I’d recently been wondering whether the archives of the Bourbaki group would be put on-line, and today noticed that there’s a project to do so, with results available here. One can read copies of “La Tribu”, internal reports on the activities of the group, up through 1953. There are a wide variety of interesting mathematical documents, often consisting of attempts to write up one subject or another, efforts that sometimes made it into the published books, often not.

One subject that Bourbaki struggled with over the years was that of how to set out the foundations of differential geometry. My colleague Hervé Jacquet likes to tell about how Chevalley at one point made an effort to do so, with the peculiar starting point of defining things in terms of “cubes”. I wasn’t sure whether to believe him, but here it is. According to Borel, in 1957 Grothendieck presented the group with his own take on the question of manifolds:

Grothendieck lost no time and presented to the next Congress, about three months later, two drafts:

Chap. 0: Preliminaries to the book on manifolds. Categories of manifolds, 98 pages

Chap. I: Differentiable manifolds, The differential formalism, 164 pages

and warned that much more algebra would be needed, e.g., hyperalgebras. As was often the case with Grothendieck’s papers, they were at points discouragingly general, but at others rich in ideas and insights. However, it was rather clear that if we followed that route, we would be bogged down with foundations for many years, with a very uncertain outcome.

I don’t see these documents on the list, perhaps documents from the later years are still to appear.

The documents often start out with some unvarnished comments, here’s an example, from Chevalley’s report on a text about semi-simple Lie algebras:

Au moment d’écrire ces observations, je me demande si ce ramassis des méthodes les plus éculées et les plus pisseuses, ces résultats les moins généraux possibles établis de la manière la plus incompréhensible possible, ne sont pas un canular intrabourbachique monté par le rédacteur. Même s’il en est ainsi, je me laisse prendre au canular et présente les observations suivantes.

éculées –down at the heels, shabby

canular — hoax

Curiosity got the better of me so I googled “babelfish” and got the following translation of that french stuff:

“At the time of writing these observations, I wonder whether this bunch of the most worn down methods and more the baby girls, these results the least general possible benches in the most incomprehensible possible way, is not a hoax intrabourbachic assembled by the writer. Even if it is thus, I let myself take with the hoax and presents the following observations.”

Perhaps our blog host can let us know how faithfully it represents the original french.

It’s deeply interesting, and extremely telling, that Bourbaki couldn’t handle the most important and profound branch of modern mathematics, namely differential geometry. They ought to inscribe this passage on Bourbaki’s tombstone.

Au moment d’écrire ces observations, je me demande si ce ramassis des méthodes les plus éculées et les plus pisseuses, ces résultats les moins généraux possibles établis de la manière la plus incompréhensible possible, ne sont pas un canular intrabourbachique monté par le rédacteur. Même s’il en est ainsi, je me laisse prendre au canular et présente les observations suivantes.

As I write these observations, I ask myself if this grab-bag of outworn and infantile methods, these results with the least possible generality established in the most incomprehensible manner, isn’t a Bourbaki in-house hoax perpetrated by the writer. But even if it is, I will let myself be taken in by the hoax and present the following observations.

I think pisseuses refers to infants of the wet-diaper type—pissy ones—an earthy expression.

Concerning the volume on differential deometry, here is a short extract from a 1990 French radio programme in which Laurent Schwartz and Jean-Pierre Serre shed some light on the matter (at least for those who understand French): http://www.fileden.com/files/2009/10/3/2591723/schwartz_serre.mp3

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Marcus’s translation is perfect, including his interpretation of “pisseuses” (sic).

The definition in Chevalley’s Bourbaki draft is in fact not contrived in any way: it simply specifies a (smooth) manifold as a space which is locally homeomorphic to an Euclidean space, and which carries a distinguished collection of “smooth” functions, in such a way that _locally_ the smooth functions match (via the local homeomorphisms) the infinitely differentiable functions on Euclidean space. Choosing (the interior of) a cube as the local model of a smooth manifold (as the “standard” piece of an Euclidean space) is certainly among the most natural choices.

Chevalley’s definition is effectively the same as that of manifolds as (locally) ringed spaces which are locally isomorphic to Euclidean space with its sheaf of smooth functions, only leaving the machinery of sheaves (of rings) out. Among other things, this definition has the benefit of putting smooth manifolds on into the same framework as schemes, analytical spaces etc – usually define as ringed spaces locally isomorphic to a model (such as the spectrum of a commutative ring in the case of a scheme). It also provides a direct route to “synthetic” differential geometry: carrying out the basic constructions such as those of tangent spaces and connection “algebraically”, in the same way they are normally done in algebraic geometry (EGA IV).

The above hints at some of the issues, not entirely straightforward, that Bourbaki was facing: how general (and generalizable) framework to choose? Minimum needed for a few future volumes (such as Lie groups) or a basis for a significant expansion of the book series into different types of geometries? In the end, they only published a “Fascicule des résultats”, a quick fix for the former use.

It seems that this outcome was to a large etent due to some of the same difficulties Bourbaki faced in writing a book on algebraic topology (which they never finished): sheaves, and category theory in particular, were difficult to embed into the structure of the series at that stage – it would have needed a major overhaul of much of the earlier volumes. Some rather interesting results of this can be seen in Algèbre Ch. 4, where Symmetric tensor functor is mentioned in subsection headings in spite of functors never having been defined nor appearing in the actual text. And the chapter (10) on homological algebra is a curious example of categorical thinking without categories. And all this in spite of many of the key people behind category theory, sheaves etc. were also key members of Bourbaki.

J,

I think the reason people don’t do it Chevalley’s way, specifying a cube in R^n (or any other figure in R^n) is that the intersection of two cubes is not a cube. Surely you can do it Chevalley’s way, as well as lots of others, but there are good reasons that way of doing things is unpopular.

Looking at all the Bourbaki drafts of a differentiable manifolds book, I can’t avoid the conclusion that the lesson is that one shouldn’t let algebraists write books on differential geometry. Bourbaki was dominated by algebraists, and I think this was one one of its major weaknesses. It led them to miss out on lots of important areas of mathematics (including the connections of mathematics to physics via geometry).

Besides differential geometry, where they were pretty hopeless, they also never really did even algebraic geometry. This wasn’t their fault though, since the foundations of that subject were in flux at the time.

Peter,

Rather too harsh criticism on Bourbaki, I would say.

First, whether one uses cubes or other shapes (or just open sets) as the local models for manifolds is not really crucial to Chevalley’s approach – even starting with cubes can always saturate by finite intersections. The inconvenience is of the same order of magnitude that one meets in defining the metric topology using balls defined by the metric: the intersection of two balls is not not a ball either.

Instead, it is the definition of smooth manifolds (or other geometric creatures) in terms of (the sheaf of) distinguished functions on them that is important. While it is to certain extent a matter of taste whether to start from the sheaf of functions or from charts, both viewpoints are quite important. Their equivalence is of course rather trivial – it is more the viewpoint that matters. One should also note that the “usual” definition in terms of charts runs into a rather similar issue with intersections: intersection of (the domains) of two charts is not in general (a domain of) a chart – unless one uses a saturated atlas. Finally, as regards this particular topic in Bourbaki, it is fair to note that they eventually settled for the “traditional” definition, and explain the sheaf viewpoint a few pages later, in the published Fascicule des résultats”.

Second, whether one likes his (differential) geometry with more or less algebra is also a matter of taste – I’ve sometimes had the feeling that differential geometry is too important to be left to people with background too much in analysis… But more seriously, expecting a completely universal coverage of all important topics in mathematics by Bourbaki would of course be unrealistic – even if they had continued at the same pace from late 60’s onwards. It is moreover clear from a careful look at the Bourbaki archives that a great many later ababdoned topics were planned for inclusion, so seeing a deliberate exclusion from the project is not really correct. Rather, the project seems to have run out of steam by the early 70’s, with many plans unrealised. Specifically for the connections between geometry and physics that you presumably have in mind, they were discovered at a time when Bourbaki had already become rather inactive. Not writing on gauge theory in the sense of Atiyah and Donaldson in the late 60’s probably should not coutn against them…

As for algebraic geometry, it is fair to note that the biggest names of the field were Bourbaki members (Grothendieck, Serre, Weil,…). While the subject being in flux was a big reason for a book on algebraic topology getting stuck, I think there was an even clearer reason it made no sense for Bourbaki to write a book on algebraic geometry: Grothendieck’s EGA (written in by him and the arch-Bourbakist Dieudonné) was already an essentially Bourbaki-style treatment of the subject. No need to duplicate that.

Considering that Bourbaki never wrote a book on geometry, it’s not too surprising that they never got around to differential geometry. For a group supposedly committed to a rigorous presentation of the foundations of mathematics, it was always a serious omission.

Bourbaki had difficulty with geometries of all kinds because they chose to work through the medium of set theory. While some may disagree with me, my own opinion is that set theory and geometry are fundamentally incompatible at the elementary level. That is to say, they are two independent and unconnected ways of doing mathematics and ne’er the twain shall meet. You really cannot talk about geometric elements like points, lines, planes and circles by describing them as sets (though many do!), and sets cannot be encompassed at all by geometrical descriptions. The fundamental disconnect seems to me to be that sets have no notion of order or position, which is what geometry is all about.

My main point is that there is no one road, royal or otherwise, to every place in mathematics. Bourbaki tried to carve a path to all topics using set theory and they unsurprisingly failed. I cannot be done. Mathematics cannot be reduced a linear axiomatic presentation and moreover we have known this since the days of Godel and before.

This isn’t to say that Bourbaki’s results have been a waste. The Bourbaki framework and style make it possible to rigorously discuss advanced concepts and structures in a way that simply wasn’t available before.

On the other hand, Bourbaki has in some way made it

impossibleto talk about anything without smothering the discussion in formality. To whit; what is a manifold? Or a group? Or better yet, what is the determinant of a matrix?The answers Bourbaki gives to these questions, while correct, leave something to be desired.

It may be, as J said, that Bourbaki “… seems to have run out of steam by the early 70’s, with many plans unrealised …”,

but

from the late 70’s onward some very geometrical works, such as the books “Manifolds All of Whose Geodesics Are Closed” (Springer 1978) and “Einstein Manifolds” (Springer 1987) were written by Arthur L. Besse whose wikipedia entry states:

“Arthur Besse is a pseudonym chosen by a group of French differential geometers, led by Marcel Berger, following the model of Nicolas Bourbaki. A number of monographs have appeared under the name.”,

so

maybe you could say that the geometric view,

in the form of exposition of concrete examples without obscuration by formal abstraction,

has been carried out by a very close relative of Nicolas Bourbaki.

For example, the preface of the 1978 book describes such visualizable geometric structures as “… les espaces projectifs sur les nombres reels, sur les nombres complexes et sur les quaternions (ainsi que l’esoterique plan projectif des octaves de Cayley) …”.

Individuals acknowledged to be related to Arthur L. Besse include:

Genevieve Averous, Pierre Berard, Lionel Berard Bergery, Marcel Berger, Jean-Pierre Bourguignon, Yves Colin de Verdiere, Andre Derdzinski, Annie Deschamps, Dennis M. DeTurck, Paul Gauduchon, Nigel J. Hitchin, Josette Houillot, Hermann Karcher, Jerry L. Kazdan, Norihito Koiso, Jacques Lafontaine, Rene Michel, Pierre Pansu, Albert Polombo, Pierrette Sentenac, John A. Thorpe, and Liane Valere.

Tony Smith

OT, but no Nobel prize rumor-mongering this year? You’re losing your touch, Peter!

weichi,

I retired from the Nobel prize predictions business after this:

http://www.math.columbia.edu/~woit/wordpress/?p=84

My only prediction for this year is that it’s not going to be in particle physics, since that was last year’s field, and there hasn’t been anything prizeworthy in particle physics for a while now.

It’s not exactly about the Bourbakis but I just learned some very sad news about another great mathematician: Israel Gelfand is no longer with us.

At 96 and after living a very long and fruitful life M.I. Gelfand (working to the very end) passed away. He has been and will be remembered as one of the few truly great mathematicians of the 20 century. Despite of being jewish, managed due to his mathematical ability to stay in Moscow university even in times of anti-semitism. His mathematical work is truly impressive. Rest in peace.

I didn’t read the definition of manifolds using cubes. I see some attempts at defending this by saying it is no worse than using anything else. I’m not sure I agree. But could someone try to articulate exactly why Bourbaki would think that using cubes was somehow *better* than using, say, arbitrary open sets of R^n?

The Nobel Prize in Physics 2009

Charles K. Kao – “for groundbreaking achievements concerning the transmission of light in fibers for optical communication” (1/2 Prize);

AND

Willard S. Boyle and George E. Smith – “for the invention of an imaging semiconductor circuit – the CCD sensor” (1/4 Prize each).

Congratulations!

I seem to be 100% on my Nobel prize predictions. This one was for something I know absolutely nothing about.

I hope someone more able to do justice to him than me will write about Gelfand. He was one of the major figures in developing twentieth century representation theory.

For starters one can look here.

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Bourbaki did write a volume on differential geometry:

Sorry it took so long for me to find it

Varietes differentielles et analytiques – Paragraphes 1-7 (Elements de…

by Nicolas Bourbaki

from

http://www.librarything.com/work/7952041

Thomas,

Yes, there is a Bourbaki text on the subject, but it’s short and just what they call a “fascicule de resultats”, not a detailed treatment with full proofs of the sort they produced for other parts of mathematics.