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Tag: Weil

9 Bourbaki founding members, really?

The Clique (Twenty Øne Piløts fanatic fanbase) is convinced that the nine Bishops of Dema were modelled after the Bourbaki-group.

It is therefore of no surprise to see a Photoshopped version circulating of this classic picture of some youthful Bourbaki-members (note Jean-Pierre Serre poster-boying for Elon Musk’s site),

replacing some of them with much older photos of other members. Crucial seems to be that there are just nine of them.

I don’t know whether the Clique hijacked Bourbaki’s Wikipedia page, or whether they were inspired by its content to select those people, but if you look at that Wikipedia page you’ll see in the right hand column:


  • Henri Cartan
  • Claude Chevalley
  • Jean Coulomb
  • Jean Delsarte
  • Jean Dieudonné
  • Charles Ehresmann
  • René de Possel
  • André Weil

Really? Come on.

We know for a fact that Charles Ehresmann was brought in to replace Jean Leray, and Jean Coulomb to replace Paul Dubreil. Surely, replacements can’t be founders, can they?

Well, unfortunately it is not quite that simple. There’s this silly semantic discussion: from what moment on can you call someone a Bourbaki-member…

The collective name ‘Nicolas Bourbaki’ was adopted only at the Bourbaki-congress in Besse in July 1935 (see also this post).

But, before the Besse-meeting there were ten ‘proto-Bourbaki’ meetings, the first one on December 10th, 1934 in Cafe Capoulade. These meetings have been described masterly by Liliane Beaulieu in A Parisian Cafe and Ten Proto-Bourbaki Meetings (1934-35) (btw. if you know a direct link to the pdf, please drop it in the comments).

During these early meetings, the group called itself ‘The Committee for the Treatise on Analysis’, and not yet Bourbaki, whence the confusion.

Do we take the Capoulade-1934 meeting as the origin of the Bourbaki group (in which case the founding-members would be Cartan, Chevalley, De Possel, Delsarte, Dieudonne, and Weil), or was the Bourbaki-group founded at the Besse-congress in 1935 (when Cartan, Chevalley, Coulomb, De Possel, Dieudonne, Mandelbrojt, and Weil were present)?

Here’s a summary of which people were present at all meetings from December 1934 until the second Chancay-congress in September 1939, taken from Gatien Ricotier ‘Projets collectifs et personnels autour de Bourbaki dans les années 1930 à 1950′:

07-1935 is the Besse-congress, 09-1936 is the ‘Escorial’-congress (or Chancay 1) and 09-1937 is the second Chancay-congress. The ten dates prior to July 1935 are the proto-Bourbaki meetings.

Even though Delsarte was not present at the Besse-1935 congress, and De Possel moved to Algiers and left Bourbaki in 1941, I assume most people would agree that the six people present at the first Capoulade-meeting (Cartan, Chevalley, De Possel, Delsarte, Dieudonne, and Weil) should certainly be counted among the Bourbaki founding members.

What about the others?

We can safely eliminate Dubreil: he was present at just one proto-Bourbaki meeting and left the group in April 1935.

Also Leray’s case is straightforward: he was even excluded from the Besse-meeting as he didn’t contribute much to the group, and later he vehemently opposed Bourbaki, as we’ve seen.

Coulomb’s role seems to restrict to securing a venue for the Besse-meeting as he was ‘physicien-adjoint’ at the ‘Observatoire Physique du Globe du Puy-de-Dome’.

Because of this he could rarely attend the Julia-seminar or Bourbaki-meetings, and his interest in mathematical physics was a bit far from the themes pursued in the seminar or by Bourbaki. It seems he only contributed one small text, in the form of a letter. Due to his limited attendance, even after officially been asked to replace Dubreil, he can hardly be counted as a founding member.

This leaves Szolem Mandelbrojt and Charles Ehresmann.

We’ve already described Mandelbrojt as the odd-man-out among the early Bourbakis. According to the Bourbaki archive he only contributed one text. On the other hand, he also played a role in organising the Besse-meeting and in providing financial support for Bourbaki. Because he was present already early on (from the second proto-Bourbaki meeting) until the Chancay-1937 meeting, some people will count him among the founding members.

Personally I wouldn’t call Charles Ehresmann a Bourbaki founding member because he joined too late in the process (March 1936). Still, purists (those who argue that Bourbaki was founded at Besse) will say that at that meeting he was put forward to replace Jean Leray, and later contributed actively to Bourbaki’s meetings and work, and for that reason should be included among the founding members.

What do you think?

How many Bourbaki founding members are there? Six (the Capoulade-gang), seven (+Mandelbrojt), eight (+Mandelbrojt and Ehresmann), or do you still think there were nine of them?

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TØP PhotoShop mysteries

Suppose you’re writing a book, and for the duration of that project you keep a certain photo as your desktop-background. I guess we might assume that picture to be inspirational for your writing process.

If you PhotoShopped it to add specific elements, might we assume these extra bits to play a crucial role in your story?

Now, let’s turn to Twenty One Pilots and the creation process of their album Trench, released on October 5, 2018

We know from this tweet (from August 19th, 2018) that Tyler Joseph’s desktop-background picture was a photoshopped version of the classic Bourbaki-1938 photo on the left below, given it Trench-yellow, and added a bearded man in the doorway (photo on the right)

And we know from this interview (from September 5th, 2018) that, apart from the bearded man, he also replaced in the lower left corner the empty chair by a sitting person (lower photo).

The original photo features on the Wikipedia page on Nicolas Bourbaki, and as Tyler Joseph has revealed that Blurryface‘s real name is Nicolas Bourbaki (for whatever reason), and that he appears in the lyrics of Morph on Trench, this may make some sense.

But, of the seven people in the picture only three were founding members of Bourbaki: Weil, Diedonne and Delsarte. Ehresmann entered later, replacing Jean Leray, and Pison and Chabauty were only guinea pigs at that moment (they later entered Bourbaki, Chabauty briefly and Pison until 1950), and finally, Simonne Weil never was a member.

There’s another strange thing about the original picture. All of them, but Andre and Simone Weil, look straight into the camera, the Weil’s seem to be more focussed on something happening to the right.

Now, TØP has something with the number 9. There are nine circles on the cover of Blurryface (each representing one of a person’s insecurities, it seems), there are nine towers in the City of Dema, nine Bishops, etc.

So, from their perspective it makes sense to Photoshop two extra people in, and looking at the original there are two obvious places to place them: in the empty doorway, and on the empty chair.

But, who are they, and what is their significance?

1. The bearded man in the doorway

As far as I know, nobody knows who he is. From a Bourbaki point of view it can only be one person: Elie Cartan.

We know he was present at the 1938 Bourbaki Dieulefit/Beauvallon meeting, and that he was kind of a father figure to Bourbaki. Among older French mathematicians he was one of few (perhaps the only one) respected by all of Bourbaki.

But, bearded man is definitely not Elie Cartan…

If bearded man exists and has a Wikipedia page, the photo should be on that page. So, if you find him, please leave a comment.

Previous in this series I made a conjecture about him, but I’m not at all sure.

2. Why, of all people, Szolem Mandelbrojt?

We know from this Twentyonepilots subReddit post that the man sitting on the previously empty chair in none other than Bourbaki founding member Szolem Mandelbrojt, shopped in from this other iconic early Bourbaki-photo from the 1937 Chancay-meeting.

Let me tell you why this surprises me.

Szolem Mandelbrojt was atypical among the first Bourbaki-gang in many ways: he was the only one who didn’t graduate from the ENS, he was a bit older than the rest, he was the only one who was a full Professor (at Clermont-Ferrand) whereas the others were ‘maitre de conference’, he was the only one who didn’t contribute actively in the Julia seminar (the proto-Bourbaki seminar) nor much to the Bourbaki-congresses either for that matter, etc. etc.

Most of all, I don’t think he would feel particularly welcome at the 1938 congress. Here’s why.

(Jacques Hadamard (left), and Henri Lebesgue (right))

From Andre Weil’s autobiography (page 120):

Hadamard’s retirement left his position open. I thought myself not unworthy of succeeding him; my friends, especially Cartan and Delsarte, encouraged me to a candidate. It seemed to me that Lebesgue, who was the only mathematician left at the College de France, did not find my candidacy out of place. He even let me know that it was time to begin my ‘campaign visits’.

But the Bourbaki-campaign against a hierarchy of scientific prizes instituted by Jean Perrin (the so called ‘war of the medals’) interfered with his personal campaign. (Perhaps more important was that Mandelbrojt did his Ph.D. under Hadamard…)

Again from Weil’s autobiography (page 121):

Finally Lebesque put an end to my visits by telling me that he had decided on Mandelbrojt. It seemed to me that my friends were more disappointed than I at this outcome.

In the spring of 1938, Mandelbrojt succeeded Hadamard at the College de France.

There’s photographic evidence that Mandelbrojt was present at the 1935 Besse-congress and clearly at the 1937-Chancay meeting, but I don’t know that he was even present at Chancay-1936.

The only picture I know of that meeting is the one below. Standing on bench: Chevalley’s nephews, seated Andre Weil and Chevalley’s mother; standing, left to right: Ninette Ehresmann, Rene de Possel, Claude Chavalley, Jacqueline Chavalley, Mirles, Jean Delsarte and Charles Ehresmann.

Of all possible people, Szolem Mandelbrojt would be the miscast at the 1938-meeting. So, why did they shop him in?

– convenience: they had an empty chair in the original picture, another Bourbaki-photo with a guy sitting on such a chair, so why not shop him in?

– mistaken identity: in the subReddit post the sitting guy was mistakenly identified as Claude Chevalley. Now, there is a lot to say about wishing to add Chevalley to the original. He is by far the most likeable of all Bourbakis, so if these nine were ever supposed to be the nine Bishops of Dema, he most certainly would be Keons. But, Chevalley was already in the US at that time, and was advised by the French consul to remain there in view of the situation in Europe. As a result, Chevalley could not obtain a French professorship before the early 50ties.

– a deep hidden clue: remember all that nonsense about Josh Dun’s ‘alma mater’ being that Ukrainian building where Nico and the niners was shot? Well, Szolem Mandelbrojt’s alma mater was the University of Kharkiv in Ukraine. See this post for more details.

3. Is it all about Simone Weil?

If you super-impose the two photographs, pinning Mandelbrojt in both, the left border of the original 1938-picture is an almost perfect mirror for both appearances of Simone Weil. Can she be more important in all of this than we think?

Previous in the Bourbaki&TØP series:

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Dema2Trench, AND REpeat

There’s this band Twenty One Pilots and they’ve woven a complicated story around some of their albums, notably Blurryface, Trench, and Scaled and Icy.

Since Trench, an important component of that story is the Bourbaki group, so I’m just curious whether the few things I know about them can help to clarify parts in the TØP- storyline.

Pretty pointless, I know, as no artistic project will follow blindly historical facts. But hey, as long as I discover new things I’ll keep going.

The story’s about a City of Dema, ruled by nine Bishops installing a terror regime called Vialism, and a land outside the city walls, called Trench, to which citizens would like to escape.

Think of Dema as an extremely toxic environment from which you need to escape to a safe place, let’s call it Trench.

Sadly, too often survivors from abusive settings later on create their own toxic environment, abusive to others.

So, Dema-escapees to Trench should always be wary of the danger of creating a new Dema for others.

It is very hard to break these Dema2Trench cycles of violence. That’s probably why the map of the City of Dema is circular.

Let’s start with these two photos not (yet) in the Dema-lore:

Both pictures are of the French mathematician Gaston Julia.

Julia graduated from the ENS in 1914, so was among the worst victims of the military regime Lavisse installed at the ENS. He was mobilised but hadn’t yet completed his second year of military training. That was shortened to just 5 months, after which we has send as a second lieutenant to the war.

In January 1915 he was seriously wounded in his face, had to undergo a series of operations and for the rest of his life he resigned himself to wearing a leather strap around the area where his nose had been.

He ran a weekly seminar from 1933 till 1939, the Seminaire Julia, to which the Bourbaki core members contributed a vast number of lectures.

Until 1937-38 (so just before the Dieulefit Bourbaki congress) the Bourbakis felt happy citizens of Julia’s Seminar/Dema. But then they discovered his political agenda and were expelled from it, or escaped from it depending on the version.

Jean Leray convinced Julia that it was a terrible mistake to let his seminar run by Bourbaki, and that things would go much better if he ran it. Julia expelled Bourbaki from the seminar, changed its name to ‘ Cercle mathématique de l’École normale supérieure’ and moved the venue from the IHP to the ENS. The attendants of this seminar were younger and less international that in the preceding years, hence more malleable to his political ideas.

Another reason for the break-up between Bourbaki and Julia was that they reproached him of attending in June 1937 the festivities of the bicentennial of the University of Gottingen, which were seen as pure propaganda for the Nazi-regime.

During WW2, Julia collaborated with the occupying Nazi-regime in that he tried to find French mathematicians to contribute to the Zentralblatt. After the war he was briefly suspended for this.

Much more on the Julia seminar and the break-up with Bourbaki can be read in the thesis by Gatien Ricotier ‘Projets collectifs et personnels autour de Bourbaki dans les années 1930 à 1950’, and Michele Audin’s book on the Julia Seminar.

Let us compare Julia’s photographs to these two in Dema-lore:

Is it a coincidence that Clancy in Trench has a scar on his nose? Is it a coincidence that the black paint on some of the Bishop’s faces looks a lot like Julia’s mask?

Can it be that victims of one Dema-era become Bishops in a next era?

This repetitiveness of Dema-environments also indicates the importance of Bishop Andre. Recall that all the Bishops’ names (except for Nico) come from concatenations of word-parts in the lyrics of the songs on the Blurryface album.

ANDRE comes from ‘..AND REpeat’ in Fairly local:

Tomorrow I’ll keep a beat
And repeat yesterday’s dance

In view of this, let’s have another look at the two Bourbaki-related photographs that appeared in the run up to the Trench-album:

On the left is the photo of the Dieulefit/Beauvallon 1938 meeting, which is on the Bourbaki Wikipedia page, and was on the desktop of Tyler Joseph.

On the right a photo of Andre Weil together with a girl, according to Wikipedia the picture dates from 1956. I’m pretty certain it was taken in the summer of 1957, and that the girl is Mireille Cartan, the second youngest daughter of Henri Cartan. Not that any of this matters, TØP-wise. A clipping of the girl was among the material originally posted at the site.

In 1938, Andre Weil was a victim of Lavisse’s Dema. His year was the last one getting a military training to become reserve officers in the French infantry/artillery (as were Cartan, Dieudonne and Delsarte).

When France would mobilise they were forced to return to Dema (military service) and lead their bataljons as second lieutenants into war. All of them, except for Weil, did this.
Weil escaped to Trench (Finland), and was taken back to Dema, and imprisonment.

In 1957, Bourbaki dominated much of French mathematical life, and certainly its influence in Paris was suffocating for aspiring math-students. A good read on this is Jacques Roubaud’s Mathematique.

Bourbaki has turned French mathematics (and beyond) into its own Dema, and Andre Weil certainly was one of the more important Bishops of it.

Previous in the Bourbaki&TØP series:

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Weil photos used in Dema-lore

On April 20th of 2018, twenty one pilots updated their store page to include a video with a hidden message at the end of it.

and with a bit of sleuthing it led to a page on the site containing:

This was immediately identified as part of the photo on the right, which is on the French Wikipedia page for Andre Weil.

The photo is clipped in such a way one cannot be certain whether the child is a boy or girl, so a logical explanation is that this is supposed to be the nine year old Clancy, shielding his eyes from the violence (vialism) he just discovered in Dema.

The full picture suggests that Clancy’s struggles might mirror some in Andre Weil’s life.

Andre Weil was born May 6th, 1906, so ‘in his ninth year’ World War 1 breaks out in 1914.

Last time we’ve seen that Bourbaki’s Dema = Ecole Normal Superieure in Paris during WW1, Vialism = militant patriotism sending ENS-graduates as trained reserve second lieutenants in the infantry to the trenches, and there getting killed ‘pour la patrie’ and the glory of the ENS and its director Ernest Lavisse, “L’instituteur national”.

Here’s a G-translation of his letter to the young French, published September 23rd 1914:

Dear children of France, You will be old one day, and, like the old, you will like to remember times past. There will come evenings when your little children, seeing you dreaming, will say to you: Tell us, grandfather. And you will tell. It will be a few episodes of the war, a long march, an alert, a bayonet assault, a cavalry charge, the feat of a battery of 75, the strewn enemy dead on the plain, or else, in the streets of a city, the serried ranks of corpses left standing for lack of room to fall; and then the death of comrades, the terrible losses of your company and your regiment, your wounds received in Belgium, in Champagne, on the banks of the Rhine, beyond the Rhine; but the joy of victories, the poles knocked down on too narrow frontiers, triumphal entries.

On those evenings, after the amazed children have gone to sleep, you will open a drawer where you will have collected precious objects, a bullet extracted from a wound, a piece of shell, a cloth where your blood will have turned pale, a cross of honour, I hope, or a military medal, at the very least a medal from the 1914 war, on the ribbon of which the silver clasps will bear the names of immortal battles. And whatever your life, happy or unhappy, you will be able to say: I lived great days such as the history of men had not yet seen. And you will be right to be proud of your youth, because you are sublime young people!

I have read your letters; I have spoken with the wounded. Through you, I know what heroism is. I had heard a lot about it, being a historian by profession, but now I see it, I touch it, and how beautiful your heroism is, embellished with grace and smiling in the French way! Young soldiers if you were given one chevron per battle, your march would not be enough to accommodate them, because at the end of the war you would count more chevrons than years;

Young soldiers you are glorious old warriors.

Oh! Thanks thanks! Thank you for the beautiful end of life that you give to the elderly who, for forty-four years have suffered so much from the abasement of the fatherland.

The 44 years refers to the Franco-Prussian war of 1870 in which Bourbaki (the general) played a dramatic role.

The next cycle of militant patriotism occurred in the years leading up to the second world war. Here, Andre Weil’s experiences mirror those of Clancy. He tried several times to escape, first from military action (although he too was a reserve officer in the French army), then from France itself. He was captured in Finland, brought back to France to face trial and imprisonment, was released on the condition that he did active military duty, escaped with the French army to England, there demobilised he refused to join de Gaulle’s troops, left England on a boat to Marseille, from where he escaped to the US.

All this, and much more, you can read in his autobiography The Apprenticeship of a Mathematician, especially Chapter VI, The War and I: A Comic Opera in Six Acts.

(for TØP-ers, note the Bishop-red cover…)

Comic or not, the book tries to ‘explain’ his actions in those years, but failed to convince the French from offering him a professorship at a French university after the war.

Perhaps it may be worth looking into a comparison between Weil’s autobiography and the collected Clancy letters.

I guess that’s the best I can do to explain the use of that Weil photo by TØP. Surely they didn’t search any deeper as to where and when this picture was taken, or who the girl was next to Weil.

In case anyone might be interested, I’ll be happy to explain my own theory about this in another post.

I’m sure the full photograph ended up in the ‘Trench-bible’, given to the director of their clip-movies. The scenery is used at the end of the Jumpsuit video when ‘Clancy’ takes out a jumpsuit from the burning car and walks away along a road very similar to that in the photo.

The boy/girl shielding his/her eyes for the violence, should have been used at about minute one into the Outside video

Now, there’s another Weil (or rather Bourbaki) photograph we know did inspire Twenty One Pilots, the classic picture at the Dieulefit/Beauvallon 1938 Bourbaki-congress

which was photoshopped in order to get Szolem Mandelbrojt in from the Chancay (quite similar to Clancy now that i type this) 1937 Bourbaki congress

Now, these were the only two Bourbaki-meetings Simone Weil (Andre’s sister) attended, and she features prominently in both pictures.

Probably this brother/sister thing struct a chord with Twenty One Pilots. But then, you quickly end up with this iconic picture of both of them, taken in the summer of 1922, just before Andre entered the Ecole Normale (he entered the ENS at age 16…)

I’d love to be send a copy of the ‘Trench bible’ because I’m fairly certain also this photograph is in it. At the end of the Nico and the niners-video you see this boy and girl (who may be around age 9 and discover the truth about Dema) finding a jumpsuit with the Bishops approaching

and they reappear a bit older at the end of the Outside-video, with a burning Dema in the background.

Previous in the Bourbaki&TØP series:

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From Weil’s foundations to schemes

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’.

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The birthplace of schemes

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

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.

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Le Guide Bourbaki : Marlotte

During the 1950ties, the Bourbakistas usually scheduled three meetings in the countryside. In the spring and autumn at places not too far from Paris (Royaumont, Celles-sur-plaines, Marlotte, Amboise…), in the summer they often went to the mountains (Pelvoux, Murols, Sallieres-les-bains,…).

Being a bit autistic, they preferred to return to the same places, rather than to explore new ones: Royaumont (6 times), Pelvoux (5 times), Celles-sur-plaine (4 times), Marlotte (3 times), Amboise (3 times),…

In the past, we’ve tried to pinpoint the exact locations of the pre-WW2 Bourbaki-conferences: in 1935 at le Station Biologique de l’Université Blaise Pascal’, Rue du Lavoir, Besse-et-Saint-Anastaise, in 1936 and 1937 at La Massotterie in Chancay, and in 1938 at l’ecole de Beauvallon (often mistakingly referred to as the ‘Dieulefit-meeting’).

Let’s try to do the same for their conferences in the 1950ties. Making use of the recent La Tribu releases for he period 1953-1960, let’s start arbitrarily with the 1955 fall meeting in Marlotte.

Three conferences were organised in Marlotte during that period:

  • La Tribu 37 : ‘Congres de la lune’, October 23-29 1955
  • La Tribu 43 : ‘Congres de la deuxieme lune’, October 6-11 1957
  • La Tribu 44 : ‘Congres des minutes de silence’, March 16-22 1958

Grothendieck was present at all three meetings, Weil at the last two. But let us return to the fight between these two (‘congres des minutes de silence’) regarding algebraic geometry/category theory in another post.

Today we’ll just focus on the location of these meetings. At first, this looks an easy enough task as on the opening page of La Tribu we read:

“The conference was held at the Hotel de la mare aux canards’ (‘Hotel of the duck pond’) in Marlotte, near Fontainebleau, from October 23rd till 29th, 1955”.

Just one little problem, I can’t find any reference to a ‘Hotel de la Mare aux Canards’ in Marlotte, neither at present nor in the past.

Nowadays, Bourron-Marlotte is mainly a residential village with no great need for lodgings, apart from a few ‘gites’ and a plush hotel in the local ‘chateau’.

At the end of the 19th century though, there was an influx of painters, attracted by the artistic ‘colonie’ in the village, and they needed a place to sleep, and gradually several ‘Auberges’ and Hotels opened their doors.

Over the years, most of these hotels were demolished, or converted to family houses. The best list of former hotels in Marlotte, and their subsequent fate, I could find is L’essor hôtelier de Bourron et de Marlotte.

There’s no mention of any ‘Hotel de la mare aux canards’, but there was a ‘Hotel de la mare aux fées’ (Hotel of the fairy pond), which sadly was demolished in the 1970ties.

There’s little doubt that this is indeed the location of Bourbaki’s Marlotte-meetings, as the text on page one of La Tribu 37 above continues as (translation by Maurice Mashaal in ‘Bourbaki a secret society of mathematicians’, page 109):

“Modest and subdued sunlight, lustrous bronze leaves fluttering in the wind, a pond without fairies, modules without end, indigestible stones, and pierced barrels: everything contributes to the drowsiness of these blasé believers. ‘Yet they are serious’, says the hotel-keeper, ‘I don’t know what they are doing with all those stones, but they’re working hard. Maybe they’re preparing for a journey to the moon’.”

Bourbaki didn’t see any fairies in the pond, only ducks, so for Him it was the Hotel of the duck pond.

In fact La mare aux fées is one of the best known spots in the forest of Fontainebleau, and has been an inspiration for many painters, including Pierre-August Renoir:

Here’s the al fresco restaurant of the Hotel de la mare aux fées:

Both photographs are from the beginning of the 20th century, but also in the 50ties it was a Hotel of some renown as celebreties, including the actor Jean Gabin, stayed there.

The exact location of the former Hotel de la mare aux fées is 83, Rue Murger in Bourron-Marlotte.

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Princeton’s own Bourbaki

In the first half of 1937, Andre Weil visited Princeton and introduced some of the postdocs present (notably Ralph Boas, John Tukey, and Frank Smithies) to Poldavian lore and Bourbaki’s early work.

In 1935, Bourbaki succeeded (via father Cartan) to get his paper “Sur un théorème de Carathéodory et la mesure dans les espaces topologiques” published in the Comptes Rendus des Séances Hebdomadaires de l’Académie des Sciences.

Inspired by this, the Princeton gang decided to try to get a compilation of their mathematical ways to catch a lion in the American Mathematical Monthly, under the pseudonym H. Petard, and accompanied by a cover letter signed by another pseudonym, E. S. Pondiczery.

By the time the paper “A contribution to the mathematical theory of big game hunting” appeared, Boas and Smithies were in cambridge pursuing their postdoc work, and Boas reported back to Tukey: “Pétard’s paper is attracting attention here,” generating “subdued chuckles … in the Philosophical Library.”

On the left, Ralph Boas in ‘official’ Pondiczery outfit – Photo Credit.

The acknowledgment of the paper is in true Bourbaki-canular style.

The author desires to acknowledge his indebtedness to the Trivial Club of St. John’s College, Cambridge, England; to the M.I.T. chapter of the Society for Useless Research; to the F. o. P., of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.

The Trivial Club of St. John’s College probably refers to the Adams Society, the St. John’s College mathematics society. Frank Smithies graduated from St. John’s in 1933, and began research on integral equations with Hardy. After his Ph. D., and on a Carnegie Fellowship and a St John’s College studentship, Smithies then spent two years at the Institute for Advanced Study at Princeton, before returning back ‘home’.

In the previous post, I assumed that Weil’s visit to Cambridge was linked to Trinity College. This should probably have been St. John’s College, his contact there being (apart from Smithies) Max Newman, a fellow of St. John’s. There are two letters from Weil (summer 1939, and summer 1940) in the Max Newman digital library.

The Eagle Scanning Project is the online digital archive of The Eagle, the Journal of St. John’s College. Last time I wanted to find out what was going on, mathematically, in Cambridge in the spring of 1939. Now I know I just had to peruse the Easter 1939 and Michaelmas 1939 volumes of the Eagle, focussing on the reports of the Adams Society.

In the period Andre Weil was staying in Cambridge, they had a Society Dinner in the Music Room on March 9th, a talk about calculating machines (with demonstration!) on April 27th, and the Annual Business Meeting on May 11th, just two days before their punting trip to Grantchester,

The M.I.T. chapter of the Society for Useless Research is a different matter. The ‘Useless Research’ no doubt refers to Extrasensory Perception, or ESP. Pondiczery’s initials E. S. were chosen with a future pun in mind, as Tukey said in a later interview:

“Well, the hope was that at some point Ersatz Stanislaus Pondiczery at the Royal Institute of Poldavia was going to be able to sign something ESP RIP.”

What was the Princeton connection to ESP research?

Well, Joseph Banks Rhine conducted experiments at Duke University in the early 1930s on ESP using Zener cards. Amongst his test-persons was Hubert Pearce, who scored an overall 40% success rate, whereas chance would have been 20%.

Pearce and Joseph Banks Rhine (1932) – Photo Credit

In 1936, W. S. Cox tried to repeat Rhine’s experiment at Princeton University but failed. Cox concluded “There is no evidence of extrasensory perception either in the ‘average man’ or of the group investigated or in any particular individual of that group. The discrepancy between these results and those obtained by Rhine is due either to uncontrollable factors in experimental procedure or to the difference in the subjects.”

As to the ‘MIT chapter of the society for useless research’, a chapter usually refers to a fraternity at a University, but I couldn’t find a single one on the list of MIT fraternities involved in ESP, now or back in the late 1930s.

However, to my surprise I found that there is a MIT Archive of Useless Research, six boxes full of amazing books, pamphlets and other assorted ‘literature’ compiled between 1900 and 1940.

The Albert G. Ingalls pseudoscience collection (its official name) comprises collections of books and pamphlets assembled by Albert G. Ingalls while associate editor of Scientific American, and given to the MIT Libraries in 1940. Much of the material rejects contemporary theories of physical sciences, particularly theoretical and planetary physics; a smaller portion builds upon contemporary science and explores hypotheses not yet accepted.

I don’t know whether any ESP research is included in the collection, nor whether Boas and Tukey were aware of its existence in 1938, but it sure makes a good story.

The final riddle, the F. o. P., of Princeton University is an easy one. Of course, this refers to the “Friends of Pondiczery”, the circle of people in Princeton who knew of the existence of their very own Bourbaki.

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Cambridge, spring 1939

One of the few certainties we have on the Bourbaki-Petard wedding invitation is that it was printed in, and distributed out of Cambridge in the spring of 1939, presumably around mid April.

So, what was going on, mathematically, in and around Trinity and St. John’s College, at that time?

Well, there was the birth of Eureka, the journal of the Archimedeans, the mathematical society of the University of Cambridge. Eureka is one of the oldest recreational mathematics publications still in existence.

Since last year the back issues of Eureka are freely available online, unfortunately missing out the very first two numbers from 1939.

Ralph Boas, one of the wedding-conspirators, was among the first to contribute to Eureka. In the second number, in may 1939, he wrote an article on “Undergraduate mathematics in America”.

And, in may 1940 (number 4 of Eureka) even the lion hunter H. Petard wrote a short ‘Letter to the editors’.

But, no doubt the hottest thing that spring in Cambridge were Ludwig Wittgenstein’s ‘Lectures on the Foundations of Mathematics’. Wittgenstein was just promoted to Professor after G.E. Moore resigned the chair in philosophy.

For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation.

These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies.

Here’s a clip from the film Wittgenstein, directed by Derek Jarman.

Missing from the list of people attending Wittgenstein’s lectures is Andre Weil, a Bourbaki member and the principal author of the wedding invitation.

Weil was in Cambridge in the spring of 1939 on a travel grant from the French research organisation for visits to the UK and Northern Europe. At that time, Weil held a position at the University of Strasbourg, uncomfortably close to Nazi-Germany.

Weil not attending Wittgenstein’s lectures is strange for several reasons. Weil was then correcting the galley proofs of Bourbaki’s first ever booklet, their own treatment of set theory, which appeared in 1939.

But also on a personal level, Andre Weil must have been intrigued by Wittgenstein’s philosophy, as it was close to that of his own sister Simone Weil

There are many parallels between the thinkers Simone Weil and Ludwig Wittgenstein. They each lived in a tense relationship with religion, with both being estranged from their cultural Jewish ancestry, and both being tempted at various times by the teachings of Catholicism.

They both underwent a profound and transformative mystical turn early into their careers. Both operated against the backdrop of escalating global conflict in the early 20th century.

Both were concerned, amongst other things, with questions of culture, ethics, aesthetics, epistemology, science, and necessity. And, perhaps most notably, they both sought to radically embody their ideas and physically ‘live’ their philosophies.

From Between Weil and Wittgenstein

Andre and Simone Weil in Knokke-Zoute, 1922 – Photo Credit

Another reason why Weil might have been interested to hear Wittgenstein on the foundations of mathematics was a debate held in Paris of few months previously.

On February 4th 1939, the French Society of Philosophy invited Albert Lautman and Jean Cavaillès ‘to define what constitutes the ‘life of mathematics’, between historical contingency and internal necessity, describe their respective projects, which attempt to think mathematics as an experimental science and as an ideal dialectics, and respond to interventions from some eminent mathematicians and philosophers.’

Among the mathematicians present and contributing to the discussion were Weil’s brothers in arms, Henri Cartan, Charles Ehresmann, and Claude Chabauty.

As Chabauty left soon afterwards to study with Mordell in Manchester, and visited Weil in Cambridge, Andre Weil must have known about this discussion.

The record of this February 4th meeting is available here (in French), and in English translation from here.

Jean Cavaillès took part in the French resistance, was arrested and shot by the Nazis on April 4th 1944. Albert Lautman was shot by the Nazis in Toulouse on 1 August 1944.

Jean Cavailles (2nd on the right) 1903-1944 – Photo Credit

A book review of Wittgenstein’s Lectures on the Foundations of Mathematics by G. Kreisel is available from the Bulletin of the AMS. Curiously, Kreisel compares Wittgenstein’s approach to … Bourbaki’s very own manifesto L’architecture des mathématiques.

For all these reasons it is strange that Andre Weil apparently didn’t show much interest in Wittgenstein’s lectures.

Had he more urgent things on his mind, like prepping for a wedding?

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What’s Pippa got to do with the Bourbaki wedding?

Last time we’ve seen that on June 3rd 1939, the very day of the Bourbaki wedding, Malraux’ movie ‘L’espoir’ had its first (private) viewing, and we mused whether Weil’s wedding card was a coded invitation to that event.

But, there’s another plausible explanation why the Bourbaki wedding might have been scheduled for June 3rd : it was intended to be a copy-cat Royal Wedding…

The media-hype surrounding the wedding of Prince William to Pippa’s sister led to a hausse in newspaper articles on iconic royal weddings of the past.

One of these, the marriage of Edward VIII, Duke of Windsor and Wallis Warfield Spencer Simpson, was held on June 3rd 1937 : “This was the scandal of the century, as far as royal weddings go. Edward VIII had just abdicated six months before in order to marry an American twice-divorced commoner. The British Establishment at the time would not allow Edward VIII to stay on the throne and marry this woman (the British Monarch is also the head of the Church of England), so Edward chose love over duty and fled to France to await the finalization of his beloved’s divorce. They were married in a private, civil ceremony, which the Royal Family boycotted.”

But, what does this wedding have to do with Bourbaki?

For starters, remember that the wedding-card-canular was concocted in the spring of 1939 in Cambridge, England. So, if Weil and his Anglo-American associates needed a common wedding-example, the Edward-Wallis case surely would spring to mind. One might even wonder about the transposed symmetry : a Royal (Betti, whose father is from the Royal Poldavian Academy), marrying an American (Stanislas Pondiczery).

Even Andre Weil must have watched this wedding with interest (perhaps even sympathy). He too had to wait a considerable amount of time for Eveline’s divorce (see this post) to finalize, so that they could marry on october 30th 1937, just a few months after Edward & Wallis.

But, there’s more. The royal wedding took place at the Chateau de Cande, just south of Tours (the A on the google-map below). Now, remember that the 2nd Bourbaki congress was held at the Chevalley family-property in Chancay (see the Escorial post) a bit to the north-east of Tours (the marker on the map). As this conference took place only a month after the Royal Wedding (from 10th till 20th of July 1937), the event surely must have been the talk of the town.

Early on, we concluded that the Bourbaki-Petard wedding took place at 12 o’clock (‘a l’heure habituelle’). So did the Edward-Wallis wedding. More precisely, the civil ceremony began at 11.47 and the local mayor had to come to the castle for the occasion, and, afterwards the couple went into the music-room, which was converted into an Anglican chapel for the day, at precisely 12 o’clock.

The emphasis on the musical organ in the Bourbaki wedding-invitation allowed us to identify the identity of ‘Monsieur Modulo’ to be Olivier Messiaen as well as that of the wedding church. Now, the Chateau de Cande also houses an impressive organ, the Skinner opus 718 organ.

For the wedding ceremony, Edward and Wallis hired the services of one of the most renowned French organists at the time : Marcel Dupre who was since 1906 Widor’s assistent, and, from 1934 resident organist in the Saint-Sulpice church in Paris. Perhaps more telling for our story is that Dupre was, apart from Paul Dukas, the most influential teacher of Olivier Messiaen.

On June 3rd, 1937 Dupre performed the following pieces. During the civil ceremony, an extract from the 29e Bach cantate, canon in re-minor by Schumann and the prelude of the fugue in do-minor of himself. When the couple entered the music room he played the march of the Judas Macchabee oratorium of Handel and the cortege by himself. During the religious ceremony he performed his own choral, adagium in mi-minor by Cesar Franck, the traditional ‘Oh Perfect Love’, the Jesus-choral by Bach and the toccata of the 5th symphony of Widor. Compare this level of detail to the minimal musical hint given in the Bourbaki wedding-invitation

“Assistent Simplexe de la Grassmannienne (lemmas chantees par la Scholia Cartanorum)”

This is one of the easier riddles to solve. The ‘simplicial assistent of the Grassmannian’ is of course Hermann Schubert (Schubert cell-decomposition of Grassmannians). But, the composer Franz Schubert only left us one organ-composition : the Fugue in E-minor.

I have tried hard to get hold of a copy of the official invitation for the Edward-Wallis wedding, but failed miserably. There must be quite a few of them still out there, of the 300 invited people only 16 showed up… You can watch a video newsreel film of the wedding.

As Claude Chevalley’s father had an impressive diplomatic career behind him and lived in the neighborhood, he might have been invited, and, perhaps the (unused) invitation was lying around at the time of the second Bourbaki-congress in Chancay,just one month after the Edward-Wallis wedding…

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