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 Elie Cartan 
Elie Cartan  (1869-1951)

The Many Faces of
Nicolas  Bourbaki
© 2000-2018   Gérard P. Michon, Ph.D.

Bourbaki  recognized only one godfather, Elie Cartan.
Pierre Cartier (b.1932)
The Continuing Silence of Bourbaki 

[We witness]  a monumental exposition of the whole of present day mathematics whose framework won't become obsolete in the near future.  It makes visible the common bond between the various branches and can easily absorb new ideas.
(Edited exerpt; Bulletin of the AMS 59, 1953 p.474)  Emil Artin (1898-1962)

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Related articles on this site:

Related Links (Outside this Site)

Nicolas Bourbaki  by  Mike Conlay  (1995-06-11).
Association des Collaborateurs de Nicolas Bourbaki (ENS, 1952-).
Bourbaki and the Foundations of Modern Mathematics (CNRS News, 2017-10-14).
25 Years with Nicolas Bourbaki  (1949-1973)  by  Armand Borel.
Nicolas Bourbaki   |   Nicolas Bourbaki  at  www.absoluteastronomy.com.
Bourbaki:   The pre-war years  &  The post-war years   at  MacTutor.
Henri Cartan, interviewed in 1999  (notices of the AMS).
Climbing Mount Bourbaki  by Harvard student  Akhil Mathew  (2010).
Wikipedia Category

The Case of the Disappearing Mathematician (24:05, audio)  by  Amir Aczel  (Ira Flatow, NPR, 2006-11-17).


The Many Faces of  Nicolas Bourbaki

Stokes' theorem is a superb generalization of the fundamental theorem of calculus.  Nicolas Bourbaki and this general result are partly  due to each other...

 Border of a Surface
òW   dw     =     ò¶W   w
 Boundary of a Volume

Some Incarnations of  Stokes' Theorem
Gradient Conservativity
ò  b
  grad f . dM     =     f(b) - f(a)
Kelvin-Stokes' Formula òòS   rot U . dS     =     òC   U . dM
Ostrogradsky's Theorem òòòV   div U  dV     =     òòS   U . dS

Nicolas Bourbaki  was born on January 14th 1935, as the collective identity of a group of several highly talented young French mathematicians, in part from the urge to elucidate the general validity of the above formula  (as reported by the key founder,  André Weil,  then 28).

The  Bourbaki  collaboration has been extremely influential in France and elsewhere.  Bourbaki brought about new riguor based on the  logical foundations  of mathematics  (along the way, it also became instrumental in some controversial reforms of mathematical education, dubiously known as  new math  in the US, which originated in a meeting at Melun in 1952  under the leadership of Jean Dieudonné, Gustave Choquet and André Lichnerowicz).

The  active  founding members of the  Nicolas Bourbaki  group were: 

The official list of founders includes four members who were less active, namely:

Szolem Mandelbrojt  was the uncle and early mentor of the mathematician Benoît Mandelbrot (1924-2010; X1944) of  fractal  fame.

Two other mathematicians had been present at preliminary meetings, before the actual foundation of the Bourbaki group:

Other noted  bourbakists,  who joined the group later, include:

The rule was that all members would have to retire from the group at the age of  50  (Grothendieck and Lang left early, in anger).  All of the above are thus retired.

According to Jean Dieudonné (interviewed on 1987-06-12)  one of the motivations for the age rule was that many mathematicians seem overly enamored with what they learned when they were young.  The founders wanted the group to remain forever receptive to new ideas.

The ambition of the founders was to put on a fresh solid foundation the entire mathematical knowledge of their time.  This has taken the form of a collection of books entitled  Elements de Mathématique  (note the militant use of the grammatically incorrect singular form of  Mathématiques).

Henri Cartier  said that, during his own tenure from 1955 to 1983, the Bourbaki group was holding three yearly meetings  (for a total of about one month per year).

One of the first items on the original agenda was the aforementioned general  Stokes Theorem,  which unifies great results of vector calculus.  However, it sparked a search for rigorous settings which would delay by many years the publication of the final presentation by Bourbaki of that particular topic...

The  (nonprofit)  Association  formally incorporated in 1952 seems alive and well,  although it's not nearly as active and/or influential as it once was.  At one points,  there were concerns that the volumes published in 1983 and 1998 would be the last ones of the collection.  However,  one major update was published in 2016.


Opinions about Nicolas Bourbaki

The hope was that a fairly small simple set of basic structures were the basis of most research.  I think it is also true that for a while, this was generally believed.  But for the last 20 years, it has been increasingly doubted:  First because Bourbaki began to drown in its own need to be general enough, and they never could be sure when to stop  (e.g., before doing the reals, they need a general theory of topological fields).  Secondly, because math began to be driven by complex theories rather than simple ones  (e.g., the Langlands conjecture at the abstract end, control theory and probability at the applied end).  These theories don't benefit much from the Bourbaki.

David B. Mumford  (b. 1937, Fields medalist 1974)  1993-04-21

A Glitch in the Popularity of Nicolas Bourbaki

At this writing  (2018-12-08)  this page consistently ranks in Google's top page for queries related to  Nicolas Bourbaki.  Its average rank has been 6.3 over the last three months spanned by the plot below,  where the light-blue curve  (scale at right)  indicates the number of daily  impressions  of whatever Google result-page  (usually the top one)  mentions the page you're now reading.  The dark-blue curve  (scale at left)  gives the number of daily visits our page received from Google.

The latter closely follows the former,  with noise due to the varying location within Google's search results  (on most days, our rank was between 5 and 8).

Either curve is a good way to monitor the number of enquiries about Bourbaki in Google.  That was less than 100 per day before the incident diagnosed below,  which caused a hundredfold peak  (and a tenfold increase in the aftermath).

 Popularity of Nicolas Bourbaki

Whatever happened in late September (see below) caused the peak of  11836  impressions observed on Sunday, October 4, 2018  (157 visits to our own page,  whose average position was 5.6 that day).  At first,  I had hoped this had been the result of some nice mathematical online discussion.  Not so.  It turns out that the name of  Nicolas Bourbaki  was just used by songwriter  Tyler Joseph (1988-)  in the third track  (entitled  Morph)  of the  Trench  album  officially released on October 5,  2018  by his  Twenty Øne Piløts  band/duo.

Tyler's  rap lyrics  just pertain to his so-called  Dema  dreamworld,  haunted by  bishops  he named  André  [Weil?]  and  Nico.  He had previously called the latter  Blurryface  before identifying him as  Nicolas Bourbaki.  The  Twenty Øne Piløts  publicity campaign spread doctored bits of iconography from the  original Bourbaki group  to send fans on a  wild-goose chase,  which they seem to enjoy.

In the wake of the above,  one such fan produced the following two videos,  which are tedious to watch for those of us who aren't interested in Tyler's ramblings.  At least,  this lady can boast some of the scientific training which Tyler lacks.    Wink

Not Even Bullshit  by  Joachim I, KruegerBrown University  (Phychology Today, 2015-12-06).


A Few Bourbakists :

Samuel Eilenberg
Born: 30 September 30, 1913, in Warsaw  (now Poland)
Died: 30 January 1998, in New-York, NY  (USA)

Topologist.  He invented Category Theory in 1945, with Saunders Mac Lane (1909-2005).

  Sammy Eilenberg
Sammy Eilenberg

Laurent Schwartz   Laurent Schwartz
 Fields Medal Born: 5 March, 1915, in Paris  (France)
Died: 4 July 2002, in Paris  (France)

Conceived the  Theory of Distributions  in 1944.
Fields Medalist in 1950.  Noted professor.
He taught this writer  Hlbertian Analysis,  in the Fall of 1977.

 Jean-Pierre Serre    Fields Medal Born: 15 September 1926, in Bâges

Youngest Fields Medalist, in 1954.
First Abel Prize recipient, in 2003.

Wikipedia   |   Collège de France

 Alexandre Grothendieck    Fields Medal Born: 28 March, 1928, in Berlin
Died: 13 November, 2014, in Saint-Girons (Ariège).
Once stateless, he carried a UN "Nansen" passport.  He was the most prominent champion of  Category Theory.

Fields Medalist in 1966.  Professor at the French IHES.
Declined the Crafoord Prize in 1988.

The Grothendieck Circle, by  Leila Schneps   |   SGA   |   NUMDAM

Pierre Cartier   Pierre Cartier
Born: 10 June, 1932, in Sedan  (Ardennes, France)

Wikipedia   |   Cartier divisor   |   Cartier-Manin operator   |   Ph.D. 1958 (Cartan & Weil)

Alain Connes   Alain Connes
 Fields Medal Born: 1 April, 1947, in Draguignan (Var, France)

Fields Medalist in 1982.

www.alainconnes.org   |   Ph.D. 1973 (under Jacques Dixmier)

 Jean-Christophe Yoccoz    Fields Medal Born: May 29, 1957, in France

Fields Medalist in 1994.

Wikipedia   |   Mathematics Genealogy   |   MacTutor


Fields Medalists

1936 :

  • Lars Ahlfors (Finland).
  • Jesse Douglas (United States).

1950 :

1954 :

1958 :

  • Klaus Roth (United Kingdom).
  • René Thom (France).

1962 :

  • Lars Hörmander (Sweden).
  • John Milnor (United States).

1966 :

1970 :

  • Alan Baker (United Kingdom).
  • Heisuke Hironaka (United States).
  • Sergei Novikov (USSR).
  • John G. Thompson (United States).

1974 :

  • Enrico Bombieri (Italy).
  • David Mumford (United States).
 Pierre Deligne

1978 :

  • Pierre Deligne (Belgium).
  • Charles Fefferman (United States).
  • Gregori Alexandrovitch Margulis (USSR).
  • Daniel Quillen (United States).

1982 :

1986 :

  • Simon Donaldson (United Kingdom).
  • Gerd Faltings (West Germany).
  • Michael Freedman (United States).

1990 :

  • Vladimir Drinfeld (USSR).
  • Vaughan F. R. Jones (United States).
  • Shigefumi Mori (Japan).
  • Edward Witten (United States).
 Jean Bourgain

1994 :

  • Jean Bourgain (Belgium).
  • Pierre-Louis Lions (France).
  • Jean-Christophe Yoccoz (France).
  • Efim Zelmanov (United States).

1998 :

  • Richard E. Borcherds (UK).
  • William T. Gowers (United Kingdom).
  • Maxim Kontsevich (Russia).
  • Curtis T. McMullen (United States).

2002 :

  • Laurent Lafforgue (France).
  • Vladimir Voevodsky (United States).

2006 :

2010 :

2014 :

2018 :

Abel Laureates

2003 :

2004 :

2005 :

  • Peter Lax

2006 :

  • Lennart Carleson

2007 :

  • S. R. Srinivasa Varadhan

2008 :

  • John G. Thompson
  • Jacques Tits

2009 :

  • Mikhail Gromov

2010 :

  • John Tate

2011 :

  • John Milnor

2012 :

  • Endre Szemerédi

2013 :

  • Pierre Deligne

2014 :

  • Yakov Sinai

2015 :

2016 :

2017 :

2018 :

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