To most mathematicians, a good LaTeX-frontend (such as TeXShop for Mac-users) is the crucial tool to get the work done. We use it to draft ideas, write papers and courses, or even to take notes during lectures.

However, after six years of blogging, my own LaTeX-routine became rusty. I rarely open a new tex-document, and when I do, I’d rather copy-paste the long preamble from an old file than to start from scratch with a minimal list of packages and definitions needed for the job at hand. The few times I put a paper on the arXiv, the resulting text resembles a blog-post more than a mathematical paper, here’s an example.

As I desperately need to get some math-writing done, I need to pull myself away from the lure of an ever-open WordPress admin browser-screen and reacquaint myself with the far more efficient LaTeX-environment.

Perhaps even my blogging will benefit from the change. Whereas I used to keep on adding to most of my tex-files in order to keep them up-to-date, I rarely edit a blog-post after hitting the ’publish’ button. If I really want to turn some of my better posts into a book, I need them in a format suitable for neverending polishing, without annoying the many RSS-feed aggregators out there.

Who better than Terry Tao to teach me a more proficient way of blogging? A few days ago, Terry announced he will soon have his 5th (!!) book out, after three years of blogging…

How does he manage to do this? Well, as far as I know, Terry blogs in LaTeX and then uses a python-script called LaTeX2WP ’a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress. This way, you can write, and preview, your post in LaTeX, then run LaTeX2WP, and post into WordPress whatever comes out.’ More importantly, one retains a pure-tex-file of the post on which one can keep on editing to get it into a (book)-publishable form, eventually.

Nice, but one can do even better, as Eric from Curious Reasoning worked out. He suggests to install two useful python-packages : WordPressLib “with this library you can control remotely a WordPress installation. Use of library is very simple, you can write a small scripts or full applications that allows you to automate publishing of articles on your blog/site powered by WordPress” and plasTeX “plasTeX is a LaTeX document processing framework written entirely in Python. It currently comes bundled with an XHTML renderer (including multiple themes), as well as a way to simply dump the document to a generic form of XML”. Installation is easy : download and extract the files somewhere, go there and issue a sudo python setup.py install to add the packages to your python.

Finally, get Eric’s own wplatex package and install it as explained there. WpLaTeX has all the features of LaTeX2WP and much more : one can add titles, tags and categories automatically and publish the post from the command-line without ever having to enter the taboo WordPress-admin page! Here’s what I’ve written by now in TeXShop

I’ve added the screenshot and the script will know where to find it online for the blog-version as well as on my hard-disk for the tex-version. Very handy is the iftex … fi versus ifblog … fi alternative which allows you to add pure HTML to get the desired effect, when needed. Remains only to go into Terminal and issue the command

wplpost -x http://www.neverendingbooks.org/xmlrpc.php ReturnToLatex.tex

(if your blog is on WordPress.com it even suffices to give its name, rather than this work-around for stand-alone wordpress blogs). The script asks for my username and password and will convert the tex-file and post it automatic.

However, the Bourbakis never made it to to Spain that summer as the Spanish civil war broke out July 17th, a few weeks before the intended conference. Still, the second Bourbaki-meeting remains often referred to as the ‘Escorial conference’. Can we GEO-tag the exact location of Bourbaki’s “Escorial”?

Claude Chevalley came up with a Plan-B and suggested they would use his parents’ place in Chançay as their venue. Chevalley’s father was a French diplomat and his house sure did possess a matching ‘grandeur’ as can be seen from the famous picture below, taken at the (second) Chançay meeting in 1937 (Weil to the left, Chevalley to the right and Weil’s sister Simonne standing).

Thanks to the Bourbaki archives we know that the meeting took place from september 16th to 28th, that each of them had to pay 16 francs for full pension and had to bring along their own sheets and towels.

But where exactly is this beautiful house? Jacques Borowczyk has written a nice paper Bourbaki et la touraine in which he describes the Bourbaki congresses of 1936 and 1937 at the Chevalley-house in Chançay and further those held in 1956, 1957 and 1959 in ‘hôtel de la Brèche’ in Amboise.

Borowczyk places the Chevalley house in the little hamlet of Chançay, called “La Massoterie”. The village files assert that in 1931 three people were living at La Massoterie : father Abel Chevalley, who took residence there after his retirement in 1931, his wife Marguerite and their son Claude. But, at the time of the Bourbaki congres in 1936, Marguerite remained the only permanent inhabitant. Sadly, Abel Chevalley, who together with Marguerite compiled the The concise Oxford French dictionary, died in 1934.

Usually when you know the name of the hamlet, of the village and add just to be certain ‘France’, Google Maps takes you there within metres. So, this was going to be a quick post, for a change… Well, much to my surprise, typing ‘La Massoterie, Chançay, France’ only produced the answer “We could not understand the location La Massoterie, Chançay, France”.

Did I spell it wrong? Or, did the name change over times? No, Googling for it the first hit gives you the map of a 10km walk around Chançay passing through la Massoterie!

Now what? Fortunately Borowczyk included in his paper an old map, from Napoleonic times, showing the exact location of La Massoterie (just above the flash-sign), facing the castle of Volmer. If you compare it with the picture below from present day Chançay (via Google earth) it is surprising how many of the landmarks have survived the changes over two centuries.

It is now easy to pinpoint the exact location and zoom into the Chavalley-house, and, you’re in for a small surprise : the place is called La Massotterie with 2 t’s…

Probably, Googles database is more reliable than the information provided by the village of Chançay, or the paper by Borowczyk as it is the same spelling as on the old Napoleonic map. Anyway, feel free to have a peek at Bourbaki’s Escorial yourself!

to those expressing and via specific universal polynomials. An attempt to capture the essence of -rings without formulas?

Lenstra’s elegant construction of the 1-power series rings requires only one identity to remember

.

Still, one can use it to show the existence of ringmorphisms , for all numbers . Consider the formal ‘logarithmic derivative’

where is the usual formal derivative of a power series. As this derivative satisfies the chain rule, we have

and so all the maps are additive. To show that they are also multiplicative, it suffices by functoriality to verify this on the special 1-series for all . But,

That is, and Lenstra’s identity implies that is indeed multiplicative! A first attempt :

hassle-free definition 1 : a commutative ring is a -ring if and only if there is a ringmorphism splitting , that is, such that .

In particular, a -ring comes equipped with a multiplicative set of ring-endomorphisms satisfying . One can then define a -ringmorphism to be a ringmorphism commuting with these endo-morphisms.

The motivation being that -rings are known to form a subcategory of commutative rings for which the 1-power series functor is the right adjoint to the functor forgetting the -structure. In particular, if is a -ring, we have a ringmorphism corresponding to the identity morphism.

But then, what is the connection to the usual one involving all the operations ? Well, one ought to recover those from .

For to be a ringmorphism will require identities among the . I hope an expert will correct me on this one, but I’d guess we won’t yet obtain all identities required. By the very definition of an adjoint we must have that is a morphism of -rings, and, this would require defining a -ring structure on , that is a ringmorphism , the so called Artin-Hasse exponential, to which I’d like to return later.

For now, we can define a multiplicative set of ring-endomorphisms from requiring that for all . Another try?

hassle-free definition 2 : is a -ring if and only if there is splitting to satisfying the compatibility relations .

But even then, checking that a map is a ringmorphism is as hard as verifying the lists of identities among the . Fortunately, we get such a ringmorphism for free in the important case when A is of ‘characteristic zero’, that is, has no additive torsion. Then, a ringmorphism exists whenever we have a multiplicative set of ring endomorphisms for all such that for every prime number the morphism is a lift of the Frobenius, that is, .

Perhaps this captures the essence of -rings best (without the risk of getting an headache) : in characteristic zero, they are the (commutative) rings having a multiplicative set of endomorphisms, generated by lifts of the Frobenius maps.

The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The addition on is then recovered by inducing an involution on the odd roots, pairing the one corresponding to x to the one corresponding to x+1.

The second uses Conway’s ’simplicity rules’ to define an addition and multiplication on the set of all ordinal numbers. Conway proves in ONAG that this becomes an algebraically closed field of characteristic two and that is the subfield of all ordinals smaller than . The finite ordinals (the natural numbers) form the quadratic closure of .

On the natural numbers the Conway-addition is binary addition without carrying and Conway-multiplication is defined by the properties that two different Fermat-powers multiply as they do in the natural numbers, and, Fermat-powers square to its sesquimultiple, that is . Moreover, all natural numbers smaller than form a finite field . Using distributivity, one can write down a multiplication table for all 2-powers.

The Knight-seating problems asks for a consistent placing of n-th Knight at an odd root of unity, compatible with the two different realizations of . Last time, we were able to place the first 15 Knights as below, and asked where you would seat

was placed at as 4 was the smallest number generating the ‘Fermat’-field (with multiplicative group of order 15) subject to the compatibility relation with the generator 2 of the smaller Fermat-field (with group of order 15) that .

To include the next Fermat-field (with multiplicative group of order 255) consistently, we need to find the smallest number n generating the multiplicative group and satisfying the compatibility condition . Let’s first concentrate on finding the smallest generator : as 2 is a generator for 1st Fermat-field and 4 a generator for the 2-nd Fermat-field a natural conjecture might be that 16 is a generator for the 3-rd Fermat-field and, more generally, that would be a generator for the next field .

However, an “exercise” in the 1978-paper by Hendrik Lenstra Nim multiplication asks : “Prove that is a primitive root in the field if and only if i=0 or 1.”

I’ve struggled with several of the ‘exercises’ in Lenstra’s paper to the extend I feared Alzheimer was setting in, only to find out, after taking pen and paper and spending a considerable amount of time calculating, that they are indeed merely exercises, when looked at properly… (Spoiler-warning : stop reading now if you want to go through this exercise yourself).

In the picture above I’ve added in red the number to each of the involutions. Clearly, for each pair these numbers are all distinct and we see that for the indicated pairing they make up all numbers strictly less than 8.

By Conway’s simplicity rules (or by checking) the pair (16,17) gives the number 8. In other words, the equation is an irreducible polynomial over having as its roots in the numbers 16 and 17. But then, 16 and 17 are conjugated under the Galois-involution (the Frobenius ). That is, we have and and hence . Now, use the multiplication table in given in the previous post (or compute!) to see that 8 is of order 5 (and NOT a generator). As a consequence, the multiplicative order of 16 is 5×17=85 and so 16 cannot be a generator in . For general i one uses the fact that and are the roots of the polynomial over and argues as before.

Right, but then what is the minimal generator satisfying ? By computing we see that the pairings of all numbers in the range 16…31 give us all numbers in the range 8…15 and by the above argument this implies that the 17-th powers of all numbers smaller than 32 must be different from 4. But then, the smallest candidate is 32 and one verifies that indeed (use the multiplication table given before).

Hence, we must place Knight at root and place the other Knights prior to the 256-th at the corresponding power of 32. I forgot the argument I used to find-by-hand the requested place for Knight 16, but one can verify that so we seat at root .

But what about Knight ? Well, by this time I was quite good at squaring and binary representations of integers, but also rather tired, and decided to leave that task to the computer.

If we denote Nim-addition and multiplication by and , then Conway’s simplicity results in ONAG establish a field-isomorphism between and the field where the satisfy the Artin-Schreier equations

and the i-th Fermat-field corresponds to . The correspondence between numbers and elements from these fields is given by taking . But then, wecan write every 2-power as a product of the and use the binary representation of numbers to perform all Nim-calculations with numbers in these fields.

Therefore, a quick and dirty way (and by no means the most efficient) to do Nim-calculations in the next Fermat-field consisting of all numbers smaller than 65536, is to use sage and set up the field by

R.< x,y,z,t > =GF(2)[]
S.< a,b,c,d >=R.quotient((x^2+x+1,y^2+y+x,z^2+z+x*y,t^2+t+x*y*z))

To find the smallest number generating the multiplicative group and satisfying the additional compatibility condition we have to find the smallest binary number (larger than 255) satisfying

(i1*a*b*c*t+i2*b*c*t+i3*a*c*t+i4*c*t+i5*a*b*t+i6*b*t+
i7*a*t+i8*t+i9*a*b*c+i10*b*c+i11*a*c+i12*c+i13*a*b+
i14*b+i15*a+i16)^257=a*c

It takes a 2.4GHz 2Gb-RAM MacBook not that long to decide that the requested generator is 1051 (killing another optimistic conjecture that these generators might be 2-powers). So, we seat Knight at root and can then arrange seatings for all Knight queued up until we reach the 65536-th! In particular, the first Knight we couldn’t place before, that is Knight , will be seated at root .

If you’re lucky enough to own a computer with more RAM, or have the patience to make the search more efficient and get the seating arrangement for the next Fermat-field, please drop a comment.

I’ll leave you with another Lenstra-exercise which shouldn’t be too difficult for you to solve now : “Prove that has three solutions in for each .”

I wouldn’t be surprised if one or more of these books are borrowed out, probably all of them to the same person. I’m afraid I’m that person in Antwerp…

Lately, there’s been a renewed interest in -rings and the endo-functor W assigning to a commutative algebra its ring of big Witt vectors, following Borger’s new proposal for a geometry over the absolute point.

However, as Hendrik Lenstra writes in his 2002 course-notes on the subject Construction of the ring of Witt vectors : “The literature on the functor W is in a somewhat unsatisfactory state: nobody seems to have any interest in Witt vectors beyond applying them for a purpose, and they are often treated in appendices to papers devoting to something else; also, the construction usually depends on a set of implicit or unintelligible formulae. Apparently, anybody who wishes to understand Witt vectors needs to construct them personally. That is what is now happening to myself.”

Before doing a series on Borger’s paper, we’d better run through Lenstra’s elegant construction in a couple of posts. Let A be a commutative ring and consider the multiplicative group of all ‘one-power series’ over it . Our aim is to define a commutative ring structure on taking as its ADDITION the MULTIPLICATION of power series.

That is, if , then we define our addition . This may be slightly confusing as the ZERO-element in will then turn be the constant power series 1…

We are now going to define a multiplication on which is distributively with respect to and turns into a commutative ring with ONE-element the series .

We will do this inductively, so consider the (classes of) one-power series truncated at term n, that is, the kernel of the natural augmentation map between the multiplicative group-units . Again, taking multiplication in as a new addition rule , we see that is an Abelian group, whence a -module.

For all elements we have a scaling operator (sending ) which is an A-ring endomorphism of , in particular multiplicative wrt. . But then, is an additive endomorphism of , so is an element of the endomorphism-RING . Because composition (being the multiplication in this endomorphism ring) of scaling operators is clearly commutative () we can define a commutative RING being the subring of generated by the operators .

The action turns into an E-module and we define an E-module morphism by .

All of this looks pretty harmless, but the upshot is that we have now equipped the image of this E-module morphism, say (which is the additive subgroup of generated by the elements ) with a commutative multiplication induced by the rule .

Explicitly, is the set of one-truncated polynomials with coefficients in such that one can find elements such that . We multiply with another such truncated one-polynomial (taking elements ) via

and using distributivity and the multiplication rule this gives the element . Being a ring-qutient of we have that is a commutative ring, and, from the construction it is clear that behaves functorially.

For rings such that we are done, but in general may be strictly smaller. The idea is to use functoriality and do the relevant calculations in a larger ring where we can multiply the two truncated one-polynomials and observe that the resulting truncated polynomial still has all its coefficients in .

Here’s how we would do this over : take two irreducible one-polynomials u(t) and v(t) of degrees r resp. s smaller or equal to n. Then over the complex numbers we have and . Then, over the field we have that and hence we can compute their product as before to be . But then, all coefficients of this truncated K-polynomial are invariant under all permutations of the roots and the roots and so is invariant under all elements of the Galois group. But then, these coefficients are algebraic numbers in whence integers. That is, . It should already be clear from this that the rings contain a lot of arithmetic information!

For a general commutative ring we will copy this argument by considering a free overring (with 1 as one of the base elements) by formally adjoining roots. At level 1, consider to be the set of all non-constant one-polynomials over and consider the ring

The idea being that every one-polynomial now has one root, namely in . Further, is a free A-module with basis elements all with .

Good! We now have at least one root, but we can continue this process. At level 2, will be the set of all non-constant one-polynomials over and we use them to construct the free overring (which now has the property that every has at least two roots in ). And, again, we repeat this process and obtain in succession the rings . Finally, we define having the property that every one-polynomial over A splits entirely in linear factors over .

But then, for all we can compute . Remains to show that the resulting truncated one-polynomial has all its entries in A. The ring contains two copies of namely and and the intersection of these two rings in exactly (here we use the freeness property and the additional fact that 1 is one of the base elements). But then, by functoriality of , the element lies in the intersection . Done!

Hence, we have endo-functors in the category of all commutative rings, for every number n. Reviewing the construction of one observes that there are natural transformations and therefore also natural transformations . Taking the inverse limits we therefore have the ‘one-power series’ endo-functor which is ‘almost’ the functor W of big Witt vectors. Next time we’ll take you through the identification using ‘ghost variables’ and how the functor can be used to define the category of -rings.

What does this have to do with finite fields? Well, consider the simplest of all finite field and consider its algebraic closure . Last year, we’ve run a series starting here, identifying the field , following John H. Conway in ONAG, with the set of all ordinals smaller than , given the Nim addition and multiplication. I know that ordinal numbers may be intimidating at first, so let’s just restrict to ordinary natural numbers for now. The Nim-addition of two numbers can be calculated by writing the numbers n and m in binary form and add them without carrying. For example, . Nim-multiplication is slightly more complicated and is best expressed using the so-called Fermat-powers . We then demand that whenever and . Distributivity wrt. can then be used to calculate arbitrary Nim-products. For example, . Conway’s remarkable result asserts that the ordinal numbers, equipped with Nim addition and multiplication, form an algebraically closed field of characteristic two. The closure is identified with the subfield of all ordinals smaller than . For those of you who don’t feel like going transfinite, the subfield is identified with the quadratic closure of .

The connection between and the odd roots of unity has been advocated by Alain Connes in his talk before a general public at the IHES : “L’ange de la géométrie, le diable de l’algèbre et le corps à un élément” (the angel of geometry, the devil of algebra and the field with one element). He describes its content briefly in this YouTube-video

At first it was unclear to me which ‘coupling-problem’ Alain meant, but this has been clarified in his paper together with Caterina Consani Characteristic one, entropy and the absolute point. The non-zero elements of can be identified with the set of all odd roots of unity. For, if x is such a unit, it belongs to a finite subfield of the form for some n, and, as the group of units of any finite field is cyclic, x is an element of order . Hence, can be identified with the set of -roots of unity, with corresponding to a generator of the unit-group. So, all elements of correspond to an odd root of unity. The observation that we get indeed all odd roots of unity may take you a couple of seconds¹.

Assuming we succeed in fixing a one-to-one correspondence between the non-zero elements of and the odd roots of unity respecting multiplication, how can we recover the addition on ? Well, here’s Alain’s coupling function, he ties up an element x of the algebraic closure to the element s(x)=x+1 (and as we are in characteristic two, this is an involution, so also the element tied up to x+1 is s(x+1)=(x+1)+1=x. The clue being that multiplication together with the coupling map s allows us to compute any sum of two elements as . For example, all information about the finite field is encoded in this identification with the 15-th roots of unity, together with the pairing s depicted as

Okay, we now have two identifications of the algebraic closure : the smaller ordinals equipped with Nim addition and Nim multiplication and the odd roots of unity with complex-multiplication and the Connes-coupling s. The question we started from asks for a general recipe to identify these two approaches.

To those of you who are convinced that finite fields (LOL, even characteristic two!) are objects far too trivial to bother thinking about : as far as I know, NOBODY knows how to do this explicitly, even restricting the ordinals to merely the natural numbers!

Please feel challenged! To get you started, I’ll show you how to place the first 15 Knights and give you a procedure (though far from explicit) to continue. Here’s the Nim-picture compatible with that above

To verify this, and to illustrate the general strategy, I’d better hand you the Nim-tables of the first 16 numbers. Here they are

It is known that the finite subfields of are precisely the sets of numbers smaller than the Fermat-powers . So, the first one is all numbers smaller than (check!). The smallest generator of the multiplicative group (of order 3) is 2, so we take this to correspond to the unit-root . The next subfield are all numbers smaller than and its multiplicative group has order 15. Now, choose the smallest integer k which generates this group, compatible with the condition that . Verify that this number is 4 and that this forces the identification and coupling given above.

The next finite subfield would consist of all natural numbers smaller than . Hence, in this field we are looking for the smallest number k generating the multiplicative group of order 255 satisfying the extra condition that which would fix an identification at that level. Then, the next level would be all numbers smaller than and again we would like to find the smallest number generating the multiplicative group and such that the appropriate power is equal to the aforementioned k, etc. etc.

Can you give explicit (even inductive) formulae to achieve this? I guess even the problem of placing Knight 16 will give you a couple of hours to think about… (to be continued).

1. If m is odd, then (2,m)=1 and so 2 is a unit in the finite cyclic group whence , so the m-roots of unity lie within those of order

Remember from last time that we identified Olivier Messiaen as the ‘Monsieur Modulo’ playing the musical organ at the Bourbaki wedding. This was based on the fact that his “modes à transposition limitée” are really about epimorphisms between modulo rings Z/12Z→Z/3Z and Z/12Z→Z/4Z.

However, Messiaen had more serious mathematical tricks up his sleeve. In two of his compositions he did discover (or at least used) one of the smaller sporadic groups, the Mathieu group of order 95040 on which we have based a whole series of Mathieu games two and a half years ago.

Messiaen’s ‘Ile de fey 2′ composition for piano (part of Quatre études de rythme (“Four studies in rhythm”), piano (1949–50)) is based on two concurrent permutations. The first is shown below, with the underlying motive rotational permutation shown.

This gives the permutation (1,7,10,2,6,4,5,9,11,12)(3,8). A second concurrent permutation is based on the permutation (1,6,9,2,7,3,5,4,8,10,11) and both of them generate the Mathieu group . This can be seen by realizing the two permutations as the rotational permutations

and identifying them with the Mongean shuffles generating . See for example, Dave Benson’s book “Music: A Mathematical Offering”, freely available online.

Clearly, Messiaen doesn’t use all of its 95040 permutations in his piece! Here’s how it sounds. The piece starts 2 minutes into the clip.

The second piece is “Les Yeux dans les Roues” (The Eyes in the Wheels), sixth piece from the “Livre d’Orgue” (1950/51).

According to Hauptwerk, the piece consists of a melody/theme in the pedal, accompanied by two fast-paced homorhythmic lines in the manuals. The pedal presents a sons-durées theme which is repeated six times, in different permutations. Initially it is presented in its natural form. Afterwards, it is presented alternatively picking notes from each end of the original form. Similar transformations are applied each time until the sixth, which is the retrograde of the first. The entire twelve-tone analysis (pitch only, not rhythm) of the pedal is shown below:

That is we get the following five permutations which again generate Mathieu 12 :

• a=(2,3,5,9,8,10,6,11,4,7,12)
• b=(1,2,4,8,9,7,11,3,6,12)(5,10)=ea
• c=(1,12,11,9,5,4,6,2,10,7)(3,8)=ed
• d=(1,11,10,8,4,5,3,7,2,9,6)
• e=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)

Considering the permutations and one obtains canonical generators of , that is, generators satisfying the defining equations of this sporadic group

I leave you to work out the corresponding dessin d’enfant tonight after a couple of glasses of champagne! It sure has a nice form. Once again, a better 2010!

The Bourbaki Code

1. The wedding invitation that nearly killed Andre Weil
2. When was the Bourbaki wedding?
3. Where is the Royal Poldavian Academy?
4. Where was the Bourbaki wedding?
5. Seriously now, where was the Bourbaki wedding?

A few days before Halloween, Norbert Dufourcq (who supposedly died on december 17th 1990…), sent me a comment, filled with useful info, and hinting I did mess up big time in the previous post…

Norbert Dufourcq, an organist and student of André Marchal, the organist-in-charge at the Saint-Germain-des-Prés church in 1939, the place where I speculated the Bourbaki wedding took place, concluded his comment with :

“P.S. Lieven, you do know about the Schola Cantorum, now, don’t you?!?”.

Euh… actually … no, I did not …

La Schola Cantorum is a private music school in Paris. It was founded in 1894 by Charles Bordes, Alexandre Guilmant and Vincent d’Indy as a counterbalance to the Paris Conservatoire’s emphasis on opera. Its alumni include many significant figures in 20th century music, such as Erik Satie and Cole Porter.

Schola Cantorum is situated 69, rue Saint Jacques, Paris, just around the corner of the Ecole Normal Superieure, home base to the Bourbakis. In fact, closer investigation reveals striking similarities and very close connections between the circle of artists at la Schola and the Bourbaki group.

In december 1934, the exact month the Bourbaki group was formed, a radical reorganisation took place at the Schola, when Nestor Lejeune became the new director. He invited several young musicians, many from the famous Dukas-class, to take up teaching positions at the Schola.

Here’s a picture of part of the Dukas class of 1929, several of its members will play a role in the upcoming events : from left to right next to the piano : Pierre Maillard-Verger, Elsa Barraine, Yvonne Desportes, Tony Aubin, Pierre Revel, Georges Favre, Paul Dukas, René Duclos, Georges Hugon, Maurice Duruflé. Seated on the right : Claude Arrieu, Olivier Messiaen.

The mid-1930s in Paris saw the emergence of two closely-related groups with a membership which overlapped : La Spirale and La Jeune France. La Spirale was founded in 1935 under the leadership of Georges Migot; its other committee members were Paul Le Flem, his pupil André Jolivet, Edouard Sciortino, Claire Delbos, her husband Olivier Messiaen, Daniel-Lesur and Jules Le Febvre. The common link between almost all of these musicians was their connection with the Schola Cantorum.

On the left : Les Jeunes Musiciens Français : André Jolivet on the Piano. Standing from left to right : Olivier Messiaen, Yves Baudrier, Daniel-Lesur.

Nigel Simeone wrote this about Messiaen and La Jeune France : “The extremely original and independent-minded Messiaen had already shown himself to be a rather unexpected enthusiast for joining groups: in December 1932 he wrote to his friend Claude Arrieu about a letter from another musician, Jacques Porte, outlining plans for a new society to be called Les Jeunes Musiciens Français. Messiaen agreed to become its vice-president, but nothing seems to have come of the project. Six months later, in June 1933, he had a frustrating meeting with Roger Désormière on behalf of the composers he described to Arrieu as ‘les quatre’, all of them Dukas pupils: Elsa Barraine, the recently-deceased Jean Cartan, Arrieu and Messiaen himself; during the early 1930s Messiaen and Arrieu organised concerts featuring all four composers.”

Finally, we’re getting a connection with the Bourbaki group! Norbert Dufourcq mentioned it already in his comment “Messiaen was also a good friend of Jean Cartan (himself a composer, and Henri’s brother)”. Henri Cartan was one of the first Bourbakis and an excellent piano player himself.

The Cartan family picture on the right : standing from left to right, father Elie Cartan (one of the few older French mathematicians respected by the Bourbakis), Henri and his mother Marie-Louise. Seated, the younger children, from left to right : Louis, Helene (who later became a mathematician, herself) and the composer Jean Cartan, who sadly died very young from tuberculoses in 1932…

The december 1934 revolution in French music at the Schola Cantorum, instigated by Messiaen and followers, was the culmination of a process that started a few years before when Jean Cartan was among the circle of revolutionados. Because Messiaen was a fiend of the Cartan family, they surely must have been aware of the events at the Schola (or because it was merely a block away from the ENS), and, the musicians’ revolt may very well have been an example to follow for the first Bourbakis…(?!)

Anyway, we now know the intended meaning of the line “with lemmas sung by the Scholia Cartanorum” on the wedding-invitation. Cartanorum is NOT (as I claimed last time) bad Latin for ‘Cartesiorum’, leading to Descartes and the Saint-Germain-des-Pres church, but is in fact passable Latin (plur. gen.) of CARTAN(us), whence the translation “with lemmas sung by the school of the Cartans”. There’s possibly a double pun intended here : first, a reference to (father) Cartan’s lemma and, of course, to La Schola where the musical Cartan-family felt at home.

Fine, but does this brings us any closer to the intended place of the Bourbaki-Petard wedding? Well, let’s reconsider the hidden ‘clues’ we discovered last time : the phrase “They will receive the trivial isomorphism from P. Adic, of the Order of the Diophantines” might suggest that the church belongs to a a religious order and is perhaps an abbey- or convent-church and the phrase “the organ will be played by Monsieur Modulo” requires us to identify this mysterious Mister Modulo, because Norbert Dufourcq rightfully observed :

“note however that in 1939, it wasn’t as common to have a friend-organist perform at a wedding as it is today: the appointed organists, especially at prestigious Paris positions, were much less likely to accept someone play in their stead.”

The history of La Schola Cantorum reveals something that might have amused Frank Smithies (remember he was one of the wedding-invitation-composers) : the Schola is located in the Convent(!) of the Brittish Benedictines…

In 1640 some Benedictine monks, on the run after the religious schism in Britain, found safety in Paris under the protection of Cardinal Richelieu and Anne of Austria at Val-de-Grace, where the Schola is now housed.

As is the case with most convents, the convent of the Brittish Benedictines did have its own convent church, now called l’église royale Notre-Dame du Val-de-Grâce (remember that one of the possible interpretations for “of the universal variety” was that the name of the church would be “Notre-Dame”…).

This church is presently used as the concert hall of La Schola and is famous for its … musical organ : “In 1853, Aristide Cavaillé-Coll installed a new organ in the Church of Sainte-geneviève which had been restored in its rôle as a place of worship by Prince President Louis-Napoléon. In 1885, upon the decision of President Jules Grévy, this church once again became the Pantheon and, six years later, according to an understanding between the War and Public Works Departments, the organ was transferred to the Val-de-Grâce, under the supervision of the organ builder Merklin. Beforehand, the last time it was heard in the Pantheon must have been for the funeral service of Victor Hugo. In 1927, a raising was carried out by the builder Paul-Marie Koenig, and the inaugural concert was given by André Marchal and Achille Philippe, the church’s organist. Added to the register of historic monument in 1979, Val-de-Grâce’s “ little great organ ”, as Cavaillé-Coll called it, was restored in 1993 by the organ builders François Delangue and Bernard Hurvy. The organ of Val-de-Grâce is one the rare parisian surviving witnesses of the art of Aristide Cavaillé-Coll, an instrument that escaped abusive and definitive transformations or modernizations. This explain why, in spite of its relatively modest scale, this organ enjoys quite a reputation, and this, as far as the United States.”

By why would the Val-de-Grace organiste at the time Achille Philip, “organiste titulaire du Val-de-Grâce de 1903 à 1950 et professeur d’orgue et d’harmonie à la Schola Cantorum de 1904 à 1950″, be called ‘Mister Modulo’ in the wedding-invitations line “L’orgue sera tenu par Monsieur Modulo”???

Again, the late Norbert Dufourcq comes to our rescue, proposing a good candidate for ‘Monsieur Modulo’ : “As for “modulo”, note that the organist at Notre-Dame at that time, Léonce de Saint-Martin, was also the composer of a “Suite Cyclique”, though I admit that this is just wordplay: there is nothing “modular” about this work. Maybe a more serious candidate would be Olivier Messiaen (who was organist at the Église de la Trinité): his “modes à transposition limitée” are really about Z/12Z→Z/3Z and Z/12Z→Z/4Z. “

Messiaen’s ‘Modes of limited transposition’ were compiled in his book ‘Technique de mon langage musical’. This book was published in Paris by Leduc, as late as 1944, 5 years after the wedding-invitation.

Still, several earlier works of Messiaen used these schemes, most notably La Nativité du Seigneur, composed in 1935 : “The work is one of the earliest to feature elements that were to become key to Messiaen’s later compositions, such as the extensive use of the composer’s own modes of limited transposition, as well as influence from birdsong, and the meters and rhythms of Ancient Greek and traditional Indian music.”

More details on Messiaen’s modes and their connection to modular arithmetic can be found in the study Implementing Modality in Algorithmic Composition by Vincent Joseph Manzo.

Hence, Messiaen is a suitable candidate for the title ‘Monsieur Modulo’, but would he be able to play the Val-de-Grace organ while not being the resident organist?

Remember, the Val-de-Grace church was the concert hall of La Schola, and its musical organ the instrument of choice for the relevant courses. Now … Olivier Messiaen taught at the Schola Cantorum and the École Normale de Musique from 1936 till 1939. So, at the time of the Bourbaki-Petard wedding he would certainly be allowed to play the Cavaillé-Coll organ.

Perhaps we got it right, the second time around : the Bourbaki-Pétard wedding was held on June 3rd 1939 in the church ‘l’église royale Notre-Dame du Val-de-Grâce’ at 12h?

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The Bourbaki Code

1. The wedding invitation that nearly killed Andre Weil
2. When was the Bourbaki wedding?
3. Where is the Royal Poldavian Academy?
4. Where was the Bourbaki wedding?
5. Seriously now, where was the Bourbaki wedding?

I’m pretty certain I got the intended date & time of the Bourbaki-Pétard wedding right : June 3rd 1939 at 12h. Finding the exact location of the wedding-ceremony is an entirely different matter. And, quite probably, we are reading way too much in these pranks of the Weil-clan.

Still, it’s fun trying to find an elegant answer, based on the (intended or imagined) clues in the text and the little we know about the early Bourbaki-days. Here, the translation of the relevant part of the wedding announcement :

“They will receive the trivial isomorphism from P. Adic, of the Order of the Diophantines, in the Principal Cohomology of the Universal Variety, on the third of Cartember, year VI, at the usual hour.
The organ will be played by Monsieur Modulo, Assintant Simplex of the Grassmannian (with lemmas sung by the Scholia Cartanorum). The collection will be donated in full to the retirement home for Poor Abstracts. Convergence will be guaranteed.”

First solution : Perhaps one might read “in the Principal Cohomology of the Universal Variety” as : “in the Principal Church of the generic type/name”. In many French cities the main church is the Cathedral and an awful lot of them are called Notre Dame, so it might mean : in the Notre Dame Cathedral. But even then, we have to choose between these two

On the left, the Notre Dame Cathedral in Paris. On the right the Cathédrale Notre-Dame-de-l’Annonciation in Nancy. As the invitation promises guests to be entertained after the ceremony by Monsieur et Madame Bourbaki at their ‘Fundamental Domains’, the choice depends on the location of the Bourbaki-household in June 1939.

‘Bourbaki’ made two applications to become an AMS-member. The first, in 1948, tells us that Bourbaki is a scientific advisor to the Hermann Publishing Co. in Paris since 1934, and, the second in 1950, that he is ‘Directeur Libre de Recherches a l’Université de Nancy’. I couldn’t find out when exactly Nicolas did change cities, and even Liliane Beaulieu’s talk Bourbaki a Nancy does not provide an answer.

Second solution : Or, one can read that sentence as a mathematical, perhaps proto-motivic, statement, and, hunt for clues elsewhere in the text. But then, what are these clues?

• Mass is celebrated by “P. Adic, of the Order of the Diophantines”. This suggests that the church itself belongs to a monastic order, and is perhaps a convent-church.
• Hymns are “sung by the Scholia Cartanorum”. Scholia Cartanorum is Latin of sorts and refers perhaps to the Paris’ Latin Quarter, le Quartier Latin.
• The collection is donated to the “retirement home for Poor Abstracts”. Perhaps the church is connected to a saint for the poor.

Let’s consider “Scholia Cartanorum” more closely. It may be Latin, admittedly very bad Latin, for ‘the Scholiums of Cartesius’, that is, ‘of Descartes‘.

One of the more famous ‘Scholia’ in scientific history is Newton’s general scholium to the Principia, which is a prime example of Descartes-bashing. Newton attacks Descartes on his vortical theory of planetary motion, his aeter to explain gravity, his God-axiom (unlike Descartes, Newton induced God from nature, rather than starting with God as an axiom) and his hypothetico-deductive method. So, there is a link between Descartes and ‘Scholium’, although the genitive form ‘Cartesiorum’ might be fairly inappropriate…

But then, Descartes died on 11 February 1650 in Stockholm (Sweden) where he was buried, so there won’t be a connection to a French or Parisian church, right? Well, not quite. The fate of Descartes’ remains is a rather strange story : “In 1666, sixteen years after his death, the bones of René Descartes were dug up in the middle of the night and transported from Sweden to France under the watchful eye of the French Ambassador. This was only the beginning of the journey for Descartes’ bones, which, over the next 350 years, were fought over, stolen, sold, revered as relics, studied by scientists, used in séances, and passed surreptitiously from hand to hand. ” For example, during the French Revolution, his remains were disinterred for burial in the Pantheon in Paris among the great French thinkers. But today, his ashes are burried in…

the abbay church of Saint-Germain-des-Prés, located in the Quartier Latin, within walking distance of the Bourbaki-café Capoulade and the Ecole Normal Superieure.

Now all the hints fall handsomely in place. St-Germain-des-Prés is the oldest church in Paris. Parts of it date to the 6th century, when a Benedictine abbey was founded on the site by Childebert, son of Clovis. Hence the sentence ‘in the Principal Cohomology of the Universal Variety’ might simply mean ‘in the first church, ever’. In medieval times, the Left Bank of Paris was prone to flooding from the Seine, so much of the land could not be built upon and the Abbey stood in the middle of fields, or prés in French, thereby explaining its appellation.

The other part of its name, Saint Germain, comes from Saint Germanus of Paris, also known as the ‘father of the poor’ (!). His remains were interred in St. Symphorien’s chapel in the vestibule of St. Vincent’s church, but in 754, when he was canonized, his relics were solemnly removed into the body of the church, in the presence of Pepin and his son, Charlemagne, then a child of seven, and the church was reconsecrated as Saint-Germain-des-Prés. That is, also the remains of the ‘father of the poor’ are buried in this church.

Here’s my best guess : the Bourbaki-Pétard wedding was held on June 3rd 1939 in the church Saint-Germain-des-Prés at 12h. Genuine aficionados of the Da Vinci code may regret it wasn’t held in the neighboring Saint-Sulpice church, but then, perhaps someone can bend the clues accordingly…

Remains this problem : who was the organist, Monsieur Modulo? Suggestions anyone?

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An excellent account of the first 10 ‘proto-Bourbaki’ meetings in the Capoulade-Café, 63 boulevard Saint- Michel, is told magnificently by Liliane Beaulieu in her 1993 paper A Parisian Café and Ten Proto-Bourbaki Meetings(1934-1935). Here we will concentrate on the preparations of the Besse congress.

At their very first meeting on december 10th 1934, they already state the importance of the upcoming summer-congress where a precise plan and a distribution of the writing-load for the first volumes will be discussed : “Aux prochaines grandes vacances aura lieu une réunion pléniere d’ou sortira un plan définitif tres précis et une répartition du travail de rédaction des différents fascicules”. The second meeting, on january 14th 1935, decides that the definite list of Bourbaki-members will consist of those (among the nine ‘possibles’ Weil, Delsarte, Mandelbrojt, Cartan, Dubreil, Dieudonné, de Possel, Chevalley and Leroy) present at the congress : “Il est entendu que la liste définitive, extraite de la précédente, sera composée des noms des membres présents a la réunion pléniere d’Aout ou Septembre prochain, réunion dans laquelle sera dressé le plan définitif et précis du traité”.

On march 25th 1935, the first precise plans are made about the location, the lodgings and the extremely important issue of the meals which will be taken in a nearby Hotel of which “la cuisine est, parait-il, fort bonne”.

René de Possel obtained a mandate to do whatever it took for the group to have their congress in Besse between 12 and 25 July, and, to enquire until what date they could still change their mind. His biography contains the following lines : “On many issues de Possel and André Weil were on opposite sides in the arguments. At the first Bourbaki congress in July 1935 de Possel was still an active member of the group and much involved with contributing but, largely due to differences with Weil, he dropped out of the project. De Possel married Yvonne Liberati on 12 August 1935; they had three children, Yann, Maya, and Daphné.”

By and large, the Bourbaki-differences between de Possel and Weil were of a professional nature. They had different mathematical interests, different mathematical talents, different ambitions, and, a different level of commitment wrt. the work ahead (Weil being the lazier one of the two). Still, it is difficult to understand the group-dynamics of the first generation Bourbakis without mentioning a personal tragedy, often ‘forgotten’ (as in the above biography) or given no more than half a sentence, in passing…

Aged 24, René de Possel marries Evelyne Gillet in 1929 and their son, Alain, is born on august 16th 1931. However, the marriage breaks up, one account dates the separation in 1933, another around the time of the Besse conference in 1935.

What is certain is that Evelyne Gillet and André Weil start a relationship no later than the autumn of 1935. At that time, Weil is concocting the Bourbaki Comptes-Rendus note and as the Academy demands a short biography of the author, he has to come up with a first name (at the Besse conference, they only decided on the name ‘N. Bourbaki’). Evelyne chooses Nicolas and is referred to ever after as ‘Bourbaki’s godmother’. Early 1936, the couple spends a vacation together in Spain.

Early 1937, the official divorce papers come through, allowing Weil and Evelyne to marry on october 30th 1937. The very same year, René de Possel remarries with Yvonne Liberati. For more information, you can traverse Evelyne’s genealogy-tree here, but bear in mind that not all information is included (for example, Evelyne died on may 24th, 1986).

Contrary to the suggestion made in the biography, there is no evidence that de Possel left the Bourbaki-group as a result of this affaire or because of his arguments with Weil. In fact, at least until the second Chançay-congress in 1938, de Possel was one of the hardest workers in the group, present at all meetings, doing his share of the write-up and even chastising his fellow-Bourbakis for not being as committed to the project as they ought to be, see for example the 7 theses of Chançay document. It was only in the fall of 1941 that de Possel asked to be transferred to the university of Algiers and left the Bourbaki-group.

At the meeting of march 25th 1935, de Possel attempts a coup d’état. He comes up with an entirely new plan for the summer-congress. Paul Valéry, the French poet, essayist, and philosopher (in the notules he is described as ‘le célebre fantaisiste’) proposed the Bourbaki-group to use his ‘centre universitaire mediterranéen’ (the proto-University of Nice) as their place of venue. They could choose any period between july and october and they wouldn’t have to pay a thing! de Possel was in contact with Valéry at that time, he was writing a 44 page booklet on game theory Sur la théorie mathématique des jeux de hasard et de réflexion, with a preface by Paul Valéry, which appeared later in 1937 via Valéry’s center for mediterranean studies.

There is one small catch though … Valéry insists that de Possel should be president of the Bourbaki-group during the meeting! Naturally, this wasn’t received enthusiastically by the others, but they didn’t rule the plan out, requesting additional information and observing that july and august may be way too hot in Nice.

The next meeting (May 6th 1935), de Possel tries to increase the pressure by asserting that the original Besse-plan is in danger because “Les naturalistes de Clermond-Ferrand semblent vouloir se servir de ce qui leur appartient” (the biologists of Clermond-Ferrand want to use their facilities themselves). But the others are not impressed and they give de Possel “pleins pouvoirs pour réagir avec violence.”

A fortnight later, Weil demands to know the latest on the Besse-negotiations and de Possel replies “en principe les biologistes de Clermond-Ferrand pourront y séjourner des le 15 juin, il y a tout lieu de présumer que ces derniers ne seront que trois ou quatre; ils seront donc fort peu génante étant donné le nombre des locaux dont nous pourrons disposer”, that is, there won’t be more than 3 or 4 biologists around, and, there’s plenty of room for everyone!

Putsch averted, the Bourbakis can start packing their suitcases, hire a secretary for the meeting, and split the costs among all committee-members. Because even this circulaire is preserved, we now know such trivia as the cost of full-pension in the Besse-Hotel with the excellent kitchen : 25 Ffr/day…