In June 2000, the Clay Mathematics Institute organized an Instructional Sym- posium on Noncommutative Geometry in conjunction with the AMS-IMS-SIAM Joint Summer Research Conference. These events were held at Mount Holyoke College in Massachusetts from June 18 to 29, 2000. The Instructional Symposium consisted of several series of expository lectures which were intended to introduce key topics in noncommutative geometry to mathematicians unfamiliar with the subject. Those expository lectures have been edited and are reproduced in this volume. Specifically, the lectures of Rosenberg and Weinberger discussed various applications of noncommutative geometry to problems in “ordinary” topology, and the lectures of Lagarias and Tretkoff discussed the Riemann hypothesis and the possible application of the methods of noncommutative geometry in number the- ory. This book also contains an account by Higson of the “residue index theorem” of Connes and Moscovici. At the conference, Higson and Roe gave an overview of noncommutative ge- ometry which was intended to provide a point of entry to the later, more advanced lectures. This preface represents a highly compressed version of that introduction. We hope that it will provide sufficient orientation to allow mathematicians new to the subject to read the later, more technical essays in this volume. Noncommutative geometry—as we shall use the term—is to an unusual extent the creation of a single mathematician, Alain Connes; his book [12] is the central text of the subject. The present volume could perhaps be regarded as a sort of extended introduction to that dense and fascinating book. With that in mind, let us review some of the central notions in Connes’ work. To see what is meant by the phrase “noncommutative geometry,” consider ordi- nary geometry: for example, the geometry of a closed surface S in three-dimensional Euclidean space R3. Following Descartes, we study the geometry of S using coor- dinates. These are just the three functions x, y, z on R3, and their restrictions to S generate, in an appropriate sense, the algebra C(S) of all continuous functions on S. All the geometry of S is encoded in this algebra C(S); in fact, the points of S can be recovered simply as the algebra homomorphisms from C(S)toC.Inthe language of physics, one might say that the transition from S to C(S) is a transition from a “particle picture” to a “field picture” of the same physical situation. This idea that features of geometric spaces are reflected within the algebras of their coordinate functions is familiar to everyone. Further examples abound: • If X is a compact Hausdorff space, then (complex) vector bundles over X correspond to finitely generated and projective modules over the ring of continuous, complex-valued functions on X. vii viii PREFACE • If M is a smooth manifold, then vector fields on M correspond to deriva- tions of the algebra of smooth, real-valued functions on M. • If Σ is a measure space, then ergodic transformations of Σ correspond to automorphisms of the algebra of essentially bounded measurable functions on Σ which fix no non-constant function. • If G is a compact and Hausdorff topological group, then the algebra of complex-valued representative functions on G (those whose left transla- tions by elements of G span a finite-dimensional space) determines, when supplemented by a comultiplication operation, not only the group G as a space, but its group operation as well. We could cite many others, and so of course can the reader. The algebra C(S) of continuous functions on the space S is commutative. The basic idea of noncommutative geometry is to view noncommutative algebras as coordinate rings of “noncommutative spaces.” Such noncommutative spaces must necessarily be “delocalized,” in the sense that there are not enough “points” (ho- momorphisms to C) to determine the coordinates. This has a natural connection with the Heisenberg uncertainty principle of quantum physics. Indeed, a major strength of noncommutative geometry is that compelling examples of noncommu- tative spaces arise in a variety of physical and geometric contexts. Connes’ work is particularly concerned with aspects of the space–algebra cor- respondence which, on the algebra side, involve Hilbert space methods, especially the spectral theory of operators on Hilbert space. Motivation for this comes from a number of sources. As we hinted above, one is physics: quantum mechanics asserts that physical observables such as position and momentum should be modeled by elements of a noncommutative algebra of Hilbert space operators. Another is a long series of results, dating back to Hermann Weyl’s asymptotic formula, connecting geometry to the spectral theory of the Laplace operator and other operators. A third comes from the application of operator algebra K-theory to formulations of the Atiyah-Singer index theorem and surgery theory. Here the notion of positivity which is characteristic of operator algebras plays a key role. A fourth motivation, actively discussed at the Mount Holyoke conference, involves the reformulation of the Riemann hypothesis as a Hilbert-space-theoretic positivity statement. Connes introduces several successively more refined kinds of geometric struc- ture in noncommutative geometry: measure theory, topology (including algebraic topology), differential topology (manifold theory), and differential (Riemannian) geometry. To begin with the coarsest of these, noncommutative measure theory means the theory of von Neumann algebras.LetH be a Hilbert space and let B(H) be the set of bounded linear operators on it. A von Neumann algebra is an involutive subalgebra M of B(H) which is closed in the weak operator topology on B(H). The commutative example to keep in mind is H = L2(X, µ), where (X, µ)is a measure space,
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