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Twenty-Five Years with Nicolas Bourbaki, 1949-1973

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Twenty-Five Years with Nicolas Bourbaki, 1949-1973 Twenty-Five Years with Nicolas Bourbaki, 1949–1973 Armand Borel he choice of dates is dictated by per- To set the stage, I shall briefly touch upon the sonal circumstances: they roughly bound first fifteen years of Bourbaki. They are fairly well the period in which I had inside knowl- documented2, and I can be brief. edge of the work of Bourbaki, first In the early thirties the situation of mathemat- through informal contacts with several ics in France at the university and research levels, T the only ones of concern here, was highly unsat- members, then as a member for twenty years, until the mandatory retirement at fifty. isfactory. World War I had essentially wiped out one Being based largely on personal recollections, generation. The upcoming young mathematicians my account is frankly subjective. Of course, I had to rely for guidance on the previous one, in- checked my memories against the available docu- cluding the main and illustrious protagonists of the mentation, but the latter is limited in some ways: so-called 1900 school, with strong emphasis on not much of the discussions about orientation and analysis. Little information was available about current developments abroad, in particular about general goals has been recorded.1 Another mem- the flourishing German school (Göttingen, Ham- ber might present a different picture. burg, Berlin), as some young French mathemati- cians (J. Herbrand, C. Chevalley, A. Weil, J. Leray) Armand Borel is professor emeritus at the School of Math- were discovering during visits to those centers.3 ematics, Institute for Advanced Study, Princeton, NJ. In 1934 A. Weil and H. Cartan were Maîtres de This article is an outgrowth of a lecture given at the Uni- Conférences (the equivalent of assistant professors) versity of Bochum, Germany, October 1995, in a Collo- quium in honor of R. Remmert, and at the International at the University of Strasbourg. One main duty Center for Theoretical Physics, Trieste, Italy, September was, of course, the teaching of differential and in- 1996. With permission of the author, it is being published tegral calculus. The standard text was the Traité concurrently in the Notices and in the Mitteilungen der d’Analyse of E. Goursat, which they found want- Deutsche Mathematiker-Vereinigung. ing in many ways. Cartan was frequently bugging 1The Archives of Bourbaki at the École Normale Weil with questions on how to present this mate- Supérieure, Paris, contain reports, surveys, successive rial, so that at some point, to get it over with once drafts or counterdrafts of the chapters, remarks on those and for all, Weil suggested they write themselves resulting from discussions, and proceedings of the Con- gresses, called “Tribus”. Those provide mainly a record 2 of plans, decisions, commitments for future drafts, as well See [2, 3, 6, 7, 8, 14]. as jokes, sometimes poems. 3For this, see pp. 134–136 of [8]. MARCH 1998 NOTICES OF THE AMS 373 a new Traité d’Analyse. This suggestion was spread ous text in the same book or to an earlier book (in around, and soon a group of about ten math- the given ordering). The title “Éléments de Math- ematicians began to meet regularly to plan this trea- ématique” was chosen in 1938. It is worth noting tise. It was soon decided that the work would be that they chose “Mathématique” rather than the collective, without any acknowledgment of indi- much more usual “Mathématiques”. The absence vidual contributions. In summer 1935 the pen of the “s” was of course quite intentional, one way name Nicolas Bourbaki was chosen.4 for Bourbaki to signal its belief in the unity of The membership varied over the years; some mathematics. people in the first group dropped out quickly, oth- The first volumes to appear were the Fascicle of ers were added, and later there was a regular Results on Set Theory (1939) and then, in the for- process of additions and retirements. I do not in- ties, Topology and three volumes of Algebra. tend to give a detailed account. At this point let At that time, as a student and later assistant at me simply mention that the true “founding fa- the E.T.H. (Swiss Federal Institute of Technology) thers”, those who shaped Bourbaki and gave it in Zurich, I read them and learned from them, es- much of their time and thoughts until they re- pecially from Multilinear Algebra, for which there tired, are: was no equivalent anywhere, but with some reser- Henri Cartan vations. I was rather put off by the very dry style, Claude Chevalley without any concession to the reader, the appar- Jean Delsarte ent striving for the utmost generality, the inflexi- Jean Dieudonné ble system of internal references and the total ab- André Weil sence of outside ones (except in Historical Notes). For many, this style of exposition represented an born respectively in 1904, 1909, 1903, 1906, 1906— alarming tendency in mathematics, towards gen- all former students at the École Normale Supérieure erality for its own sake, away from specific prob- in Paris.5 lems. Among those critics was H. Weyl, whose A first question to settle was how to handle opinion I knew indirectly through his old friend and references to background material. Most existing former colleague M. Plancherel, who concurred, books were found unsatisfactory. Even B. v.d. Waer- at a time I was the latter’s assistant. den’s Moderne Algebra, which had made a deep im- In fall 1949 I went to Paris, having received a fel- pression, did not seem well suited to their needs lowship at the C.N.R.S. (Centre National de la (besides being in German). Moreover, they wanted Recherche Scientifique), benefiting from an ex- to adopt a more precise, rigorous style of exposi- change convention just concluded between the tion than had been traditionally used in France, so C.N.R.S. and the E.T.H. I quickly got acquainted with they decided to start from scratch and, after many some of the senior members (H. Cartan, discussions, divided this basic material into six J. Dieudonné, L. Schwartz) and, more usefully for “books”, each consisting possibly of several vol- informal contacts, with some of the younger ones, umes, namely: notably Roger Godement, Pierre Samuel, Jacques I Set Theory Dixmier, and, most importantly, Jean-Pierre Serre, II Algebra the beginning of intense mathematical discussions III Topology and a close friendship. Of course, I also attended IV Functions of One Real Variable the Bourbaki Seminar, which met three times a V Topological Vector Spaces year, offering each time six lectures on recent de- VI Integration velopments. These books were to be linearly ordered: refer- Those first encounters quickly changed my vi- ences at a given spot could only be to the previ- sion of Bourbaki. All these people—the elder ones, of course, but also the younger ones—were 4See [3] for the origin of the name. very broad in their outlook. They knew so much and knew it so well. They shared an efficient way 5They all contributed in an essential way. For Cartan, Chevalley, Dieudonné, and Weil I could witness it at first- to digest mathematics, to go to the essential hand, but not for Delsarte, who was not really active any- points, and reformulate the math in a more com- more when I came on board. But his importance has been prehensive and conceptual way. Even when dis- repeatedly stressed to me by Weil in conversations. See also cussing a topic more familiar to me than to them, [14] and comments by Cartan, Dieudonné, Schwartz in [3, their sharp questions often gave me the impres- pp. 81–83]. In particular, he played an essential role in sion I had not really thought it through. That transforming into a coherent group, and maintaining it methodology was also apparent in some of the lec- so, a collection of strong, some quite temperamental, in- tures at the Bourbaki seminar, such as Weil’s on dividuals. Besides, obviously, Book IV, Functions of one real theta functions (Exp. 16, 1949) or Schwartz’s on variable, owes much to him. Some other early members, notably Szolem Mandelbrojt and René de Possel, also con- Kodaira’s big Annals paper on harmonic integrals tributed substantially to the work of the group in its ini- (Exp. 26, 1950). Of course, special problems were tial stages. not forgotten—in fact, were the bread and but- 374 NOTICES OF THE AMS VOLUME 45, NUMBER 3 ter of most discussions. The writing of the books The underlying thought was apparently that re- was obviously a different matter. ally new, groundbreaking ideas were more likely Later I was invited to attend (part of) a Bourbaki to arise from confrontation than from an orderly Congress and was totally bewildered. Those meet- discussion. When they did emerge, Bourbaki mem- ings (as a rule three per year: two of one week, one bers would say, “the spirit has blown” (“l’esprit a of about two) were private affairs, devoted to the soufflé”), and it is indeed a fact that it blew much books. A usual session would discuss a draft of more often after a “spirited” (or should I say some chapter or maybe a preliminary report on a stormy) discussion than after a quiet one. topic under consideration for inclusion, then or Other rules of operation also seemed to mini- later. It was read aloud line by line by a member, mize the possibility of a publication in a finite and anyone could at any time interrupt, comment, time: ask questions, or criticize. More often than not, this Only one draft was read at a given time, and “discussion” turned into a chaotic shouting match.
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