# Category: tBC

In the Grothendieck meets Lacan-post we did mention that Alain Connes wrote a book together with Patrick Gauthier-Lafaye “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient”, on the potential use of Grothendieck’s toposes for the theory of unconsciousness, proposed by the French psychoanalyst Jacques Lacan.

A bit more on that book you can read in the topos of unconsciousness. For another take on this you can visit the blog of l’homme quantique – Sur les traces de Lévi-Strauss, Lacan et Foucault, filant comme le sable au vent marin…. There is a series of posts dedicated to the reading of ‘A l’ombre de Grothendieck et de Lacan’:

Alain Connes isn’t the first (former) Bourbaki-member to write a book together with a Lacan-disciple.

In 1984, Henri Cartan (one of the founding fathers of Bourbaki) teamed up with the French psychoanalyst (and student of Lacan) Jean-Francois Chabaud for “Le Nœud dit du fantasme – Topologie de Jacques Lacan”.

(Chabaud on the left, Cartan on the right, Cartan’s wife Nicole in the mddle)

“Dans cet ouvrage Jean François Chabaud, psychanalyste, effectue la monstration de l’interchangeabilité des consistances de la chaîne de Whitehead (communément nommée « Noeud dit du fantasme » ou du « Non rapport sexuel » dans l’aire analytique), et peut ainsi se risquer à proposer, en s’appuyant sur les remarques essentielles de Jacques Lacan, une écriture du virage, autre nom de la passe. Henri Cartan (1904-2008), l’un des Membres-fondateur de N. Bourbaki, a contribué à ce travail avec deux réflexions : la première, considère cette monstration et l’augmente d’une présentation ; la seconde, traite tout particulièrement de l’orientation des consistances. Une suite de traces d’une séquence de la chaîne précède ce cahier qui s’achève par : « L’en-plus-de-trait », une contribution à l’écriture nodale.”

Lacan was not only fascinated by the topology of surfaces such as the crosscap (see the topos of unconsciousness), but also by the theory of knots and links.

The Borromean link figures in Lacan’s world for the Real, the Imaginary and the Symbolic. The Whitehead link (that is, two unknots linked together) is thought to be the knot (sic) of phantasy.

In 1986, there was the exposition “La Chaine de J.H.C. Whitehead” in the
Palais de la découverte in Paris (from which also the Chabaud-Cartan picture above is taken), where la Salle de Mathématiques was filled with different models of the Whitehead link.

In 1988, the exposition was held in the Deutches Museum in Munich and was called “Wandlung – Darstellung der topologischen Transformationen der Whitehead-Kette”

The set-up in Munich was mathematically more interesting as one could see the link-projection on the floor, and use it to compute the link-number. It might have been even more interesting if the difference in these projections between two subsequent models was exactly one Reidemeister move

You can view more pictures of these and subsequent expositions on the page dedicated to the work of Jean-Francois Chabaud: La Chaîne de Whitehead ou Le Nœud dit du fantasme Livre et Expositions 1980/1997.

Part of the first picture featured also in the Hommage to Henri Cartan (1904-2008) by Michele Audin in the Notices of the AMS. She writes (about the 1986 exposition):

“At the time, Henri Cartan was 82 years old and retired, but he continued to be interested in mathematics and, as one sees, its popularization.”

More than ten years ago, when I ran a series of posts on pre-WW2 Bourbaki congresses, I knew most of the existing B-literature. I’m afraid I forgot most of it, thereby missing opportunities to spice up a dull post (such as yesterday’s).

Right now, I need facts about the infamous ACNB and its former connection to Nancy, so I reread Liliane Beaulieu’s Bourbaki a Nancy:

(page 38) : “Like a theatrical canvas, “La Tribu” often carries as its header a subtitle, the product of its editor’s imagination, which brings out the theme of the congress, if necessary. There is thus a “De Nicolaıdes” congress in Nancy, “Du banc public” (reference to Brassens) that of the “Universites cogerees” (in October 68, at the time of co-management).”

The first La Ciotat congress (February 27 to March 6, 1955) was called ‘the congress of the public bench’ (‘banc public’ in French) where Serre and Cartan tried to press Bourbaki to opt for the by now standard approach to varieties (see yesterday), and the following Chicago-congress retaliated by saying that there were also public benches nearby, but of little use.

What I missed was the reference to French singer-songwriter George Brassens. In 1953, he wrote, composed and performed Bancs Public (later called ‘Les Amoureux des bancs publics’).

If you need further evidence (me, I’ll take Liliane’s word on anything B-related), here’s the refrain of the song:

“Les amoureux qui s’bécotent sur les bancs publics,
Bancs publics, bancs publics,
En s’foutant pas mal du regard oblique
Des passants honnêtes,
Les amoureux qui s’bécotent sur les bancs publics,
Bancs publics, bancs publics,
En s’disant des “Je t’aime'” pathétiques,
Ont des p’tits gueules bien sympathiques!

(G-translated as:
‘Lovers who smooch on public benches,
Public benches, public benches,
By not giving a damn about the sideways gaze
Honest passers-by,
The lovers who smooch on the public benches,
Public benches, public benches,
Saying pathetic “I love you” to each other,
Have very nice little faces!‘)

Compare this to page 3 of the corresponding “La Tribu”:

“Geometrie Algebrique : elle a une guele bien sympathique.”

(Algebraic Geometry : she has a very nice face)

More Bourbaki congresses got their names rather timely.

In the summer of 1959 (from June 25th – July 8th) there was a congress in Pelvout-le-Poet called ‘Congres du cerceau’.

‘Cerceau’ is French for Hula Hoop, whose new plastic version was popularized in 1958 by the Wham-O toy company and became a fad.

(Girl twirling Hula Hoop in 1958 – Wikipedia)

The next summer it was the thing to carry along for children on vacation. From the corresponding “La Tribu” (page 2):

“Le congres fut marque par la presence de nombreux enfants. Les distractions s’en ressentirent : baby-foot, biberon de l’adjudant (tres concurrence par le pastis), jeu de binette et du cerceau (ou faut-il dire ‘binette se jouant du cerceau’?) ; un bal mythique a Vallouise faillit faire passer la mesure.”
(try to G-translate it yourself…)

Here’s another example.

The spring 1949 congress (from April 13th-25th) was held at the Abbey of Royaumont and was called ‘le congres du cocotier’ (the coconut-tree congress).

From the corresponding “La Tribu 18”:

“Having absorbed a tough guinea pig, Bourbaki climbed to the top of the Royaumont coconut tree, and declared, to unanimous applause, that he would only rectify rectifiable curves, that he would treat rational mechanics over the field $\mathbb{Q}$, and, that with a little bit of vaseline and a lot of patience he would end up writing the book on algebraic topology.”

The guinea pig that congress was none other than Jean-Pierre Serre.

A year later (from April 5th-17th 1950) there was another Royaumont-congress called ‘le congres de la revanche du cocotier’ (the congress of the revenge of the coconut-tree).

From the corresponding La Tribu 22:

“The founding members had decided to take a dazzling revenge on the indiscipline young people; mobilising all the magical secrets unveiled to them by the master, they struck down the young people with various ailments; rare were those strong enough to jump over the streams of Royaumont.”

Here’s what Maurice Mashaal says about this in ‘Bourbaki – a secret society of mathematicians’ (page 113):

“Another prank among the members was called ‘le cocotier’ (the coconut tree). According to Liliane Beaulieu, this was inspired by a Polynesian custom where an old man climbs a palm tree and holds on tightly while someone shakes the trunk. If he manages to hold on, he remains accepted in the social group. Bourbaki translated this custom as the following: some members would set a mathematical trap for the others. If someone fell for it, they would yell out ‘cocotier’.”

May I be so bold as to suggest that perhaps this sudden interest in Polynesian habits was inspired by the recent release of L’ile aux cocotiers (1949), the French translation of Robert Gibbing’s book Coconut Island?

Last time, we’ve seen that the first time ‘schemes’ were introduced was in ‘La Tribu’ (the internal Bourbaki-account of their congresses) of the May-June 1955 congress in Chicago.

Here, we will focus on the events leading up to that event. If you always thought Grothendieck invented the word ‘schemes’, here’s what Colin McLarty wrote:

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea)

What were Weil’s foundations of algebraic geometry?

Well, let’s see how Weil defined an affine variety over a field $k$. First you consider a ‘universal field’ $K$ containing $k$, that is, $K$ is an algebraically closed field of infinite transcendence degree over $k$. A point of $n$-dimensional affine space is an $n$-tuple $x=(x_1,\dots,x_n) \in K^n$. For such a point $x$ you consider the field $k(x)$ which is the subfield of $K$ generated by $k$ and the coordinates $x_i$ of $x$.

Alternatively, the field $k(x)$ is the field of fractions of the affine domain $R=k[z_1,\dots,z_n]/I$ where $I$ is the prime ideal of all polynomials $f \in k[z_1,\dots,z_n]$ such that $f(x) = f(x_1,\dots,x_n)=0$.

An affine $k$-variety $V$ is associated to a ‘generic point’ $x=(x_1,\dots,x_n)$, meaning that the field $k(x)$ is a ‘regular extension’ of $k$ (that is, for all field-extensions $k’$ of $k$, the tensor product $k(x) \otimes_k k’$ does not contain zero-divisors.

The points of $V$ are the ‘specialisations’ of $x$, that is, all points $y=(y_1,\dots,y_n)$ such that $f(y_1,\dots,y_n)=0$ for all $f \in I$.

Perhaps an example? Let $k = \mathbb{Q}$ and $K=\mathbb{C}$ and take $x=(i,\pi)$ in the affine plane $\mathbb{C}^2$. What is the corresponding prime ideal $I$ of $\mathbb{Q}[z_1,z_2]$? Well, $i$ is a solution to $z_1^2+1=0$ whereas $\pi$ is transcendental over $\mathbb{Q}$, so $I=(z_1^2+1)$ and $R=\mathbb{Q}[z_1,z_2]/I= \mathbb{Q}(i)[z_2]$.

Is $x=(i,\pi)$ a generic point? Well, suppose it were, then the points of the corresponding affine variety $V$ would be all couples $(\pm i, \lambda)$ with $\lambda \in \mathbb{C}$ which is the union of two lines in $\mathbb{C}^2$. But then $i \otimes 1 + 1 \otimes i$ is a zero-divisor in $\mathbb{Q}(x) \otimes_{\mathbb{Q}} \mathbb{Q}(i)$. So no, it is not a generic point over $\mathbb{Q}$ and does not define an affine $\mathbb{Q}$-variety.

If we would have started with $k=\mathbb{Q}(i)$, then $x=(i,\pi)$ is generic and the corresponding affine variety $V$ consists of all points $(i,\lambda) \in \mathbb{C}^2$.

If this is new to you, consider yourself lucky to be young enough to have learned AG from Fulton’s Algebraic curves, or Hartshorne’s chapter 1 if you were that ambitious.

By 1955, Serre had written his FAC, and Bourbaki had developed enough commutative algebra to turn His attention to algebraic geometry.

La Ciotat congress (February 27th – March 6th, 1955)

With a splendid view on the mediterranean, a small group of Bourbaki members (Henri Cartan (then 51), with two of his former Ph.D. students: Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist), Jacques Dixmier (then 31), and Pierre Samuel (then 34), a former student of Zariski’s) discussed a previous ‘Rapport de Geometrie Algebrique'(no. 206) and arrived at some unanimous decisions:

1. Algebraic varieties must be sets of points, which will not change at every moment.
2. One should include ‘abstract’ varieties, obtained by gluing (fibres, etc.).
3. All necessary algebra must have been previously proved.
4. The main application of purely algebraic methods being characteristic p, we will hide nothing of the unpleasant phenomena that occur there.

(Henri Cartan and Jean-Pierre Serre, photo by Paul Halmos)

The approach the propose is clearly based on Serre’s FAC. The points of an affine variety are the maximal ideals of an affine $k$-algebra, this set is equipped with the Zariski topology such that the local rings form a structure sheaf. Abstract varieties are then constructed by gluing these topological spaces and sheaves.

At the insistence of the ‘specialistes’ (Serre, and Samuel who had just written his book ‘Méthodes d’algèbre abstraite en géométrie algébrique’) two additional points are adopted, but with some hesitation. The first being a jibe at Weil:
1. …The congress, being a little disgusted by the artificiality of the generic point, does not want $K$ to be always of infinite transcendent degree over $k$. It admits that generic points are convenient in certain circumstances, but refuses to see them put to all the sauces: one could speak of a coordinate ring or of a functionfield without stuffing it by force into $K$.
2. Trying to include the arithmetic case.

The last point was problematic as all their algebras were supposed to be affine over a field $k$, and they wouldn’t go further than to allow the overfield $K$ to be its algebraic closure. Further, (and this caused a lot of heavy discussions at coming congresses) they allowed their varieties to be reducible.

The Chicago congress (May 30th – June 2nd 1955)

Apart from Samuel, a different group of Bourbakis gathered for the ‘second Caucus des Illinois’ at Eckhart Hall, including three founding members Weil (then 49), Dixmier (then 49) and Chevalley (then 46), and two youngsters, Armand Borel (then 32) and Serge Lang (then 28).

Their reaction to the La Ciotat meeting (the ‘congress of the public bench’) was swift:

(page 1) : “The caucus discovered a public bench near Eckhart Hall, but didn’t do much with it.”
(page 2) : “The caucus did not judge La Ciotat’s plan beyond reproach, and proposed a completely different plan.”

They wanted to include the arithmetic case by defining as affine scheme the set of all prime ideals (or rather, the localisations at these prime ideals) of a finitely generated domain over a Dedekind domain. They continue:

(page 4) : “The notion of a scheme covers the arithmetic case, and is extracted from the illustrious works of Nagata, themselves inspired by the scholarly cogitations of Chevalley. This means that the latter managed to sell all his ideas to the caucus. The Pope of Chicago, very happy to be able to reject very far projective varieties and Chow coordinates, willingly rallied to the suggestions of his illustrious colleague. However, we have not attempted to define varieties in the arithmetic case. Weil’s principle is that it is unclear what will come out of Nagata’s tricks, and that the only stable thing in arithmetic theory is reduction modulo $p$ a la Shimura.”

“Contrary to the decisions of La Ciotat, we do not want to glue reducible stuff, nor call them varieties. … We even decide to limit ourselves to absolutely irreducible varieties, which alone will have the right to the name of varieties.”

The insistence on absolutely irreducibility is understandable from Weil’s perspective as only they will have a generic point. But why does he go along with Chevalley’s proposal of an affine scheme?

In Weil’s approach, a point of the affine variety $V$ determined by a generic point $x=(x_1,\dots,x_n)$ determines a prime ideal $Q$ of the domain $R=k[x_1,\dots,x_n]$, so Chevalley’s proposal to consider all prime ideals (rather than only the maximal ideals of an affine algebra) seems right to Weil.

However in Weil’s approach there are usually several points corresponding to the same prime ideal $Q$ of $R$, namely all possible embeddings of the ring $R/Q$ in that huge field $K$, so whenever $R/Q$ is not algebraic over $k$, there are infinitely Weil-points of $V$ corresponding to $Q$ (whence the La Ciotat criticism that points of a variety were not supposed to change at every moment).

According to Ralf Krömer in his book Tool and Object – a history and philosophy of category theory this shift from Weil-points to prime ideals of $R$ may explain Chevalley’s use of the word ‘scheme’:

(page 164) : “The ‘scheme of the variety’ denotes ‘what is invariant in a variety’.”

Another time we will see how internal discussion influenced the further Bourbaki congresses until Grothendieck came up with his ‘hyperplan’.

Wikipedia claims:

“The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski’s ideas.”

and refers to the lecture by Chevalley ‘Les schemas’, given on December 12th, 1955 at the ENS-based ‘Seminaire Henri Cartan’ (in fact, that year it was called the Cartan-Chevalley seminar, and the next year Chevalley set up his own seminar at the ENS).

Items recently added to the online Bourbaki Archive give us new information on time and place of the birth of the concept of schemes.

From May 30th till June 2nd 1955 the ‘second caucus des Illinois’ Bourbaki-congress was held in ‘le grand salon d’Eckhart Hall’ at the University of Chicago (Weil’s place at that time).

Only six of the Bourbaki members were present:

• Jean Dieudonne (then 49), the scribe of the Bourbaki-gang.
• Andre Weil (then 49), called ‘Le Pape de Chicago’ in La Tribu, and responsible for his ‘Foundations of Algebraic Geometry’.
• Claude Chevalley (then 46), who wanted a better, more workable version of algebraic geometry. He was just nominated professor at the Sorbonne, and was prepping for his seminar on algebraic geometry (with Cartan) in the fall.
• Pierre Samuel (then 34), who studied in France but got his Ph.D. in 1949 from Princeton under the supervision of Oscar Zariski. He was a Bourbaki-guinea pig in 1945, and from 1947 attended most Bourbaki congresses. He just got his book Methodes d’algebre abstraite en geometrie algebrique published.
• Armand Borel (then 32), a Swiss mathematician who was in Paris from 1949 and obtained his Ph.D. under Jean Leray before moving on to the IAS in 1957. He was present at 9 of the Bourbaki congresses between 1955 and 1960.
• Serge Lang (then 28), a French-American mathematician who got his Ph.D. in 1951 from Princeton under Emil Artin. In 1955, he just got a position at the University of Chicago, which he held until 1971. He attended 7 Bourbaki congresses between 1955 and 1960.

The issue of La Tribu of the Eckhart-Hall congress is entirely devoted to algebraic geometry, and starts off with a bang:

“The Caucus did not judge the plan of La Ciotat above all reproaches, and proposed a completely different plan.

I – Schemes
II – Theory of multiplicities for schemes
III – Varieties
IV – Calculation of cycles
V – Divisors
VI – Projective geometry
etc.”

In the spring of that year (February 27th – March 6th, 1955) a Bourbaki congress was held ‘Chez Patrice’ at La Ciotat, hosting a different group of Bourbaki members (Samuel was the singleton intersection) : Henri Cartan (then 51), Jacques Dixmier (then 31), Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist).

In the La Ciotat-Tribu,nr. 35 there are also a great number of pages (page 14 – 25) used to explain a general plan to deal with algebraic geometry. Their summary (page 3-4):

“Algebraic Geometry : She has a very nice face.

Chap I : Algebraic varieties
Chap II : The rest of Chap. I
Chap III : Divisors
Chap IV : Intersections”

There’s much more to say comparing these two plans, but that’ll be for another day.

We’ve just read the word ‘schemes’ for the first (?) time. That unnumbered La Tribu continues on page 3 with “where one explains what a scheme is”:

So, what was their first idea of a scheme?

Well, you had your favourite Dedekind domain $D$, and you considered all rings of finite type over $D$. Sorry, not all rings, just all domains because such a ring $R$ had to have a field of fractions $K$ which was of finite type over $k$ the field of fractions of your Dedekind domain $D$.

They say that Dedekind domains are the algebraic geometrical equivalent of fields. Yeah well, as they only consider $D$-rings the geometric object associated to $D$ is the terminal object, much like a point if $D$ is an algebraically closed field.

But then, what is this geometric object associated to a domain $R$?

In this stage, still under the influence of Weil’s focus on valuations and their specialisations, they (Chevalley?) take as the geometric object $\mathbf{Spec}(R)$, the set of all ‘spots’ (taches), that is, local rings in $K$ which are the localisations of $R$ at prime ideals. So, instead of taking the set of all prime ideals, they prefer to take the set of all stalks of the (coming) structure sheaf.

But then, speaking about sheaves is rather futile as there is no trace of any topology on this set, then. Also, they make a big fuss about not wanting to define a general schema by gluing together these ‘affine’ schemes, but then they introduce a notion of ‘apparentement’ of spots which basically means the same thing.

It is still very early days, and there’s a lot more to say on this, but if no further documents come to light, I’d say that the birthplace of ‘schemes’, that is , the place where the first time there was a documented consensus on the notion, is Eckhart Hall in Chicago.

Rereading the Grothendieck-Serre correspondence I found a letter from Serre to Grothendieck, dated October 22nd 1958, which forces me to retract some claims from the previous La Ciotat post.

Serre writes this ten days after the second La Ciotat-congress (La Tribu 46), held from October 5th-12th 1958:

“The Bourbaki meeting was very pleasant; we all stayed in the home of a man called Guérin (a friend of Schwartz’s – a political one, I think); Guérin himself was in Paris and we had the whole house to ourselves. We worked outside most of the time, the weather was beautiful, we went swimming almost every day; in short, it was one of the best meetings I have ever been to.”

So far so good, we did indeed find Guérin’s property ‘Maison Rustique Olivette’ as the location of Bourbaki’s La Ciotat-congresses. But, Serre was present at both meetings (the earlier one, La Tribu 35, was held from February 27th – March 6th, 1955), so wouldn’t he have mentioned that they returned to that home when both meetings took place there?

From La Tribu 35:

“The Congress was held “chez Patrice”, in La Ciotat, from February 27 to March 6, 1955. Present: Cartan, Dixmier, Koszul, Samuel, Serre, le Tableau (property, fortunately divisible, of Bourbaki).”

In the previous post I mentioned that there was indeed a Hotel-Restaurant “Chez Patrice” in La Ciotat, but mistakingly assumed both meetings took place at Guérin’s property.

Can we locate this place?

On the backside of this old photograph

“Chez Patrice”
seul au bord de la mer
Hotel Restaurant tout confort
Spécialités Provençales
Plage privée Parc auto
Ouvert toute l’année
Sur la route de La Ciota-Bandol
Tel 465
La Ciota (B.-d.-R.)

So it must be on the scenic coastal road from La Ciotat to Bandol. My best guess is that “Chez Patrice” is today the one Michelin-star Restaurant “La Table de Nans”, located at 126 Cor du Liouquet, in La Ciotat.

Their website has just this to say about the history of the place:

“Located in an exceptional setting between La Ciotat and Saint Cyr, the building of “l’auberge du Revestel” was restored in 2016.”

And a comment on a website dedicated to the nearby Restaurant Roche Belle confirms that “Chez Patrice”, “l’auberge du Revestel” and “table de Nans” were all at the same place:

“Nous sommes locaux et avons découverts ce restaurant seulement le mois dernier (suite infos copains) alors que j’ai passé une partie de mon enfance et adolescence “chez Patrice” (Revestel puis chez Nans)!!!”

I hope to have it right this time: the first Bourbaki La Ciotat-meeting in 1955 took place “Chez Patrice” whereas the second 1958-congress was held at ‘Maison Rustique Olivette’, the property of Schwartz’s friend Daniel Guérin.

Still, if you compare Serre’s letter to this paragraph from Schwartz’s autobiography, there’s something odd:

“I knew Daniel Guérin very well until his death. Anarchist, close to Trotskyism, he later joined Marceau Prevert’s PSOP. He had the kindness, after the war, to welcome in his property near La Ciotat one of the congresses of the Bourbaki group. He shared, in complete camaraderie, our life and our meals for two weeks. I even went on a moth hunt at his house and caught a death’s-head hawk-moth (Acherontia atropos).”

Schwartz was not present at the second La Ciotat-meeting, and he claims Guérin shared meals with the Bourbakis whereas Serre says he was in Paris and they had the whole house to themselves.

Moral of the story: accounts right after the event (Serre’s letter) are more trustworthy than later recollections (Schwartz’s autobiography).

Dear Collaborators of Nicolas Bourbaki, please make all Bourbaki material (Diktat, La Tribu, versions) publicly available, certainly those documents older than 50 years.

Perhaps you can start by adding the missing numbers 36 and 49 to your La Tribu: 1940-1960 list.

Thank you!

Bourbaki held His Spring-Congresses between 1952 and 1954 in Celles-sur-Plaine in the Vosges department.

• La Tribu 27, ‘Congres croupion des Vosges’ (March 8th-16th, 1952)
• La Tribu 30, ‘Congres nilpotent’ (March 1st-8th, 1953)
• La Tribu 33, ‘Congres de la tangente’ (March 28th-April 3rd, 1954)

As we can consult the Bourbaki Diktat of the first two meetings, there is no mystery as to their place of venue. From Diktat 27:

“The Congress of March 1952 will be held as planned in Celles-sur-Plaine (Vosges) at the Hotel de la Gare, from Sunday March 9 at 2 p.m. to Sunday March 16 in the evening. A train leaves Nancy on Sunday morning at 8:17 a.m., direction Raon-l’Etappe, where we arrive at 9:53 a.m.; from there a bus leaves for Celles-sur-Plaine (11 km away) at 10 am. Please bring big shoes for the walks (there will probably be a lot of snow on the heights).”

Even though few French villages have a train station, most have a ‘Place de la Gare’, indicating the spot where the busses arrive and leave. Celles-sur-Plaine is no exception, and one shouldn’t look any further to find the ‘Hotel de la Gare’.

This Hotel still exists today, but is now called ‘Hotel des Lacs’.

At the 1952 meeting, Grothendieck is listed as a ‘visitor’ (he was a guinea-pig earlier and would only become a Bourbaki-member in 1955). He was invited to settle disputes over the texts on EVTs (Topological Vector Spaces). In the quote below from La Tribu 27 ‘barrel’ refers of course to barreled space:

“But above all a drama was born from the laborious delivery of the EVTs. Eager to overcome the reluctance of the opposition, the High Commissioner attempted a blackmail tactic: he summoned Grothendieck! He hoped to frighten the Congress members to such an extent that they would be ready to swallow barrel after barrel for fear of undergoing a Grothendieckian redaction. But the logicians were watching: they told Grothendieck that, if all the empty sets are equal, some at least are more equal than others; the poor man went berserk, and returned to Nancy by the first train.”

The 1953 meeting also had a surprise guest, no doubt on Weil’s invitation, Frank Smithies, who we remember from the Bourbaki wedding joke.

Frank Smithies seated in the middle, in between Ralph Boas (left) and Andre Weil (right) at the Red Lion, Grantchester in 1939.

At the 1954 meeting we see a trace of Bourbaki’s efforts to get a position for Chevalley at the Sorbonne.

“Made sullen by the incessant rain, and exhausted by the electoral campaigns of La Sorbonne and the Consultative Committee, the faithful poured out their indecisive bile on the few drafts presented to them, and hardly took any serious decisions.”

Two Bourbaki-congresses were organised at the Côte d’Azur, in La Ciotat, claiming to have one of the most beautiful bays in the world.

• La Tribu 35, ‘Congres du banc public’ (February 27th – March 6th, 1955)
• La Tribu 46, ‘Congres du banquet auxiliaire’ (October 5th-12th, 1958)

As is the case for all Bourbaki-congresses after 1953, we do not have access to the corresponding Diktat, making it hard to find the exact location.

The hints given in La Tribu are also minimal. In La Tribu 34 there is no mention of a next conferences in La Ciotat, in La Tribu 45 we read on page 11:

“October Congress: It will take place in La Ciotat, and will be a rump congress (‘congres-croupion’). On the program: Flat modules, Fiber carpets, Schwartz’ course in Bogota, Chapter II and I of Algebra, Reeditions of Top. Gen. III and I, Primary decomposition, theorem of Cohen and consorts, Local categories, Theorems of Ad(o), and (ritually!) abelian varieties.”

“The Congress was held “chez Patrice”, in La Ciotat, from February 27 to March 6, 1955.
Presents: Cartan, Dixmier, Koszul, Samuel, Serre, le Tableau (property, fortunately divisible, of Bourbaki).
The absence, for twenty-four hours, of any founding member, created a euphoric climate, consolidated by the aioli, non-cats, and sunbathing by the sea. We will ask Picasso for a painting on the theme ‘Bourbaki soothing the elements’. However, some explorations were disturbed by barbed wire, wardens, various fences, and Samuel, blind with anger, declared that he could not find ‘la patrice de massage’.”

The last sentence seems to indicate that the clue “chez Patrice” is a red herring. There was, however, a Hotel-Restaurant Chez Patrice in La Ciotat.

But, we will find out that the congress-location was elsewhere. (Edit August 4th : wrong see the post La Ciotat (2).

As to that location, La Tribu 46 has this to say:

“The Congress was held in a comfortable villa, equipped with a pick-up, rare editions, tasty cuisine, and a view of the Mediterranean. In the deliberation room, Chevalley claimed to see 47 fish (not counting an object, in the general shape of a sea serpent which served as an ashtray); this prompted him to bathe; but, indisposed by a night of contemplation in front of Brandt’s groupoid, he pretended to slip all his limbs into the same hole in Bruhat’s bathing suit.”

Present in 1958 were : Bruhat, Cartan, Chevalley, Dixmier, Godement, Malgrange
and Serre.

So far, we have not much to go on. Luckily, there are these couple of sentences in Laurent Schwartz’ autobiography Un mathématicien aux prises avec le siècle:

“I knew Daniel Guérin very well until his death. Anarchist, close to Trotskyism, he later joined Marceau Prevert’s PSOP. He had the kindness, after the war, to welcome in his property near La Ciotat one of the congresses of the Bourbaki group. He shared, in complete camaraderie, our life and our meals for two weeks. I even went on a moth hunt at his house and caught a death’s-head hawk-moth (Acherontia atropos).”

Daniel Guérin is known for his opposition to Nazism, fascism, capitalism, imperialism and colonialism. His revolutionary defense of free love and homosexuality influenced the development of queer anarchism.

Now we’re getting somewhere.

But there are some odd things in Schwartz’ sentences. He speaks of ‘two weeks’ whereas both La Ciotat-meetings only lasted one week. Presumably, he takes the two together, so both meetings were held at Guérin’s property.

Stranger seems to be that Schwartz was not present at either congress (see above list of participants). Or was he? Yes, he was present at the first 1955 meeting, masquerading as ‘le Tableau’. On Bourbaki photos, Schwartz is often seen in front of their portable blackboard, as we’ve seen in the Pelvoux-post. Here’s another picture from that 1951-conference with Weil and Schwartz discussing before ‘le tableau’. (Edit August 12th : wrong, La Tribu 37 lists both Schwartz and ‘Le Tableau’ among those present).

Presumably, Bourbaki got invited to La Ciotat via Schwartz’ connection with Guérin in 1955, and there was a repeat-visit three years later.

But, where is that property of Daniel Guérin?

I would love to claim that it is La Villa Deroze, (sometimes called the small Medici villa in La Ciotat), named after Gilbert Deroze. From the website:

“Gilbert Deroze’s commitment to La Ciotat (he will be deputy mayor in 1947) is accompanied by a remarkable cultural openness. The house therefore becomes a place of hospitality and artistic and intellectual convergence. For example, it is the privileged place of reception for Daniel Guérin, French revolutionary writer, anti-colonialist, activist for homosexual emancipation, theoretician of libertarian communism, historian and art critic. But it also receives guests from the place that the latter had created nearby, the Maison Rustique Olivette, a real center of artistic residence which has benefited in particular from the presence of Chester Himes, Paul Célan, the “beat” poet Brion Gysin, or again of the young André Schwarz-Bart.”

Even though the Villa Deroze sometimes received guests of Guérin, this was not the case for Bourbaki as Schwartz emphasises that the congress took place in Guérin’s property near La Ciotat, which we now have identified as ‘Maison (or Villa) Rustique Olivette’.

From the French wikipedia entry on La Ciotat:

“In 1953 the writer Daniel Guérin created on the heights of La Ciotat, Traverses de la Haute Bertrandière, an artists’ residence in his property Rustique Olivette. In the 1950s, he notably received Chester Himes, André Schwartz-Bart, in 1957, who worked there on his book The Last of the Righteous, Paul Celan, Brion Gysin. Chester Himes returned there in 1966 and began writing his autobiography there.”

Okay, now we’re down from a village (La Ciotat) to a street (Traverses de la Haute Bertrandière), but which of these fabulous villas is ‘Maison Rustique Olivette’?

I found one link to a firm claiming to be located at the Villa Rustique Olivette, and giving as its address: 130, Traverses de la Haute Bertrandière.

If this information is correct, we have now identified the location of the two last Bourbaki congress in La Ciotat as ‘Maison Rustique Olivette’,

with coordinates 43.171122, 5.597150.

Browsing through La Tribu (the internal report of Bourbaki-congresses), sometimes you’ll find an answer to a question you’d never ask?

Such as: “When did Grothendieck decide to change his looks?”

Photo on the left is from 1951 taken by Paulo Ribenboim, on a cycling tour to Pont-a-Mousson (between Nancy and Metz). The photo on the right is from 1965 taken by Karin Tate.

From La Tribu 43, the second Bourbaki-congress in Marlotte from October 6th-11th 1957:

“The congress gave an enthusiastic welcome to Yul Grothendieck, who arrived in his Khrushchev haircut, in order to enjoy more comfortably the shadow of the sputniks. Seized with jealousy, Dixmier and Samuel rushed to the local hairdresser, who was, alas, quite unable to imitate this masterpiece.”

This Marlotte-meeting was called ‘Congres de la deuxieme lune’, because at their first congress in Marlotte, the hotel-owner thought this group of scientists was preparing for a journey to the moon. Bourbaki was saddened to find out that ownership of the ‘Hotel de la mare aux fées’ changed over the two years between meetings, for He hoped to surprise her with a return visit just at the time the first Sputnik was launched (October 4th, 1957).

Given the fact that the 1957-summer Bourbaki-congress lasted until July 7th, and that most of the B’s may have bumped into G over the summer, I’d wager that the answer to this most important of questions is: late summer 1957.

At least six Bourbaki-congresses were held in ‘Royaumont’:

• La Tribu 18 : ‘Congres oecumenique du cocotier’, April 13th-25th 1949
• La Tribu 22 : ‘Congres de la revanche du cocotier’, April 5th-17th 1950
• La Tribu without number : ‘Congres de l’horizon’, October 8th-15th 1950
• La Tribu 26 : ‘Congres croupion’, October 1st-9th 1951
• La Tribu 31 : ‘Congres de la revelation du reglement’, JUne 6th-19th 1953
• La Tribu 32 : ‘Congres du coryza’, October 2nd-9th 1953

All meetings were pre-1954, so the ACNB generously grants us all access to the corresponding Bourbaki Diktats. From Diktat 31:

“The next congress will be held at the Abbey of Royaumont, from Saturday June 6th (not from June 5th as planned) to Saturday June 20th.
We meet at 10 a.m., June 6 at the Gare du Nord before the ticket-check. Train to Viarmes (change at Monsoult at 10.35 a.m.). Do not bring a ticket: one couch can transport 4 delegates.
Bring the Bible according to the following distribution:
Cartan: livre IV. Dixmier: Alg. 3, livre VI. Godement: Alg.4-5, Top. 1-2. Koszul: Top. 5-6-7-8-9. Schwartz: Top. 10, Alg. 1-2. Serre: Top. 3-4, livre V. Weil: Alg. 6-7, Ens. R.”

Royaumont Abbey is a former Cistercian abbey, located near Asnières-sur-Oise in Val-d’Oise, approximately 30 km north of Paris, France.

How did Bourbaki end up in an abbey? From fr.wikipedia Abbaye de Royaumont:

In 1947, under the direction of Gilbert Gadoffre, Royaumont Abbey became the “International Cultural Center of Royaumont”, an alternative place to traditional French university institutions. During the 1950s and 1960s, the former abbey became a meeting place for intellectual and artistic circles on an international scale, with numerous seminars, symposiums and conferences under the name “Cercle culturel de Royaumont”. Among its illustrious visitors came Nathalie Sarraute, Eugène Ionesco, Alain Robbe-Grillet, Vladimir Jankélévitch, Mircea Eliade, Witold Gombrowicz, Francis Poulenc and Roger Caillois.

And… less illustrious, at least according to the French edition of Wikipedia, the Bourbaki-gang.

The preparations for the unique Bourbaki-congress in Murols, start already in La Tribu 32 (fall 1953). On page 3:

“Summer 54: To suit Phileas Chevalley, Sammy Fogg and eventual Mexicans and Colombians, this Congress will be held from August 17 to 30. Samuel will look for a hotel in Auvergne, but everyone is asked to also prospect the hotels in his region.”

One should recall that the ICM 1954 was held in Amsterdam from September 2nd-9th. It was convenient for Chevalley and Eilenberg (who were in the US) and for possible more foreigners to have Bourbaki’s summer congress just before the ICM.

(Of course, Phileas Fogg is the main character in Jules Verne’s Around the World in 80 days.)

A lot of people attended the Murols-meeting (La Tribu 34, ‘Congres super-oecumenique du frigidaire et des revetements troues’).

Apart from the regular crowd (Cartan, Chevalley, Delsarte, Dieudonne, Dixmier, Godement, Koszul, Sammy (=Eilenberg), Samuel, Schwartz, Serre and Weil), there was a guinea-pig (Serge Lang), an ‘efficiency expert’ (Saunders MacLane), two ‘foreign visitors’ (Hochschild and John Tate) and two ‘honorable foreign visitors’ (Iyanaga and Kosaku Yosida).

Probably because of this, extremely detailed travel instructions were given in La Tribu 33 (page 2):

“Next congress: will be held at the Hotel des Pins, in Murols (Puy-de-Dome) from August 17 to 30.
There is at least one night train departing from Paris, going to Clermont or Issoire, followed by a bus-ride to Murols; details will be given as soon as we know the summer schedules.
For motorized people not coming from the South by the N.9, nor from the West by the N.89: go to Clermont-Ferrand, leave it by the N.9 (route d’Issoire), turn right about 17 kms further (after the village of Veyre) to take the N.678 towards Champeix; in Champeix take (on the right) the N.496 (direction of St-Nectaire, Murols and Mont-Dore).
For those coming from the South by the N.9: turn left at Issoire to take the N.496 towards Campais, St-Nectaire, Murols. For those coming from the West by the N.89: leave it a little before Lequeille to take (on the right) the N.122, turn left 2 km further to take the N.496 towards Mont-Dore, the Chambon lake and Murols (road continuing towards Champeix and Issoire).”

If you follow this route on the map, you’ll know that the congress was not held in Murols (departement de l’Aveyron), but in Murol (departement du Puy-de-Dome).

This time we do not have to search long for the place of venue as Hotel des Pins a Murol is still in operation.

Note the terras on the first floor, and the impressive line of trees in front of the hotel.

At first I felt frustrated as I couldn’t figure out where this well-known photograph of the Murol-meeting was taken.

From left to right, Godement, Dieudonne, Weil, MacLane, and a smug looking Serre (he knew he would be awarded a Fields medal in a few days time).

Today it is impossible to have this view from the hotel-terras because of the trees in front. Still, the picture was taken from the terras, and the imposing building in the background is the late Turing Hotel in Murol.

Here’s a picture of it with the Hotel des Pins in the background.

We’ve encountered the Murol-congress before on this blog when trying to piece together the history of the Yoneda lemma (Iyanaga was Yoneda’s Ph.D. advisor, and probably on his advice MacLane met with Yoneda in the Gare du Nord to hear about his lemma).

On MacLane’s role as ‘efficiency expert’ we have this in La Tribu 34:

“Frightened by the disorder of the discussions, some members had brought a world-renowned efficiency expert from Chicago. This one, armed with a hammer, tried hard and with good humor, but without much result. He quickly realized that it was useless, and turned, successfully this time, to photography.”

As we’ve seen in Amboise and Pelvoux, Bourbaki likes to have His summer venues close to places of great sentimental value.

Murol is very close to Besse-et-Saint Anastaise, the place of the very first Bourbaki-meeting in 1935.

As always, this asks for a little pilgrimage. From La Tribu 34 (page 2):

“Despite the incessant rain, Bourbaki was attracted by the waters, and went to explore lots of Auvergnian lakes. Besse and its Lac Pavin were naturally entitled to a pilgrimage. Courageous founding-fathers and lower-members, braving the rain and fog, rushed across to the lake of Guery where their dripping pants aroused the suspicions of a bar maid, and beat the motorized elements there by several lengths. Others swam and rowed. Even the Japanese were entitled to their lake.”