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Tag: Chevalley

Grothendieck’s Café

“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)

Finding that particular café in Paris, presumably in the 5th arrondissement, seemed like looking for a needle in a haystack.

Until now.

In trying to solve the next riddle in Bourbaki’s death announcement:

A reception will be held at the Bar ‘The Direct Products’, at the crossroads of the Projective Resolutions (formerly Koszul square)

I’ve been reading Mathematics, a novel by Jacques Roubaud (the guy responsible for the announcement) on Parisian math-life in the 50ties and 60ties.

It turns out that the poor Bourbakistas had very little choice if they wanted to have a beer (or coffee) after attending a seminar at the IHP.

On page 114, Roubaud writes:

“Père Plantin presided over his bar, which presided over the Lhomond/Ulm street corner. It is an obvious choice. rue Pierre-et-Marie-Curie had no bars; rue d’Ulm had no bars in eyeshot either. If we emerged, as we did, on this side of the Institut Henri Poincaré (for doing so on the other side would have meant fraternizing with the Spanish and Geography students in the cafés on rue Saint-Jacques, which was out of the question), we had no choice. Café Plantin had a hegemony.”

It is unclear to me whether Plantin was once actually the name of the café, or that it’s just Roubaud’s code-word for it. At other places in the book, e.g. on pages 82 and 113, he consistently writes “Plantin”, between quotes.

Today, the café on the crossroads of rue d’Ulm (where the Ecole Normal Superieure is located) and de rue Lhomond is the Interlude Café

and here’s what Roubaud has to say about it, or rather about the situation in 1997, when the French version of his book was published:

the thing that would currently be found at the very same corner of rues Lhomond/Ulm would not be what I am here terming “Plantin”.”

So, we can only hope that the Café ‘Aux Produits Directs’ was a lot cosier, way back then.

But let us return to Grothendieck’s “What is a scheme?” story.

Now that we have a fair idea of location, what about a possible date? Here’s a suggestion: this happened on monday december 12th, 1955, and, one of the friends present must have been Cartier.

Here’s why.

The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.

The lecturer was Claude Chevalley, and the date was december 12th 1955.


I’m fairly certain Grothendieck and Cartier attended and that Cartier was either briefed before or understood the stuff at once (btw. he gave another talk on schemes, a year later at the Chevalley seminar).

A couple of days later, on december 15th, Grothendieck sends a letter to his pal Serre (who must have been out of Paris for otherwise they’d phone each other) ending with:


.
Note the phrase: I am exploiting him most profitably. Yes, by asking him daft questions over a pint at Café “Plantin”

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the birthday of Grothendieck topologies

This is the story of the day the notion of ‘neighbourhood’ changed forever (at least in the geometric sense).

For ages a neighbourhood of a point was understood to be an open set of the topology containing that point. But on that day, it was demonstrated that the topology of choice of algebraic geometry, the Zariski topology, needed a drastic upgrade.

This ultimately led to the totally new notion of Grothendieck topologies, which aren’t topological spaces at all.

Formally, the definition of Grothendieck topologies was cooked up in the fall of 1961 when Grothendieck visited Zariski, Mike Artin and David Mumford in Harvard.

The following spring, Mike Artin ran a seminar resulting in his lecture notes on, yes, Grothendieck topologies.

But, paradigm shifts like this need a spark, ‘une bougie d’allumage’, and that moment of insight happened quite a few years earlier.

It was a sunny spring monday afternoon at the Ecole Normal Superieure. Jean-Pierre Serre was giving the first lecture in the 1958 Seminaire Claude Chevalley which that year had Chow rings as its topic.

That day, april 21st 1958, Serre was lecturing on algebraic fibre bundles:

He had run into a problem.

If a Lie group $G$ acts freely on a manifold $M$, then the set of $G$-orbits $M/G$ is again a manifold and the quotient map $\pi : M \rightarrow M/G$ is a principal $G$-fibre bundle meaning that for sufficiently small open sets $U$ of $M/G$ we have diffeomorphisms

$\pi^{-1}(U) \simeq U \times G$

that is, locally (but not globally) $M$ is just a product manifold of $G$ with another manifold and the $G$-orbits are all of the form $\{ u \} \times G$.

The corresponding situation in algebraic geometry would be this: a nice, say reductive, algebraic group $G$ acting freely on a nice, say smooth, algebraic variety $X$. In this case one can form again an orbit space $X/G$ which is again a (smooth) algebraic variety but the natural quotient map $\pi : X \rightarrow X/G$ rarely has this local product property…

The reason being that the Zariski topology on $X/G$ is way too coarse, it doesn’t have enough open sets to enforce this local product property.

(For algebraists: let $A$ be an Azumaya algebra of rank $n^2$ over $\mathbb{C}[X]$, then the representation variety $\mathbf{rep}_n(A)$ is a principal $\mathbf{PGL}_n$-bundle over $X$ but is only local trivial in the Zariski topology when $A$ is a trivial Azumaya algebra, that is, $End_{\mathbb{C}[X]}(P)$ for a rank $n$ projective module $P$ over $\mathbb{C}[X]$.)

But, Serre had come up with a solution.

He was going to study fibre bundles which were locally ‘isotrivial’, meaning that they had the required local product property but only after extending them over an unamified cover $Y \rightarrow X$ (what we now call, an etale cover) and he was able to clasify such fibre bundles by a laborious way (which we now call the first etale cohomology group).

The story goes that Grothendieck, sitting in the public, immediately saw that these etale extensions were the correct generalization of the usual (Zariski) localizations and that he could develop a cohomology theory out of them in all dimensions.

According to Colin McLarty Serre was ‘absolutely unconvinced’, since he felt he had ‘brutally forced’ the bundles to yield the $H^1$’s.

We will never known what Serre actually wrote on the blackboard on april 21st 1958.

The above scanned image tells it is an expanded version of the original talk, written up several months later after the ICM-talk by Grothendieck in Edinburgh.

By that time, Grothendieck had shown Serre that his method indeed gives cohomology in all dimensions,and convinced him that this etale cohomology was likely to be the “true cohomology needed to prove the Weil conjectures”.

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If Bourbaki=WikiLeaks then Weil=Assange

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.

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Who was ‘le P. Adique’?

Last year we managed to solve the first few riddles of the Bourbaki code, but several mysteries still remain. For example, who was the priest performing the Bourbaki-Petard wedding ceremony? The ‘faire part’ identifies him as ‘le P. Adique, de l’Ordre des Diophantiens’.

As with many of these Bourbaki-jokes, this riddle too has several layers. There is the first straightforward mathematical interpretation of the p-adic numbers $latex \hat{\mathbb{Z}}_p$ being used in the study of Diophantine problems.

For example, the local-global, or Hasse principle, asserting that an integral quadratic form has a solution if and only if there are solutions over all p-adic numbers. Helmut Hasse was a German number theorist, held in high esteem by the Bourbaki group.

After graduating from the ENS in 1929, Claude Chevalley spent some time at the University of Marburg, studying under Helmut Hasse. Hasse had come to Marburg when Kurt Hensel (who invented the p-adic numbers in 1902) retired in 1930.

Hasse picked up a question from E. Artin’s dissertation about the zeta function of an algebraic curve over a finite field and achieved the first breakthrough establishing the conjectured property for zeta functions of elliptic curves (genus one).

Extending this result to higher genus was the principal problem Andre Weil was working on at the time of the wedding-card-joke. In 1940 he would be able to settle the general case. What we now know as the Hasse-Weil theorem implies that the number N(p) of rational points of an elliptic curve over the finite field Z/pZ, where p is a prime, can differ from the mean value p+1 by at most twice the square root of p.

So, Helmut Hasse is a passable candidate for the first-level, mathematical, decoding of ‘le P. adique’.

However, there is often a deeper and more subtle reading of a Bourbaki-joke, intended to be understood only by the select inner circle of ‘normaliens’ (graduates of the Ecole Normale Superieure). Usually, this second-level interpretation requires knowledge of events or locations within the 5-th arrondissement of Paris, the large neighborhood of the ENS.

For an outsider (both non-Parisian and non-normalien) decoding this hidden message is substantially harder and requires a good deal of luck.

As it happens, I’m going through a ‘Weil-phase’ and just started reading the three main Weil-biographies : Andre Weil the Apprenticeship of a Mathematician, Chez les Weil : André et Simone by Sylvie Weil and La vie de Simone Weil by Simone Petrement.


[abp:3764326506] [abp:2283023696] [abp:2213599920]

From page 35 of ‘Chez les Weil’ : “Après la guerre, pas tout de suite mais en 1948, toute la famille avait fini par revenir à Paris, rue Auguste-Comte, en face des jardins du Luxembourg.” Sylvie talks about the Parisian apartment of her grandparents (father and mother of Andre and Simone) and I wanted to know its exact location.

More details are given on page 103 of ‘La vie de Simone Weil’. The apartment consists of the 6th and 7th floor of a building on the Montagne Sainte-Geneviève. The Weils bought it before it was even built and when they moved in, in may 1929, it was still unfinished. Compensating this, the apartment offered a splendid view of the Sacre-Coeur, the Eiffel-tower, la Sorbonne, Invalides, l’Arc de Triomphe, Pantheon, the roofs of the Louvre, le tout Paris quoi…

As to its location : “Juste au-dessous de l’appartement se trouvent l’Ecole des mines et les serres du Luxembourg, avec la belle maison ancienne où mourut Leconte de Lisle.” This and a bit of googling allows one to deduce that the Weils lived at 3, rue Auguste-Comte (the W on the map below).

Crossing the boulevard Saint-Michel, one enters the 5-th arrondissement via the … rue de l’Abbe de l’Epee…
We did deduce before that the priest might be an abbot (‘from the order of the Diophantines’) and l’Epee is just ‘le P.’ pronounced in French (cheating one egue).

Abbé Charles-Michel de l’Épée lived in the 18th century and has become known as the “Father of the Deaf” (compare this to Diophantus who is called “Father of Algebra”). Épée turned his attention toward charitable services for the poor, and he had a chance encounter with two young deaf sisters who communicated using a sign language. Épée decided to dedicate himself to the education and salvation of the deaf, and, in 1760, he founded a school which became in 1791 l’Institution Nationale des Sourds-Muets à Paris. It was later renamed the Institut St. Jacques (compare Rue St. Jacques) and then renamed again to its present name: Institut National de Jeunes Sourds de Paris located at 254, rue Saint-Jacques (the A in the map below) just one block away from the Schola Cantorum at 269, rue St. Jacques, where the Bourbaki-Petard wedding took place (the S in the map).

Completing the map with the location of the Ecole Normale (the E) I was baffled by the result. If the Weil apartment stands for West, the Ecole for East and the Schola for South, surely there must be an N (for N.Bourbaki?) representing North. Suggestions anyone?

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