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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…

Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened.

3. When did Mazur write his unpublished preprint?

According to his own website, Barry Mazur did write the paper Remarks on the Alexander polynomial in 1963 or 1964. A quick look at the references gives us a coarse lower- and upper-estimate.

Apart from a paper by Iwasawa and one by Milnor, all references predate 1962 giving us a lower-bound. More interesting is reference (14) to David Mumford’s Geometric Invariant Theory (GIT) which was first published in 1965 and is referred to as ‘in preparation’, so the paper was written no later than 1965. If we look a bit closer we see than some GIT-references are very precise

indicating that Mazur must have had the final version of GIT to consult, making it rather difficult to believe that the preprint was written late 1963 or early 1964.

Mazur’s dating of the preprint is probably based on this penciled note on the frontpage of the only surviving copy of the preprint

It reads : “Date from about 63/64, H.R. Morton”. Hugh Morton of Liverpool University confirms that it is indeed his writing on the preprint.

Further, he told me that early 64 Christopher Zeeman held a Topology Symposium in Cambridge UK, where Hugh was a graduate student at the time and, as far as he could recall, Mazur attended that conference and gave him the preprint on that occasion, whence the 63/64 dating. Hugh kindly offered to double-check this with Terry Wall who cannot remember Mazur attending that particular conference.

In fact, we will see that a more correct dating of the Mazur-preprint will be : late 1964 or early 1965.

4. The birthday : July 10th 1964

Clearly, Mumford’s insight predates the Mazur-preprint. In the first section, Mazur mentions ‘Grothendieck cohomology groups’ rather than ‘Etale cohomology groups’.

At the time, Artin’s seminar notes on Grothendieck topologies (spring 1962) were widely distributed, and Artin and Grothendieck were in the process of developing etale cohomology in their Paris 1963/64 seminar SGA 4, while Mumford was working on GIT in Harvard.

Mike Artin, David Mumford and Jean-Louis Verdier all attended the Woods Hole conference from july 6 till july 31 1964, famous for producing the Atiyah-Bott fixed point theorem (according to Fulton first proved by Verdier at the conference).

Etale cohomology was a hot topic at that conference. On july 10th there were three talks, Artin spoke on ‘Etale cohomology of schemes’, Verdier on ‘A duality theorem in the etale cohomology of schemes’ and John Tate on ‘Etale cohomology over number fields’.

After a first week of talks, more informal seminars were organized, including the Atiyah-Bott seminar leading to the ‘Woods hole duality theorem’ and one by Lubin-Tate and Serre on elliptic curves and formal groups. Two seminars adressed Etale Cohomology.

Artin and Verdier ran a seminar on the etale cohomology of number fields leading to their duality result, and, three young turks : Daniel Quillen, Steve Kleiman and Robin Hartshorne ran a Baby Seminar on Etale cohomology

Probably it is safe to say that the talks by Artin, Verdier and Tate on July 10th sparked the primes=knots idea, and if not then, a couple of days later.

5. The birthplace : the Whitney Estate

The ‘Woods Hole’ conference took place at the Whitney Estate and all the lectures took place in the rustic rooms of the main building and the participants (and their families) were housed in rented cottages in the neighborhood, for the duration of the summer.

The only picture i managed to find from the Whitney house comes from a rather surprising source : Gardeners and Caretakers ofWoods Hole. Anyway, here it is :

Probably, the knots=primes analogy was first dreamed up inside, or in the immediate neighborhood, on a walk to or from the cottages, overlooking the harbor.

The papers by Liliane Beaulieu on the history of the Bourbaki-group are genuine treasure troves of good stories. Though I’m mostly interested in the pre-war period, some tidbits are just too good not to use somewhere, sometime, such as here on a lazy friday afternoon …

In her paper Bourbaki’s art of memory she briefly mentions these two pearls of wisdom from the jolly couple Henri Cartan (left) and Sammy Eilenberg (right) in relation to their seminal book Homological Algebra (1956).

Brigitte Bardot and the Hom-Tensor relation

For the youngsters among you, Brigitte Bardot, or merely B.B., was an iconic French actress and sex-goddess par excellence in the 60ties and 70ties. She started her acting career in 1952 and became world famous for her role in Et Dieu… créa la femme from 1956, the very same year Cartan-Eilenberg was first published.

The tensor-hom adjunction in homological algebra (see II.5.2 of Cartan-Eilenberg for the original version) asserts that

when and are rings, $A$ a right $R$-module, $C$ a left $S$-module and $B$ an $R-S$-bimodule.

Surely no two topics can be farther apart than these two? Well not quite, Beaulieu writes :

“After reading a suggestive movie magazine, Cartan tried to show the formula

$Hom(B,Hom(B,B)) = Hom(B \otimes B,B)$

in which “B. B.” are the initials of famous French actress and 1950s sex symbol Brigitte Bardot and “Hom” (pronounced ‘om as in homme, the French word for man) designates, in mathematics, the homomorphisms – a special kind of mapping – of one set into another.”

Miniskirts and spectral sequences

I’d love to say that the miniskirt had a similar effect on our guys and led to the discovery of spectral sequences, but then such skirts made their appearance on the streets only in the 60ties, well after the release of Cartan-Eilenberg. Besides, spectral sequences were introduced by Jean Leray, as far back as 1945.

Still, there’s this quote : “The spectral sequence is like the mini-skirt; it shows what is interesting while hiding the essential.”

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid.

In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of the approach or an appreciation of its importance, rather than giving a visual illustration of the varieties over $\mathbb{F}_1$ the approach proposes.

Some time ago, we’ve had a couple of posts trying to depict noncommutative varieties, first the illustrations used by Shahn Majid and Matilde Marcolli, and next my own mental picture of it.

In this post, we’ll try to do something similar for affine varieties over the absolute point. To simplify things drastically, I’ll divide the islands in the Lopez Pena-Lorscheid map of $\mathbb{F}_1$ land in two subsets : the former approaches (all but the $\Lambda$-schemes) and the current approach (the $\Lambda$-scheme approach due to James Borger).

The former approaches : Francis Bacon “The Pope” (1953)

The general consensus here was that in going from $\mathbb{Z}$ to $\mathbb{F}_1$ one looses the additive structure and retains only the multiplicative one. Hence, ‘commutative algebras’ over $\mathbb{F}_1$ are (commutative) monoids, and mimicking Grothendieck’s functor of points approach to algebraic geometry, a scheme over $\mathbb{F}_1$ would then correspond to a functor

$h_Z~:~\mathbf{monoids} \longrightarrow \mathbf{sets}$

Such functors are described largely by combinatorial data (see for example the recent blueprint-paper by Oliver Lorscheid), and, if the story would stop here, any Rothko painting could be used as illustration.

Most of the former approaches add something though (buzzwords include ‘Arakelov’, ‘completion at $\infty$’, ‘real place’ etc.) in order to connect the virtual geometric object over $\mathbb{F}_1$ with existing real, complex or integral schemes. For example, one can make the virtual object visible via an evaluation map $h_Z \rightarrow h_X$ which is a natural transformation, where $X$ is a complex variety with its usual functor of points $h_X$ and to connect both we associate to a monoid $M$ its complex monoid-algebra $\mathbb{C} M$. An integral scheme $Y$ can then be said to be ‘defined over $\mathbb{F}_1$’, if $h_Z$ becomes a subfunctor of its usual functor of points $h_Y$ (again, assigning to a monoid its integral monoid algebra $\mathbb{Z} M$) and $Y$ is the ‘best’ integral scheme approximation of the complex evaluation map.

To illustrate this, consider the painting Study after Velázquez’s Portrait of Pope Innocent X by Francis Bacon (right-hand painting above) which is a distorded version of the left-hand painting Portrait of Innocent X by Diego Velázquez.

Here, Velázquez’ painting plays the role of the complex variety which makes the combinatorial gadget $h_Z$ visible, and, Bacon’s painting depicts the integral scheme, build up from this combinatorial data, which approximates the evaluation map best.

All of the former approaches more or less give the same very small list of integral schemes defined over $\mathbb{F}_1$, none of them motivically interesting.

The current approach : Jackson Pollock “No. 8″ (1949)

An entirely different approach was proposed by James Borger in $\Lambda$-rings and the field with one element. He proposes another definition for commutative $\mathbb{F}_1$-algebras, namely $\lambda$-rings (in the sense of Grothendieck’s Riemann-Roch) and he argues that the $\lambda$-ring structure (which amounts in the sensible cases to a family of endomorphisms of the integral ring lifting the Frobenius morphisms) can be viewed as descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

The list of integral schemes of finite type with a $\lambda$-structure coincides roughly with the list of integral schemes defined over $\mathbb{F}_1$ in the other approaches, but Borger’s theory really shines in that it proposes long sought for mystery-objects such as $\mathbf{spec}(\mathbb{Z}) \times_{\mathbf{spec}(\mathbb{F}_1)} \mathbf{spec}(\mathbb{Z})$. If one accepts Borger’s premise, then this object should be the geometric object corresponding to the Witt-ring $W(\mathbb{Z})$. Recall that the role of Witt-rings in $\mathbb{F}_1$-geometry was anticipated by Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$.

But, Witt-rings and their associated Witt-spaces are huge objects, so one needs to extend arithmetic geometry drastically to include such ‘integral schemes of infinite type’. Borger has made a couple of steps in this direction in The basic geometry of Witt vectors, II: Spaces.

To depict these new infinite dimensional geometric objects I’ve chosen for Jackson Pollock‘s painting No. 8. It is no coincidence that Pollock-paintings also appeared in the depiction of noncommutative spaces. In fact, Matilde Marcolli has made the connection between $\lambda$-rings and noncommutative geometry in Cyclotomy and endomotives by showing that the Bost-Connes endomotives are universal for $\lambda$-rings.

In an interview with readers of the Guardian, December 3rd 2010, Julian Assange made a somewhat surprising comparison between WikiLeaks and Bourbaki, sorry, The Bourbaki (sic) :

“I originally tried hard for the organisation to have no face, because I wanted egos to play no part in our activities. This followed the tradition of the French anonymous pure mathematians, who wrote under the collective allonym, “The Bourbaki”. However this quickly led to tremendous distracting curiosity about who and random individuals claiming to represent us. In the end, someone must be responsible to the public and only a leadership that is willing to be publicly courageous can genuinely suggest that sources take risks for the greater good. In that process, I have become the lightening rod. I get undue attacks on every aspect of my life, but then I also get undue credit as some kind of balancing force.”

Analogies are never perfect, but perhaps Assange should have taken it a bit further and studied the history of the pre-war Bourbakistas in order to avoid problems that led to the eventual split-up.

Clearly, if Bourbaki=WikiLeaks, then Assange plays the role of Andre Weil. Both of them charismatic leaders, convincing the group around them that for the job at hand to succeed, it is best to work as a collective so that individual contributions cannot be traced.

At first this works well. Both groups make progress and gain importance, also to the outside world. But then, internal problems surface, questioning the commitment of ‘the leader’ to the original project.

In the case of the Bourbakis, Claude Chevalley and Rene de Possel dropped that bombshell at the second Chancay-meeting in 1937 with a 2 page pamphlet 7 theses de Chancay.

“Criticism on the state of affairs :

• in general, a certain aging of Bourbaki, which manifests itself in a tendency to neglect internal lively opposition in favor of pursuing visible external succes ((failed) completion of versions, artificial agreement among members of the group).
• in particular, often the working method appears to be that of suffocating any objections in official meetings (via interruptions, not listening, etc. etc.). This tendency didn’t exist at the Besse meeting, began to manifest itself at the Escorial-meeting and got even worse here at Chancay. Bourbaki-members don’t pay attention to discussions and the principle of unanimous decision-making is replaced in reality by majority rule.”

Sounds familiar? Perhaps stretching the analogy a bit one might say that Claude Chevalley’s and Rene de Possel’s role within Bourbaki is similar to that of respectively Birgitta Jónsdóttir and Daniel Domscheit-Berg within WikiLeaks.

This criticism will be neglected and at the following Bourbaki-meeting in Dieulefit (neither Chevalley nor de Possel were present) hardly any work gets done, largely due to the fact that Andre Weil is more concerned about his personal safety and escapes during the meeting for a couple of days to Switserland, fearing an imminent invasion.

After the Dieulefit-meeting, even though Bourbaki’s fame is spreading, work on the manuscripts is halted because all members are reserve-officers in the French army and have to prepare for war.

Except for Andre Weil, who’s touring the world with a clear “Bourbaki, c’est moi!” message, handing out Bourbaki name-cards or invitations to Betti Bourbaki’s wedding… That Andre and Eveline Weil are traveling as Mr. and Mrs. Bourbaki is perhaps best illustrated by the thank-you note, left on their journey through Finland.

If it were not for the fact that the other members had more pressing matters to deal with, Weil’s attitude would have resulted in more people dropping out of the group, or continuing the work under another name, a bit like what happens to WikiLeaks and OpenLeaks today.