Nicolas Bourbaki

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Book cover, 1970 edition

Nicolas Bourbaki is the collective pseudonym of a group ( collective of authors ) of predominantly French mathematicians who have been working on a multi-volume textbook on mathematics in French since 1934 - the Éléments de mathématique  - and who held seminars on their joint book project several times a year at various locations in France . Allegedly Bourbaki worked at the University of Nancago (pseudonym: moved together from Nancy and Chicago , the universities where some of the leading Bourbakists were at that time, Jean Dieudonné called his villa in Nice the Villa Nancago ). The publications follow the tradition of the axiomatic foundation of mathematics.

Task

Bourbaki did not see it as his job to create new mathematical knowledge. Rather, existing mathematical findings should be reprocessed and placed in a stringent context. The axiomatic representation of set theory , based on the school of David Hilbert , served as the basis, and there was no doubt of its outstanding performance at the time of the founding of Bourbaki.

The structure and notation of the work of the Bourbaki group are extraordinarily rigid. The argument basically goes from the general to the particular. Everything that is said is based on what has been said before. The reference system in the first six books is absolutely linear: each reference refers to an earlier Bourbaki text. References to other works are considered superfluous.

The original goal was to only deal with topics that were necessary for a systematic development of the fundamentals of mathematics. Eliminated were then the lattice theory , the theory of numbers and the total applied mathematics . The geometry is regarded as done with the treatment of the topological vector spaces .

In a 1997 interview, Bourbaki member Pierre Cartier responded to later criticism of educational deficiencies in the presentation :

“The misunderstanding was that a lot of people thought that it should be taught as it was presented in the books. One can think of Bourbaki's first books as an encyclopedia of mathematics containing all the information necessary. That's a good description. If you look at it as a textbook, it's a disaster ”.

Working method

The basic rules of the group included the anonymous publication under the common pseudonym, the merciless discussion of every editorial proposal and the retirement at the age of fifty. The composition of the group and its working methods remained shrouded in mystery for a long time; It was only when they were old that the founding members began to speak publicly about Bourbaki. It is now known that Jean Dieudonné had the largest share in the first draft and in the final editing of the published volumes.

At their meetings, the group often discussed drafts of individual textbook chapters very vigorously, decided on countless changes, and then handed the manuscripts over to new authors for further processing. At the next meeting, however, no one was bound by the resolutions previously made; it was criticized again and a new revision was decided. Each chapter typically underwent ten revisions that stretched out over eight to twelve years. Every member had the right of veto.

Members met three times a year, often at country hotels and resorts, with recreational activities and accommodations being funded by increasing income from book sales. The news magazine La Tribu was used for communication within the group.

Bourbaki meeting in Dieulefit 1938. From left: Simone Weil , Charles Pisot , André Weil , Jean Dieudonné (seated), Claude Chabauty , Charles Ehresmann , Jean Delsarte .

Results

In 1939 the first of a total of 40 volumes was published, which in turn are summarized in six books:

This corresponded to the general plan of 1939 and even 1950 for the first part (there were a part IIIb geometric topology (with overlaps , the fundamental group , fiber acknowledge homotopies designed by Serre), and more VII manifolds , VIII Analytical functions and IX Lie groups planned, followed by Part 2 Commutative Algebra, Part 3 Algebraic Topology and Applications, Part 4 Functional Analysis ). Jean Dieudonné originally had much more extensive plans, for example with a draft in 27 books in 1940 and later regular suggestions for further books. The first six books were essentially complete by the 1950s. In 1957, Alexander Grothendieck proposed an expansion of books VII (Homological Algebra), VIII (Elementary Topology) and IX (Manifolds) and also wanted to expand the algebraic fundamentals, but his plans did not get through because the other members feared for so long Getting stuck in the basics. Grothendieck then went his own way and left Bourbaki in the late 1950s.

It also appeared:

Spectral theory (Volume IX) was added in 1983 as the last volume for a long time. In 1998, Chapter 10 of Commutative Algebra was published. In 2012, the eighth chapter (modules and semi-simple rings) of algebra appeared in a completely new edition, and in 2016 a new volume was published by Springer Verlag:

The most successful volumes were those on Lie groups and commutative algebra. Jean Dieudonné published the repeatedly interspersed digressions on the history of mathematics separately as Bourbakis Eléments d'histoire de mathématique (1960, 1969). He also wrote an overview of modern mathematics from “Bourbaki's perspective”: Panorama of pure mathematics as seen by Bourbaki 1982.

history

Charles Denis Bourbaki (around 1870)
André Weil (1956)

The group's six founding members were Henri Cartan , Claude Chevalley , Jean Delsarte , René de Possel , Jean Dieudonné and André Weil . You had recently graduated from the École normal supérieure and are now teaching at French provincial universities. In their teaching activities, they found the available textbooks inadequate and hopelessly out of date, especially in comparison to the simultaneously flourishing German axiomatic school around David Hilbert and Emmy Noether in Göttingen and Emil Artin in Hamburg, where some of the founding members had studied. At the center of mathematical research in France was the traditionally strong analysis there, represented for example by Jacques Hadamard , while algebra and number theory were hardly cultivated. At their occasional meetings, they decided to write their own textbook on analysis and soon came to the conclusion that they had to actually rewrite all of the basics of mathematics. They originally estimated it would take three years to do this. In fact, it took four years before even the first chapter appeared. At one of their first meetings, the group chose the name Bourbaki, after a student joke that has become legend at the École Normale Supérieure and indirectly after General Charles Denis Bourbaki from the Franco-German War of 1870/71.

Soon after the group was founded, Szolem Mandelbrojt was called in , in the 1940s Laurent Schwartz , Samuel Eilenberg , Jean Leray (who was only a brief member) and Jean-Pierre Serre . In later years newcomers were recruited from among the most gifted students of the members: the young mathematicians took part on a trial basis in a meeting of the group where they were expected to actively contribute to the discussion, which was often passionate and seemingly chaotic. In the second half of the 20th century, Bourbaki also included Paul Dubreil (briefly), Jean Coulomb (briefly), Charles Ehresmann , Pierre Cartier , Pierre Samuel , Alexander Grothendieck , Jacques Dixmier , Jean-Louis Koszul , Roger Godement , Armand Borel , Alain Connes , Serge Lang , François Bruhat , John Tate , Pierre Deligne , Adrien Douady , Bernard Teissier , Michel Demazure , Jean-Louis Verdier , Arnaud Beauville , Daniel Bennequin , Gérard Ben Arous , Jean-Christophe Yoccoz , Charles Pisot , Claude Chabauty , Hyman Bass , Michel Raynaud , Joseph Oesterlé , Guy Henniart , Marc Rosso , Olivier Mathieu , Georges Skandalis , Pierre Julg , Patrick Gérard and the Nobel Laureate in Economics Gérard Debreu . Both Grothendieck and Lang left the group prematurely because of differences of opinion. However, many Grothendieck students dominated the group from the 1960s onwards.

At the turn of the millennium it seemed as if there would be no more significant publications. Towards the end of the twentieth century, Cartier said Bourbaki was a dinosaur with its head too far from its tail.

The group's slow decline has a number of reasons that can perhaps be summarized as follows:

  • Bourbaki did its job:
    • With the first six or eight books, the original task of writing down the essential basics of mathematics was done.
    • There is now, crucially under the influence of Bourbaki, a large selection of modern textbooks in axiomatic style.
  • Bourbaki is outdated:
    • The rigid notation makes it extremely difficult to include new mathematical developments. In addition, Bourbaki decided early on not to follow the category-theoretical structure, although important members of Bourbaki such as Serre, Cartan and Grothendieck helped develop these and other innovations (such as sheaf theory ) in the 1950s. But important members like Weil did not want to rewrite the elements. The category theory but played a fundamental role in the further development of algebraic geometry and algebraic topology.
    • The claim to represent all mathematics in a closed system has proven to be impractical for practical reasons. After Dieudonné left, there was no one left who really had an overview of the entire corpus published so far.

In addition, there was a long, unpleasant legal battle with the publisher in the late 1970s.

To this day there is the Association des Collaborateurs de Nicolas Bourbaki (“Society of NB Collaborators ”), which organizes the famous Bourbaki Seminars ( Séminaire Nicolas Bourbaki ) three times a year - international conferences usually attended by more than 200 mathematicians. They are taking place today at the Institut Henri Poincaré .

In 2016 a new volume on algebraic topology was published.

Aftermath

Bourbaki's strictly logical style has had a decisive influence on today's mathematics.

Specifically, we owe Bourbaki the symbol for the empty set , the symbol for the implication, the abbreviations N , Z , Q , R , C for the sets of natural, whole, rational, real and complex numbers (in addition to the notation with the double prime ) as well as the words bijective , injective and surjective for properties of functions.

In France , Bourbak's axiomatics often still dominates the entire university teaching in mathematics as a major or minor; Foreign observers like Vladimir Igorewitsch Arnold consider this dogmatic formalism to be a crime against students ("the (I would say criminal) formalization of mathematics and of mathematical education").

In the 1960s, the Bourbaki trend also spread to school lessons (“ New Mathematics ”), one of the initiators being Jean Dieudonné.

The eléments de mathématique

First the French editions are listed. First editions, publishers and revised new editions are given as well as the numbers used by Bourbaki. ASI stands for Actualités Scientifiques et Industrielles , a series by Hermann.

  • Théorie des ensembles, set theory (Book 1)
    • 1939 Fascicule de résultats, Théorie des ensembles, Hermann (No. 1, ASI 846), new editions 1951, 1958
    • 1954 Chapter 1 (Description de la mathématique formelle), Chapter 2 (Théorie des ensembles), Hermann (No. 17, ASI 1212), new editions 1960, 1966
    • 1956 Chapter 3 (Ensembles ordonnés, cardinaux, nombres entiers), Hermann (No. 20, ASI 1243), new edition 1963
    • 1956 Chapter 4 (Structures), Hermann (No. 22, ASI 1258), new edition 1966
    • 1970, chapters 1 to 4, Hermann 1970 (reprint 1998)
  • Algèbre, Algebra (Book 2)
    • 1942 Chapter 1 (Structures algébriques), Hermann (No. 4, ASI 934), new edition 1960, 1966
    • 1947 Chapter 2 (Algèbre linéaire), Hermann (No. 6, ASI 1032), new edition 1962
    • 1948 Chapter 3 (Algèbres tensorielles, algèbres extérieures, algèbres symmétriques), Hermann (No. 7, ASI 1044), new edition 1958
    • 1950 Chapter 4 (Polynomes et fractions rationelles), 5 (Corps commutatifs), Hermann (No. 11, ASI 1102)
    • 1952 Chapter 6 (Groupes et corps ordonnés), Chapter 7 (Modules sur les anneaux principaux), Hermann (No. 11, ASI 1179)
    • 1958 Chapter 8 (Modules et anneaux semi-simples), Hermann (No. 23, ASI 1261), new edition 1981, completely revised new edition Springer 2012
    • 1959 Chapter 9 (Formes sesquilinéaires et formes quadratiques), Hermann (No. 24, ASI 1272)
    • 1970 New edition of Algebra Chapters 1-3 by Hermann, reprint Springer 2007
    • 1980 Chapter 10 (Algèbre homologique), Masson (no no.)
    • 1981 Chapters 4-7, Masson
  • Topology générale, general topology (Book 3)
    • 1940 Chapter 1 (Structures topologiques), 2 (Structures uniformes), Hermann (No. 2, ASI 858 and later 1142), new edition 1950, 1961
    • 1942 Chapter 3 (Groupes topologiques), 4 (Nombres réels), Hermann (No. 3, ASI 916 and later 1143), new edition 1951, 1960
    • 1947 Chapter 5 (Groupes à un paramétre), 6 (Espaces numériques et espaces projectifs), 7 (Les groupes additifs ), 8 (Nombres complexes), Hermann (No. 5, ASI 1029 and later 1235), new edition 1955, 1958
    • 1948 Chapter 9 (Utilisations des nombres réels en topologie générale), Hermann (No. 8, ASI 1045), new edition 1958
    • 1949 Chapter 10 (Espaces foncionelles), Dictionnaire, Hermann (No. 10, ASI 1084)
    • 1953 Fasicule des résultats (No. 16, ASI 1196)
    • 1971 Chapters 1-4, Hermann (reprint 1998)
    • 1974 Chapters 5-10 and Fasicule des résultats, Hermann, new edition Springer 2007
  • Fonctions d'une variable réelle - Théorie élémentaire, functions of a real variable (Book 4)
    • 1949 Chapter 1 (Derivées), 2 (Primitives et intégrales), 3 (Fonctions élémentaires), Hermann (No. 9, ASI 1074), new edition 1958
    • 1951 Chapter 4 (Equations différentielles), 5 (Étude locale des fonctions), 6 (Développements tayloriens généralisés; formule sommatoire d'Euler-MacLaurin), 7 (La fonction gamma), Dictionnaire, Hermann (No. 12, ASI 1132)
    • 1976 Chapters 1 to 7, Hermann, Springer reprint 2006
  • Espaces vectoriels topologiques, Topological vector spaces (Book 5)
    • 1953 Chapter 1 (Espaces vectoriels topologiques sur un corps valué), 2 (Ensembles convexes et espaces localement convexes), Hermann (No. 15, ASI 1189), new edition 1966
    • 1955, Chapter 3 (Espaces d'applications linaires continues), 4 (La dualité dans les espaces vectorielles topologiques), 5 (Espaces hilbertienne, Théorie élémentaire), Dictionnaire, Hermann (No. 18, ASI 1229)
    • 1955 Fasicule des résultats (No. 19, ASI 1230)
    • 1981 Chapters 1-5, Masson, Springer 2007 reprint
  • Integration (Book 6)
    • 1952 Chapter 1 (Inégalités de convexité), 2 (Espaces de Riesz), 3 (Mesures sur les espaces localement compactes), 4 (Prolongement d'une mesure, Espaces ), Hermann (No. 13, ASI 1175), new edition 1965
    • 1956 Chapter 5 (Intégration des mesures), Hermann (No. 21, ASI 1244), new edition 1967
    • 1959 Chapter 6 (Intégration vectorielle), Hermann, 2nd edition 1967 (No. 25, ASI 1281)
    • 1963 Chapter 7 (Mesure de Haar), 8 (Convolution et représentations), Hermann (No. 29, ASI 1306),
    • 1969 Chapter 9 (Intégration sur les espaces topologiques séparés), Hermann (No. 35, ASI 1343)
  • Algèbre commutative, Commutative Algebra (Book 7)
    • 1961 Chapter 1 (Modules plats), 2 (Localization), Hermann (No. 27, ASI 1290)
    • 1961 Chapter 3 (Graduations, filtrations et topologiques), 4 (Idéaux premieres associés et decomposition primaire), Hermann (No. 28, ASI 1293)
    • 1964 Chapter 5 (Entiers), 6 (Valuations), Hermann (No. 30, ASI 1308)
    • 1965 Chapter 7 (Diviseurs), Hermann (No. 31, ASI 1314)
    • 1983 Chapter 8 (Dimension), 9 (Anneaux locaux noethériens complets), Masson (no no.)
    • 1985 Chapters 1-4, Masson
    • 1985, chapters 5-7, Masson
    • 1998 Chapter 10 (Profondeur, Régularité, Dualité), Masson (no no.)
  • Groupes et algèbres de Lie, Lie groups and algebras, (Book 8)
    • 1960 Chapter 1 (Algébres de Lie), Hermann (No. 26, ASI 1285), new edition 1971
    • 1972 Chapter 2 (Algébres de Lie libre), 3 (Groupes de Lie), Hermann (No. 36, ASI 1349)
    • 1968 Chapter 4 (Groupes de Coxeter et systémes de Tits), 5 (Groupes engendrés par des refléxions), 6 (Systèmes de racines), Hermann (No. 34, ASI 1337), new edition Masson 1981
    • 1969 Chapter 7 (Sous-algébres de Cartan, éléments réguliers), Chapter 8 (Algèbres de Lie semi-simples déployées), Hermann (No. 38, ASI 1364), new edition 1975, reprint 1998
    • 1982 Chapter 9 (Groupes de Lie réels compacts), Masson (no no.)
  • Théorie spectrales, Spectral Theory (Book 9)
    • 1967 Chapter 1 (Algèbres normées), 2 (Groups localement compacts commutatifs), Hermann (No. 32, ASI 1332), reprint Springer 2007, new edition 2019
  • Variétés différentielles et analytiques (Fasicule de résultats), Differentiable and analytical manifolds, Hermann, 1968 (Paragraph 1 to 7, No. 33, ASI 1333), 1971 (Paragraph 8 to 15, No. 36, ASI 1347), new edition 1998 and Springer 2007 (Book 10)
  • Algèbre Topologique, Algebraic Topology
    • 2016 Chapter 1 (Revêtements), 2 (Groupoïdes), 3 (Homotopie et groupoïde de Poincaré), 4 (Espaces délaçables), Springer
  • Élements d'histoire de mathématique (summary of the historical introductions to the volumes), Hermann 1960, 2nd edition 1969, 3rd edition 1974, reprinted by Masson 1984

The books have been published by Springer since 2006 (including new editions).

English translations:

  • Topological Vector Spaces, Chapters 1-5, Springer 1987, 2003
  • Algebra I, chapters 1-3, Springer 1989
  • Algebra II, Chapters 4-7, 1990, Springer 2003
  • Commutative Algebra, Chapters 1-7, Addison-Wesley, Springer 1989
  • Lie Groups and Lie Algebras, Chapters 1-3, Springer 1989
  • General Topology 1, Chapters 1-4, Springer 1989, 1995
  • General Topology 2, Chapters 5-10, Springer 1989
  • Elements of the history of mathematics, Springer 1994
  • Integration I, Springer 2004
  • Integration II, Chapters 7-9, Springer 2004
  • Theory of Sets, Addison-Wesley / Hermann 1968, Springer 2004
  • Functions of a real variable: Elementary Theory, Springer 2004

German editions:

  • Elements of math history . Vandenhoeck and Ruprecht, 1971

Essays published by Bourbaki

  • 1935: Sur un théorème de Carathéodory et la mesure dans les espaces topologiques . In: Comptes Rendus de l'Académie des Sciences de Paris , 201, 1309–1311 (by André Weil)
  • 1938: Sur les espaces de Banach . In: Comptes Rendus de l'Académie des Sciences de Paris , Volume 206, pp. 1701–1704 (by Jean Dieudonné)
  • 1939: Note de tératopologie II . In: Revue scientifique (called Revue rose), pp. 180-181 (by N. Bourbaki and J. Dieudonné, the first article in the series was by Dieudonné, the third by Dieudonné and Henri Cartan)
  • 1941: Espaces minimaux et espaces complètement séparés . In: Comptes Rendus de l'Académie des Sciences de Paris , Volume 212, pp. 215-218 (by Jean Dieudonné or André Weil)
  • 1948: L'architecture des mathématiques . In: François Le Lionnais (ed.): Les grands courants de la pensée mathématique . Actes Sud, Paris, Collection L'humanisme scientifique de demain , pp. 35–47 (by Jean Dieudonné)
  • 1949: Foundations of Mathematics for the Working Mathematician . In: Journal of Symbolic Logic , Volume 14, pp. 1–8 (by André Weil)
  • 1949: Sur le théorème de Zorn . In: Archive of Mathematics , Volume 2, pp. 433–437 (by Henri Cartan or Jean Dieudonné)
  • 1951: Sur certains espaces vectoriels topologiques . In: Annales de l'Institut Fourier , Volume 3, pp. 3, 5-16 (by Jean Dieudonné and Laurent Schwartz)

References in pop culture

Bourbaki is mentioned in the song "Morph" from the album Trench by the duo twenty one pilots . The song "Nico and the Niners" from the same album is also a reference to Bourbaki.

literature

  • Amir Aczel: The artist and the mathematician. The story of Nicholas Bourbaki, the genius mathematician who never existed . High Stakes Publishing, 2007. , Review of Michael Atiyah, Notices AMS, October 2007
  • David Aubin: The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics . In: Science in Context . Volume 10, 1997, pp. 297–342, ( math.jussieu.fr , PDF)
  • L. Beaulieu: A parisian Café and ten pro-Bourbaki meetings . In: Mathematical Intelligencer . 1993, No. 1 (on the early years 1934/5, detailed in his dissertation in Montreal 1989)
  • Armand Borel: 25 years with Bourbaki 1949–1973 . In: Communications DMV . 1998, Notices AMS 1998, ams.org
  • Borel, Cartier, Chern, Iyanaga, Chandrasekharan: “André Weil Obituary”. In: Notices AMS . 1998 No. 4, and H. Cartan ibid. 1999, No. 6
  • Pierre Cartier: The continuing silence of Bourbaki (interview with M. Senechal). In: Mathematical Intelligencer . 1998, No. 1
  • Jean A. Dieudonné: The work of Nicolas Bourbaki . In: American Mathematical Monthly . Volume 77, 1970, p. 134, maa.org
  • Walther L. Fischer: Historical note on the genealogy of Mr. Nicolas Bourbaki . In: Physical sheets. Monthly for basic questions and peripheral problems in physics , 20th year, issue 4, 1967, pp. 157–161, doi: 10.1002 / phbl.19670230403 (joke article containing Henri Cartan's version of the origin of the name)
  • Guedj: Bourbaki - a collective mathematician , interview with Claude Chevalley. In: Mathematical Intelligencer . 1985, No. 7
  • Paul Halmos Nicolas Bourbaki , Scientific American May 1957
  • Christian Houzel: Bourbaki and after . In: Mathematical-Physical Semester Reports . Volume 49, 2002, p. 1
  • Gottfried Köthe : NB - The new order of mathematics. In: Schwerte & Spengler (ed.): Researchers and scientists in today's Europe 1: Physicists, ..., mathematicians. Series: Gestalter Our Time, Volume 3. Stalling, Oldenburg 1958, pp. 367–375
  • Maurice Mashaal: Bourbaki - a secret society of Mathematicians . American Mathematical Society, 2006, Review by Michael Atiyah, Notices AMS, October 2007
  • André Weil: The Apprenticeship of a mathematician . 1992 (his autobiography)
  • François Le Lionnais (ed.): Les grands courants de la pensée mathématique , Cahiers du Sud 1948, reprint at Hermann 1997 (the popular science collection also contains articles by the Bourbaists André Weil, Jean Dieudonne, Roger Godement, in which they explain their conception of mathematics expound)
  • Henri Cartan: Nicolas Bourbaki and Today's Mathematics , Westdeutscher Verlag 1959, Working Group for Research of the State of North Rhine-Westphalia 76
  • Siobhan Roberts: King of Infinite Space , Walker and Company 2006 (biography of HSM Coxeter , a major antagonist of the Bourbaki Current)

In addition to the French original edition of the elements (originally by Hermann), English translations of many volumes have also been published (by Springer Verlag).

German translations by Bourbaki:

  • Elements of math history . Vandenhoeck and Ruprecht, 1971
  • The architecture of mathematics , part 1,2. In: Physikalische Blätter , Volume 17, 1961, pp. 161-166, 212-218; also in M. Otte (Ed.): Mathematicians on Mathematics , Berlin 1974, pp. 140–159 (originally published in French in Les grands courants de la pensée mathématique , Cahiers du Sud, Marseille 1948, English translation by Arnold Dresden. In : American Mathematical Monthly , Volume 57, 1950, p. 221), doi: 10.1002 / phbl.19610170403 - part 1, doi: 10.1002 / phbl.19610170503 - part 2
  • Elements of math . In: W. Büttemeyer (Ed.): Philosophy of Mathematics . Freiburg / Br. 2003, 3rd edition 2009, pp. 163–171 (originally the French Introduction to Bourbakis Éléments de mathématique , Part I, Book I. Paris 1939, 2nd edition 1960, pp. 1–9)

Web links

Individual evidence

  1. ↑ In addition, according to official legend, Bourbaki was a member of the Academy of "Poldavia". Karl Strubecker: Introduction to Bourbaki essay. In: Physikalische Blätter , 1961, doi: 10.1002 / phbl.19610170402
  2. ↑ When deciding what to include, the members' subjective interests may also have played a role, for example in the Lie groups (Armand Borel)
  3. ↑ However, until 1939 the geophysicist Coulomb also took part in the meetings, who was supposed to oversee the applied part. This was never realized.
  4. ↑ However, other textbooks by group members created a certain balance, in particular the multi-volume Éléments d'Analyse by Jean Dieudonné, the books by Serge Lang, and the Bourbaki seminars, in which current mathematical research was reported.
  5. ↑ Cartier: “The misunderstanding was that many people thought that it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics, containing all the necessary information. That is a good description. If you consider it as a textbook, it's a disaster. " ega-math.narod.ru
  6. ^ Siobhan Roberts: King of Infinite Space , p. 153
  7. ^ John McCleary: Bourbaki and Algebraic Topology . (PDF)
  8. ↑ Bourbaki, Part 2 McTutor
  9. ^ In the series of the Cours d'Analyse traditional in France , z. B. by Camille Jordan or Édouard Goursat .
  10. ↑ Raoul Husson, ENS student, disguised himself as a Swedish professor Holmgoren in November 1923 and gave a lecture to freshmen with a fake beard. He wrote a sentence that should come from "Nicolas Bourbaki". According to another tradition, André Weil chose the name after the statue of the general in Nancy, where Jean-Pierre Serre taught.
  11. ↑ Mashaal, Bourbaki, AMS 2006, p. 17, lists an attendance list at a meeting of Bourbaki on October 20, 1995: Bernard Teissier (lead), Arnaud Beauville, Pierre Julg, Patrick Gérard, Joseph Oesterlé, Daniel Bennequin, Gérard Ben Arous, Guy Henniart, Marc Rosso, Olivier Mathieu, George Skandalis, Jean-Christophe Yoccoz.
  12. ↑ However, Grothendieck stayed on good terms with many Bourbaki members. His farewell letter to Bourbaki from 1960 is printed in Notices of the AMS, April 2016, p. 406. According to the memories of Hyman Bass (Notices AMS, March 2016, p. 249), Grothendieck was upset after one of the usual pointed remarks by André Weil and Grothendieck did not show up for days. Despite the efforts of Serge Lang and John T. Tate, he did not attend Bourbaki meetings.
  13. ↑ Interview, Notices American Mathematical Society 1997, No. 4, online as a PDF file here: ams.org
  14. ↑ Archives Bourbaki ( Memento of the original from January 15, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / archives-bourbaki.ahp-numerique.fr
  15. ↑ Gerard Eguether: French editions of Bourbaki. (PDF)
  16. ^ Bourbaki website
  17. ↑ Information from Archives Bourbaki, loc. cit.
  18. ^ Archives Bourbaki, loc. cit.
  19. ↑ twenty one pilots - Morph. Accessed October 6, 2018 (English).