71 search results for "bourbaki"

Hasse = “le P. Adique, de l’Ordre des Diophantiens”

Published October 12, 2016 by lievenlb

The Bourbaki wedding invitation is probably the most effective branding- and marketing-campaign in the history of mathematics.

It contains this, seemingly opaque, paragraph:

The trivial isomorphism will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.

It was pretty easy to decode the date of the wedding “3 Cartember, year VI” to be June 3rd, 1939, and (a bit more difficult) the wedding place “the Principal Cohomology of the Universal Variety” as the l’église royale Notre-Dame du Val-de-Grâce in Paris.

The identity of the celebrating priest “P. Adic, of the Diophantine Order” remained unclear. The most likely suspect was Helmut Hasse, but I couldn’t place him in Paris on June 3rd, 1939.

Hasse is the central figure in the picture above, taken in Oberwolfach in 1952, before one of his cars. Here’s another picture of car-freak Hasse (trains were to Andre Weil what cars were to Helmut Hasse). Both pictures are from the MFO photo collection.

Thanks to Peter Roquette’s publishing of Helmut Hasse’s letters we can now prove that Hasse was not in Paris on that particular day (however, he was there a couple of days earlier) but Weil had every reason to believe he might be there at the time he wrote the wedding invitation.

When was the wedding invitation written?

Frank Smithies recalls the spring 1939 period in Cambridge as follows :

“The climax of the academic year, as far as we were concerned, came in the Easter term. André Weil, Claude Chabauty, and Louis Bouckaert (from Louvain) were all in Cambridge, and the proposal was mooted that a marriage should be arranged between Bourbaki’s daughter Betti and Hector Pétard; the marriage announcement was duly printed in the canonical French style – on it Pétard was described as the ward of Ersatz Stanislas Pondiczery – and it was circulated to the friends of both parties. A couple of weeks later the Weils, Louis Bouckaert, Max Krook (a South African astrophysicist), Ralph and myself made a river excursion to Grantchester by punt and canoe to have tea at the Red Lion; there is a photograph of Ralph and myself, with our triumphantly captured lion between us and André Weil looking benevolently on.”

We know that this picture is taken on May 13th 1939 so the wedding-invitation was drawn up around mid april 1939.

“What did Weil know about Hasse’s visit to Paris?”

Hasse had been invited by Julia to give a series of lectures at the Institut Henri Poincare in 1938, but Hasse postponed his trip to Paris until May 1939.

In his letter to Hasse of January 20th 1939, Andre Weil writes:

“It is quite unfortunate that you couldn’t accept your invitation to Paris before this year, because last year all our number-theorists would have been present. By a sad coincidence all of us will be on travel this coming May (except for Chevalley perhaps who might have returned from the US by then). Pisot will be in Gottingen, Chabauty in Manchester visiting Mordell and I will be in Cambridge as I obtained a travel grant for England and Scandinavia.”

Clearly, Weil was aware of the upcoming visit of Hasse to Paris at the end of May, and there was no reason for him to assume that he wouldn’t be able to stay a weekend longer.

What do we know of Hasse’s visit to Paris?

Because Julia was exhausted and was on a three months sick leave, Elie Cartan took over the job of organising Hasse’s lecture series. In a letter of April 25th 1939 he proposes some possible dates, to which Hasse replies on April 30th 1939:

In it he fixes for the first time the dates of his talks which will be on “New results in the arithmetic of algebraic function fields” and consist of three lectures:

– On Friday 19th 1939: “Generalities: the group of divisor classes and the multiplier ring”

– On Saturday 20th 1939: “Rational and integral points on algebraic curves over the integers”

– On Tuesday 23rd 1939: “Rational points on algebraic curves with coefficient mod p”

He also mentions that he would stay for 15 days in Paris, arriving on May 17th, in time for the Jubilee Conference for Elie Cartan, scheduled on May 18th.

Weil must have known that Hasse would be present at the Cartan-fest and give a series of lectures in the following weeks. He had every reason to believe that Hasse would still be in Paris on Saturday June 3rd.

Where was Hasse on June 3rd 1939?

Back at home, as on that very day he wrote a letter to Henri Cartan, thanking him for an enjoyable day’s stay in Strasbourg, on the way back from Paris, on June 1st 1939:

If you want to catch up with previous posts on the Bourbaki wedding, you might want to download the booklet The Bourbaki Code.

Toposes alive and kicking at IHES

Published October 8, 2016 by lievenlb

After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES.

At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now on YouTube (via +David Roberts).

How did topos theory, a topic considered by most to be far too abstract to be useful to main stream mathematics, suddenly return in such force?

It always helps when a couple of world-class mathematicians become interest in the topic, for their own particular reasons. Clearly, the topic gathers considerable momentum if these people are all permanent members of the IHES.

A lot of geometric information is contained in the category of all sheaves on the geometric object. Topos theory offers a way to construct ‘geometries’ out of nothing, that is, out of arbitrary categories.

Take your favourite category $\mathbf{C}$, then “presheaves” on $\mathbf{C}$ are defined to be contravariant functors $\mathbf{C} \rightarrow \mathbf{Sets}$. For any Grothendieck topology on $\mathbf{C}$ one can then restrict to the sub-category of “sheaves” for this topology, and that’s your typical topos.

Alain Connes got interested in topos theory because he observed that even for the most trivial of categories, such as the monoid category with just one object and endomorphisms the multiplicative semigroup $\mathbb{N}_{\geq 1}^{\times}$, and taking the coarsest of all Grothendieck topologies, one gets interesting objects of baffling complexity.

One of the ‘invariants’ one can associate to a topos is its collection of “points”. Together with Katia Consani, Connes computed in Geometry of the Arithmetic Site that the collection of points of this simple presheaf topos is exactly the set of adele classes $\mathbb{Q}^{\ast}_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \hat{\mathbb{Z}}^{\ast}$.

Here’s what Connes himself said about this revelation (followed by an attempted translation):

——————————————————

(50.36)

And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.

(52.27)

——————————————————-

Pierre Cartier has a very wide interest in mathematical theories, the wilder the better: Witt rings, motifs, cosmic Galois groups, toposes…

He must have been one of the first people to speak about toposes at the Bourbaki seminar. In february 1978 he gave the talk Logique, categories et faisceaux, d’apres F. Lawvere et M. Tierney (and dedicated to Grothendieck’s 50th birthday).

He also gave the opening lecture of the Topos a l’IHES conference.

In this fragment of an interview with Stephane Dugowson and Anatole Khelif in 2014 he plays down his own role in the development of topos theory, compared to his contributions in other fields, such as motifs.

——————————————————-

(46:24)

Well, I didn’t invest much time in topos theory.

Except, I once gave a talk at the Bourbaki seminar on the use of topos theory in logic, such as the independence of the axiom of choice, that is, on the idea of forcing.

But, it was just this talk, I didn’t do anything original in it.

Then there is nonstandard analysis, where one can formulate certain things in terms of topos theory. When I got interested in nonstandard analysis, I had this possible application of topos theory in mind.

At the moment when you have a nonstandard model of the integers or more generally of set theory, then one has two models of set theory, that is two different toposes, and then one obviously tries to compare them.

In that sense, I was completely aware of the fact that everything I was doing could be expressed in the language of toposes,or at least in the philosophy of toposes.

I haven’t made any important contributions in that theory, for me it merely remained a tool.

(47:49)

——————————————————-

Laurent Lafforgue says he spend hundredths and hundredths of hours talking to Olivia Caramello about topos theory.

She must have been quite convincing. The last couple of years Lafforgue is a fierce advocate of Caramello’s work.

Her basic idea is that the same topos can arise from two very different mathematical settings (that is, two different categories with Grothendieck topologies can have equivalent categories of sheaves).

The hope then is to translate results from one theory to the other, or as she expresses it, toposes can be used as “bridges” between different mathematical topics.

At the moment though, is seems a bit far fetched for this idea to be relevant to the Langlands programme.

Caramello and Lafforgue have just a paper out: Sur la dualit´e des topos et de leurs pr´esentations et ses applications : une introduction.

The paper is based on a lecture Lafforgue gave in April in Nantes. Here’s the video:

In the introduction they write:

“It is our conviction that the theory of toposes and their representations, with its essential and structural ambiguity, is destined to have an impact on mathematics comparable to the impact group theory has had from the moment, some decades after its discovery by Galois, the mathematical community began to understand it.”

according to Groth. IV.22

Published October 1, 2016 by lievenlb

At the Bourbaki Seminar in November 1968 the participants were handed the following (premature) announcement of Bourbaki’s death.

The French text can be found at the Canulars Bourbaki, and the English translation below is from Maurice Mashaal’s book Bourbaki, a secret society of mathematicians, page 115.

I’ve underlined a couple of riddles in the text.

———-

The Cantor, Hilbert, and Noether families;
The Cartan, Chevalley, Dieudonne, and Weil families;
The Bruhat, Dixmier, Godement, Samuel, and Schwartz families;
The Cartier, Grothendieck, Malgrange, and Serre families;
The Demazure, Douady, Giraud, and Verdier families;
The Right-Filtering and Strict-Epimorphism families;
Mesdemoiselles Adele and Idele;

regret to announce the death of Monsieur

NICOLAS BOURBAKI

Respectively their father, brother, son, grandson, great-grandson, and grand-cousin.

He died piously in his home on November 11, 1968 (on the anniversary of great victory) in his home in Nancago.

The burial will take place in the cemetery for Random Functions (metro stations Markov and Goedel) on Saturday, November 23, 1968 at 3 o’clock in the afternoon.

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

Following the wish of the departed, His Eminence the Cardinal Aleph I will hold a mass in Our Lady of Universal Problems in the presence of representatives from all equivalence classes and from all (algebraically closed) fields. The students from l’Ecole Normale Superieure and the Chern classes will observe a minute of silence.

No flowers or wreath products.

“For God is the Alexandrov compactification of the universe.” Groth. IV.22

———-

This announcement is clearly inspired by the faire-part of Betti Bourbaki’s wedding (with Hector Petard), written by Andre Weil and Claude Chabauty in the spring of 1939.

Some years ago I wrote a couple of posts on possible solutions of the riddles contained in that faire-part, a pdf-version can be downloaded as the Bourbaki code. (Note to self: repost some of those and add new material!)

Whereas the wedding announcement was concocted by members of Bourbaki, this is not the case for this death announcement. It was written by the mathematician and writer Jacques Roubaud, a member of the literary group OuLiPo.

In 1997 he wrote the novel ‘Mathematique’ (now available in English translation). In it, he recalls his mathematical years, from his first lecture at the IHP in 1952 till the 70ties. It contains an insiders view on Parisian mathematics in the 50ties and 60ties, dominated largely by Bourbaki, and offers clues to decrypt some of the riddles in the death announcement.

Today, we’ll consider the final one

For God is the Alexandrov compactification of the universe.Groth. IV.22

Can we make sense of the ‘reference’ Groth. IV.22?

Does it refer to EGA IV?

Roubaud’s motif (pardon the expression) for writing the announcement of Bourbaki’s death in 1968 can be read between the lines in his book Mathematics, a novel from which all quotes below are taken.

page 146: “I was invited by Raymond Queneau to join the Oulipo and I met FLL in the fall of 1966. By then, I had reached the end of my passion for Bourbaki, after being one of their most faithful and credulous readers for many years.”

page 73: “The “biography” of that many-headed beast, Bourbaki, is still to be written. It would be a fascinating but arduous task. Here, I shall say only what is strictly necessary to my own entreprise. Having reached his dotage after 1968, “he” is for all intents and purposes now dead.”

By 1968, Bourbaki had become an institution dominating French mathematics and so had to die after the May 1968 revolt.

But, Roubaud had found a new prophet to follow…

page 284: “It was a book of mathematics. It had just been published. It was in a large format, with a blue cover. Its title was Elements of Algebraic Geometry (affectionately and familiarly abbreviated, in French, to EGA). Its author: Grothendieck.

page 285: “For I had so immersed mself in Bourbakism that such a text, the fruit of its final flowering, the monumental work of he who could be considered as Dr. Frankenstein-Bourbaki’s Monster, and which had been drafted according to the group’s inimitable stylistic norms, here applied, in its prose, in a heightened, frenetic way, ran through my mind like honey, no, like nectar, an intellectual ambrosia. Just thinking about it now fills me with stupefaction. I was someone who managed to read EGA with pleasure – worse, with delight. For any normal mathematician today, such an affirmation would seem as perverse as adoring an American soft drink.”

Roubaud was reading EGAs like others would read Nicki French thrillers, one per year:

(1960) : “Éléments de géométrie algébrique: I. Le langage des schémas”

(1961) : “Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes”

(1961) : “Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie”

(1963) : “Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie”

(1964) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie”

(1965) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie”

(1966) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie”

(1967) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie”

It was now november 1968, and Roubaud was hoping that another sequel would be published soon. As the last one ended with section IV.21, this new volume would start with IV.22, and, no doubt, contain more divine mathematics…

However, Pieter Belmans objected that it was planned from the outset for EGA4 to consist of 21 chapters, and no more. Surely, Roubaud knew about this…

ADDED october 4th: Pieter has done some further digging on this issue in his post According to Groth IV.22.

Can it refer to SGA IV?

Luckily, there is another option. Grothendieck ran the Séminaire de Géométrie Algébrique du Bois Marie at the IHES from 1962 to 1969.

SGA4 was about “Théorie des topos et cohomologie étale des schémas” (Topos theory and étale cohomology) and ran in 1963–1964. A decade later the notes were published in Springer’s Lecture Notes in Mathematics 269, 270 and 305, 1972/3.

The topic of SGA4 (topos theory) is clearly closer in spirit to the fake biblical quote on the topological nature of God than that of EGA4 which was about the local structure of schemes and their morphisms.

The original notes were published in fascicles by the IHÉS, most of which went through two or three revisions, and were published as the seminar proceeded. So, Roubaud had access to them in the later 60ties.

The original versions, as well as their re-published LaTeX versions can be found here.

Again, we face the problem that there are not enough chapters, only 19 in this case.

Fortunately, we can search the LaTeX-ed version for references to the Alexandroff compactification, and there is just a single one:

This is in the first lecture on Presheaves by Grothendieck and Verdier. More precisely, it is in section 2 (Univers et espèces de structures) of the Appendix, which is labeled

II. Appendice : Univers (by N. Bourbaki (*))

So, the paragraph on the Alexandroff compactification is in SGA IV,II.2, or, if we read 22 as II.2 this might explain Groth. IV.22.

We have found a reference in SGA IV including “Bourbaki”, “the univers” and “Alexandroff compactification”.

But then, who dreamed up this topological definition of God?

Jean-Paul Benzecri

Dieu est le compactifié d’Alexandrof de l’univers.Jean-Paul Benzecri

Jean-Paul Benzécri is a French statistician who has been professor at Université Pierre-et-Marie-Curie in Paris. In the 60ties he was a professor at the university of Rennes where he was a colleague of Roubaud.

Jacques Roubaud has another book on his reminiscences as a mathematician, Impératif catégorique. Unfortunately, this book is not (yet) translated into English.

In section 80, La déesse Fortune ne se montra pas envers moi avare de ses bienfaits, he tells about his years at the University of Rennes where also his friend and topos-theorist Jean Bénabou was at the time. Bénabou and Benzécri knew each other from their student days at the Ecole Normale.

Benzécri had a very strict catholic family background, and in the 50ties he attended the Centre Richelieu des étudiants catholiques.

He liked to explain his axiom as follows:

“Of course, God created the univers. But, he created it locally compact and not compact. That it, left on its own, the universe would suffer a serious structural defect which could only be repaired by introducing a point at infinity, which marks the presence of the divine.”