A Panorama of Pure Mathematics As Seen by N . Bourbaki Jean Dieudonne Membre de I’Institut Translated by I. G. Macdonald Department of Pure Mathematics Queen Mary College University of London London, England 1982 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Pans San Diego San Francisco S%o Paul0 Sydney Tokyo Toronto COPYRIGH@T 1 982, BY ACADEMIPCRE SS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWl 7DX Library of Congress Cataloging in Publication Data Dieudonne', Jean Alexandre, Date. A panorama of pure mathematics (as seen by N. Bourbaki) (Pure and applied mathematics) Translation of: Panorama des math6matiques pures. Bibliography: p. Includes index. . 1. Mathematics--1961- I. Title. 11. Series. QA37.2 ~5313 51 0 80-2330 ISBN 0-12-215560-2 AACR2 This is the English language translation of Panorama des mathematiques pures. Le choix bourbachique 0 Bordas, I9II PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85 9 8 7 6 5 4 3 2 1 Introduction This book is addressed to readers whose mathematical knowledge extends at least as far as the first two years of a university honors course. Its aim is to provide an extremely sketchy survey of a rather large area of modern mathe- matics, and a guide to the literature for those who wish to embark on a more serious study of any of the subjects surveyed. By “Bourbaki mathematics” I mean, with very few exceptions, the set of topics covered in the exposes of the Seminaire Bourbaki. Since the beginning of their collective work, the collaborators of N. Bourbaki have taken a definite view of mathematics, inherited from the tradition of H. Poincark and E. Cartan in France, and Dedekind and Hilbert in Germany. The “Elements de Mathematique” have been written in order to provide solid foundations and convenient access to this aspect of mathematics, in a form sufficiently general for use in as many contexts as possible. From 1948 onward, the Bourbaki group has organized a seminar, con- sisting in principle of 18 lectures each year. The purpose of these lectures is to describe those recent results that appear to the organizers to be of most interest and importance. These lectures, almost all of which have been pub- lished, now exceed 500 in number, and collectively constitute a veritable encyclopedia of these mathematical theories. * * * No publication under the name of N. Bourbaki has ever described how the topics for exposition in the seminar have been chosen. One can therefore only attempt to discern common features by examining these choices from outside, and their relation to the totality of the mathematical literature of our age. I wish to make it clear that the conclusions I have drawn from this examina- tion are my own, and do not claim in any way to represent the opinions of the collaborators of N. Bourbaki. The history of mathematics shows that a theory almost always originates in efforts to solve a specific problem (for example, the duplication of the cube 2 INTRODUCTION in Greek mathematics). It may happen that these efforts are fruitless, and we have our first category of problems: (I) Stillborn problems (examples: the determination of Fermat primes, or the irrationality of Euler’s constant). A second possibility is that the problem is solved but does not lead to progress on any other problem. This gives a second class: (11) Problems without issue (this class includes many problems arising from “combinatorics”). A more favorable situation is one in which an examination of the techniques used to solve the original problem enables one to apply them (perhaps by making them considerably more complicated) to other similar or more difficult problems, without necessarily feeling that one really understands why they work. We may call these (111) Problems that beget a method (analytic number theory and the theory of finite groups provide many examples). In a few rather rare cases the study of the problem ultimately (and perhaps only after a long time) reveals the existence of unsuspected underlying structures that not only illuminate the original question but also provide powerful general methods for elucidating a host of other problems in other areas; thus we have (IV) Problems that belong to an active and fertile general theory (the theory of Lie groups and algebraic topology are typical examples at the present time). However, as Hilbert emphasized, a mathematical theory cannot flourish without a constant influx of new problems. It has often happened that once the problems that are of the greatest importance for their consequences and their connections with other branches of mathematics have been solved, the theory tends to concentrate more and more on special and isolated questions (possibly very difficult ones). Hence we have yet another category: (V) Theories in decline (at least for the time being: invariant theory, for example, has passed through this phase several times). Finally, if a happy choice of axioms, motivated by specific problems, has led to the development of techniques of great efficacy in many areas of mathematics, it may happen that attempts are made with no apparent motive to modify these axioms somewhat arbitrarily, in the hope of repeating the success of the original theory. This hope is usually in vain, and thus we have, in the phrase of P6lya and Szegot t G. Polya and G. Szego, “Problems and Theorems in Analysis,” Springer-Verlag. Berlin and New York, 1972. INTRODUCTION 3 (VI) Theories in a state of dilution (following the example of these authors, we shall cite no instances of this). In terms of this classification, it appears to me that the majority of the topics expounded in the Seminaire Bourbaki belong to category (IV) and (to a lesser extent) category (111). This is, I believe, as objective an opinion as I can form, and I shall abstain from further comment. * * * Since the number and variety of the lectures in the Seminaire make them difficult to use, I have grouped them into sections under a fairly small number of headings, each of which contains a closely related group of subjects. One of the characteristics of Bourbaki mathematics is its extraordinary unity: there is hardly any idea in one theory that does not have notable repercussions in several others, and it would therefore be absurd, and contrary to the very spirit of our science, to attempt to compartmentalize it with rigid boundaries, in the manner of the traditional division into algebra, analysis, geometry, etc. now completely obsolete. The reader should therefore attach no im- portance to this grouping, which is purely a matter of convenience; its aim is to provide a clear overall view, halfway between the chaos of the chronological order of the lectures, and fragmentation into a dust-cloud of minitheories.A t the beginning of each section I have inserted an organization “ chart” designed to illustrate graphically its connections with the others, with arrows to indicate the direction of influence. Each section contains, to the extent that it is feasible, a rapid didactic exposition of the main questions to be considered. With a few exceptions, only those are mentioned that have been covered in the Seminaire Bourbaki; the order followed is not in general the historical order, and the infrequent historical indications make no pretence of being systematic. At the end of each section will be found a list of the mathematicians who have made significant contributions to the theories described, and a brief mention of the connections (where they exist) between these theories and the natural sciences. Each section or heading is designated by a boldface capital letter followed by a Roman numeral. This designation refers to the place occupied by the heading in the Table ofsubjects (p. 5), the capital letter indicating the level at which the heading is placed. These levels range from top to bottom, roughly speaking in decreasing order of what might be called their “Bourbaki density,” that is to say (without pretension to numerical accuracy, which would be absurd), the proportion of the topics covered by the Seminaire Bourbaki to the total mathematical literature relating to the heading con- cerned. 4 INTRODUCTION * * * The references have been organized in such a way as to serve as a guideline to readers who wish to learn more. References to the Skminaire Bourbaki are indicated by the letter B followed by the number of the exposk. They are augmented by references to : (i) the Seminaires H. Cartan, denoted by the letter C followed by the year; (ii) the expository lectures organized by the American Mathematical Society and published in its Bulletin; these are indicated by the letters BAMS followed by the volume number of the Bulletin and the name of the lecturer; (iii) the Symposia organized by the American Mathematical Society, denoted by the letters SAMS followed by a roman numeral and (sometimes) the author’s name; (iv) the lectures given at the recent International Congresses of Mathe- maticians at Stockholm (1962), Nice (1970), and Vancouver (1974); these are indicated by the name of one of these cities and the lecturer’s name (in the case of the Nice Congress, the figure I indicates a one-hour lecture, and an indication of the section of the Congress a half-hour lecture); (v) the “Lecture Notes in Mathematics” published by Springer-Verlag, denoted by the letters LN followed by a number (and by an author’s name, in the case of a colloquium or symposium); (vi) various articles and books, denoted by the letter or the number in brackets under which they are listed in the bibliography. No reference is given for mathematical terms currently used in the first two years of a university honors course. For others, either a brief explicit definition is given, or a reference to a textbook in the bibliography. The headings at level D in the table of subjects are those of Bourbaki density zero. They refer to theories that have in part been fixed for a con- siderable time, and constitute, in the etymological sense of the word, the classicc’ ?art of mathematics, which serves as a basis for the rest of the edifice. The reader will find these theories expounded in the volumes of the “Eltments de Mathkmatique that have already been published. Research still continues ” in these various theories, about which I shall say nothing except to remark on the curious historical phenomenon of a science divided into two parts that in practice ignore each other, without apparently causing the least impediment to their respective developments. I wish to thank readers whose comments enabled me to correct certain errors and omissions in the second (French) edition. At the end of each section I have appended a list of references given in the text, together with some additional ones for the reader’s benefit. TABLE OF SUBJECTS Levels A A1 A I1 A I11 A IV AV Algebraic and Differential Ordinary differential Ergodic Partial differential differential topology geometry equations theory equations A VI A VII A VIII A IX AX Noncommutative Automorphic and Analytic Algebraic Theory of harmonic analysis modular forms geometry geometry numbers B BI B I1 B I11 B IV BV B VI B VII Homological Lie “Abstract” Commutative Von Neumann Mathematical Probability algebra groups groups harmonic analysis algebras logic theory C CI c I1 c I11 Categories Commutative Spectral theory and sheaves algebra of operators D DI D I1 D 111 D IV DV D VI Set General General Classical Topological Integration theory algebra topology analysis vector spaces A1 Differential Partial differential Automorphic and Analytic Algebraic geometry modular forms geometry geometry Lie groups Categories Commutative and sheaves algebra General topology Algebraic and differential topology It may already be predicted without great likelihood of error that the 20th century will come to be known in the history of mathematics as the century of topology, and more precisely of what used to be called “combinatorial” topology, and which has developed in recent times into algebraic topology and dzfleerential topology. These disciplines were created in the last years of the 19th century by H. PoincarC, in order to provide a firm mathematical basis for the intuitive ideas of Riemann. At first they developed rather slowly, and it was not until the 1930s that they took wing. Since then they have multiplied, diversified, and refined their methods, and have progressively infiltrated all other parts of mathematics; and there is as yet no indication of any slowing down of this conquering march. 1. Techniques The initial problem of algebraic topology, roughly speaking, is to “classify” topological spaces: two spaces are to be put in the same “class” if they are homeomorphic. The general idea is to attach to each topological space “invariants,” which may be numbers, or objects endowed with algebraic structures (such as groups, rings, modules, etc.) in such a way that homeo- morphic spaces have the same “invariants” (up to isomorphism, in the case of algebraic structures). The ideal would be to have enough “invariants” to be able to characterize a “class” of homeomorphic spaces, but this ambition has been realized in only a very small number of cases (for recent progress, see Vancouver (Sullivan) and T. Price, Math. Chronicle 7 (1978)). This original problem may be reformulated as the study of continuous mappings that are bijective and bicontinuous. In this form it is merely one of a whole series of problems of existence of continuous mappings subjected to other conditions, such as to be injective, or surjective, or to be sections or retractions of given continuous mappings, or extensions of given continuous mappings, etc. [171 bis]. All these problems are amenable to the methods of algebraic topology. 7 8 A I ALGEBRAIC AND DIFFERENTIAL TOPOLOGY The idea of homeomorphism is related to, but distinct from, the more intuitive notion of “deformation.” In order to formulate mathematically the idea that a subspace Y, of a topological space X can be “deformed” into another subspace Y,, one is led to the following definition: denoting by I the interval [O, I] in R, there exists a continuous mapping (y, t)H F (y, t)o f Y, x I into X such that (i) F(y, 0) = y for all y E Y,,( ii) for each t E I, the mapping y w F(y, t) is a homeomorphism of Y onto a subspace of X, and (iii) when t = 1, this subspace is Y, . The mapping F is said to be an isotopy of Y onto Y, . The notion of isotopy is thus a strengthening of the notion of homeomorphism. The study of isotopy is difficult and has only recently led to substantial results (B 157,245,373; [86]). Homotopy (C 1949,1954; [50], [78], [170]). The notion that has be- come the most important in topology is a weakening of the notion of isotopy. Two continuous mappings g, h of a space X into a space Y are said to be homotopic if there exists a continuous mapping F : X x I --t Y such that F(x, 0) = g(x) and F(x, 1) = h(x), but with no conditions imposed on the mapping x I-+ F(x, t) for t # 0, 1. F is called a hornotopy from g to h. The property of being homotopic is an equivalence relation on the set %(X, Y) of all continuous mappings of X into Y, and the set [X, Y] of classes of homotopic mappings is evidently an “invariant” of the two spaces X, Y. It is functorial (C I) in X and Y: if a:X , + X (resp. fl: Y + YJ is a continuous mapping, and if g, h E %(X, Y) are homotopic, then so also are g 0 a and h 0 ct (resp. /3 0 g and P 0 h); whence we have a mapping a* : [X, Y] -+ [X,, Y] (resp. P* [X,y 1 [X, Yll). + The notion of homotopy leads to a “classification of topological spaces ” that is coarser than classification by homeomorphism, but is much easier to handle. A continuous mappingf : X + Y is called a homotopy equivalence if there exists a continuous mapping g : Y --t X such that g 0 f : X + X is homotopic to the identity mapping of X and f 0 g : Y + Y homotopic to the identity mapping of Y. If there exists a homotopy equivalencef : X + Y, the spaces X and Y are said to have the same homotopy type. Most of the “invari- ants” of algebraic topology are invariants of homotopy type (and not merely invariants under homeomorphisms). For example, R” (or more generally any topological vector space over R) and a space consisting of a single point have the same homotopy type (spaces having the homotopy type of a single point are said to be contractible). Besides the general notion of homotopy, there are more restrictive notions, such as the simple homotopy equivalence of J. H. C. Whitehead for spaces endowed with a “cellular” subdivision (such spaces are called CW-complexes or cell-complexes [170] ; they are generalizations of polyhedra (B 392; LN 48; BAMS 72 (Milnor)). Another variant is to consider homotopies (x, t)H F(x, t)t hat are independent oft in a given subspace A of X; this leads