70 search results for "bourbaki"

#cestGrothendieck

Published September 30, 2011 by lievenlb

In Belgium the hashtag-craze of the moment is #cestjoelle. Joelle Milquet is perceived to be the dark force behind everything, from the crisis in Greece, over DSK, to your mother-in-law coming over this weekend? #cestjoelle.

Sam Leith used the same meme in his book the coincidence engine.

A hurricane assembling a passenger jet out of old bean-cans? #cestGrothendieck

All shops in Alabama out of Chicken & Broccoli Rica-A-Roni? #cestGrothendieck

Frogs raining down on Atlanta? #cestGrothendieck

As this is a work of fiction, Alexandre Grothendieck‘s name is only mentioned in the ‘author’s note’:

“It is customary to announce on this page that all resemblances to characters living or dead are entirely coincidental. It seems only courteous to acknowledge, though, that in preparing the character of Nicolas Banacharski I was inspired by the true-life story of the eminent mathematician Alexandre Grothendieck.”

The name ‘Nicolas Banacharski’ is, of course, chosen on purpose (the old Bourbaki NB-joke even makes an appearance). The character ‘Isla Holderness’ is, of course, Leila Schneps, the ‘Banacharski ring’ is, of course, the Grothendieck circle. But, I’d love to know the name of the IRL-‘Fred Nieman’, who’s described as ‘an operative for the military’.

Sam Leith surely knows all the Grothendieck-trivia which shouldn’t come as a surprise because he wrote in 2004 a piece for the Spectator on the ‘what is a metre?’ incident (see also this n-category cafe post).

The story of ‘the coincidence engine’ is that Grothendieck did a double (or was it triple) bluff when he dropped out of academia in protest of military money accepted by the IHES. He went into hiding only to work for a weapons company and to develop a ‘coincidence bomb’. As more and more unlikely events happen during a car-ride by a young Cambridge postdoc though the US (to propose to his American girlfriend), the true Grothendieck-aficianado (and there are still plenty of them in certain circles) will no doubt begin to believe that the old genius succeeded (once again) and that Ana’s (Grothendieck’s mother) $\infty$-ring is this devilish (pun intended) device…

However,

“There was no coincidence engine. Not in this world. It existed only in Banacharski’s imagination and in the imaginations he touched. But there was a world in which it worked, and this world was no further than a metre from our own. Its effect spilled across, like light through a lampshade.

And with that light there spilled, unappeased and peregrine, fragments of any number of versions of an old mathematician who had become his own ghost. Banacharski was neither quite alive nor quite dead, if you want the truth of it. He was a displaced person again, and nowhere was his home.”

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From WordPress to ePublishing (1)

Published July 11, 2011 by lievenlb

Perhaps, the tips and tricks I did receive to turn a selection of wordpress-posts into a proper ePub-file may be of use to others, so I will describe the procedure here in some detail.

It makes a difference whether or not some of the posts contain TeX. This time, I’ll sketch the process for non-LaTeX posts and hence we will turn the Bourbaki-code posts into ePub-format to read on your iPad, rather than merely into a pdf-file as last time. Next time, I’ll add some tricks to repeat this when some of your posts do contain LaTeX.

1. Install the ePub export plugin

Get the epub export wordpress plugin, install and activate it in the usual way. ePub Export automatically creates an ePub file when a post or page is published or updated. The ePubs are stored in the uploads directory. For later use, remember that your uploads directories are located under BLOGHOME/wp-content/uploads.

2. Update the posts you want to include

Decide which posts you want to include in your eBook, edit them (or not) and press for each one of them the update-button. This will populate today’s uploads-directory with a number of epub-files. Transfer them all, for example using Transmit, to a directory on your home-computer, named say MyFirstEBook.

3. Unpack the epub-files

The crucial fact to remember about epub-files is that they are really zipped archives containing xhtml-, css- and other files and directories. As we want to edit some of those, we first have to unpack the directories. So, change the .epub extensions to .zip and double-click on them to create the directories.

The crucial files in each directory are preface.xhtml (containing title author and blog-name), text.html (containing the blog-post) and the images-directory (containing copies of all the images used in the post).

4. Rename the directories user-friendly

As all the posts will be chapters in our eBook in some specific order, we will rename the numbers of the directories to something more user-friendly such as shortened blog-titles. To do this, double-click in each of the directories on the preface.xhtml file. This will open Safari and will show the title of the blog-post. Use it to rename that directory. For convenience let us call the directory corresponding to the first chapter in our book MasterDirectory

5. Move all images to the master directory

For each of the other chapter-directories, drag all the files contained in the images-subdirectory to MasterDirectory/images.

6. Edit the MasterDirectory/text.xhtml file

Because we will have to open and copy-paste all the text.xhtml files of the different directories, it is perhaps best to rename momentarily the MasterDirectory/text.xhtml file to something like master.xhtml.

Now, open this file with a text-processor such as TextWrangler. Edit it to remove unwanted html-code (such as links to other posts at the start if you are using the series-plugin, or previous/next post links at the end). Also add the title of the blog-post between h1-tags (and if you want to include a table of contents later, give it an anchor-name).

Go to the directory of your second chapter, open that text.xhtml file and copy/paste only the post-content over to the master-xhtml file at the appropriate place. As before, add title/anchor before the copied post-content.

Repeat this procedure, in order, for all the chapters of your eBook.

Once finished, doubleclick the master.xhtml file and correct remaining errors (as it is an xhtml-file, it is rather picky about opening and closing tags) and see whether all your images are included. If you’re satisfied with it, rename the master.xhtml to text.xhtml (don’t forget this!).

7. Edit the MasterDirectory/preface.xhtml file

Open the preface.xhtml file and change the first blogpost-title to the title of your booklet, alter your name (by default it uses your wp-nick) and add a frontipiece-picture if you so desire.

8. Re-package the directory into an epub-file

This is the (only) tricky part. E-book readers require that the mimetype file is the first one in the zip document. What’s more, to be fully compliant, this file should start at a very specific point – a 30-byte offset from the beginning of the zip file (so that the mimetype text itself starts at byte 38).

Here’s how to do this on a Mac (Linux-users being the geeks they are will have given up on reading this post a while ago and as to Windows-users, yeah well …). Open Terminal.app and cd to your MasterDirectory. Now type:

zip -X MyFirstEBook.epub mimetype

Next, type:

zip -rg MyFirstEBook.epub * -x *.DS_Store

(of course you’ll have to change your book-title to whatever you want). If you want to know more about these 2 magical commands, read this post.

9. Edit metadata

Get Calibre and add the MyFirstEBook.epub to Calibre by clicking on the ‘Add Books’ button.

You can preview your eBook by clicking on the ‘View’-button. Next, click the ‘Edit metadata’-button and alter the title and author entries (and whatever else you want to include) and click the OK button. Then click ‘Save to disk’.

10. Read your eBook on your iPad

Finally, we want to see how it looks on the iPad. Mail MyFirstEBook.epub to yourself as attachement, open it with iBooks and enjoy!

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eBook ‘geometry and the absolute point’ v0.1

Published July 8, 2011 by lievenlb

In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis.

The idea being to edit these posts thoroughly, add much more detail (and proofs) and also add some extra sections on Borger’s work and Witt rings (and possibly other stuff).

For those of you who prefer to (re)read these posts on paper or on a tablet rather than perusing this blog, you can now download the very first version (minimally edited) of the eBook ‘geometry and the absolute point’. All comments and suggestions are, of course, very welcome. I hope to post a more definite version by mid-september.

I’ve used the thesis-documentclass to keep the same look-and-feel of my other course-notes, but I would appreciate advice about turning LaTeX-files into ‘proper’ eBooks. I am aware of the fact that the memoir-class has an ebook option, and that one can use the geometry-package to control paper-sizes and margins.

Soon, I will be releasing a LaTeX-ed ‘eBook’ containing the Bourbaki-related posts. Later I might also try it on the games- and groups-related posts…

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Brigitte Bardot, miniskirts and homological algebra

Published May 20, 2011 by lievenlb

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

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

Brigitte Bardot and the Hom-Tensor relation

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

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

$Hom_R(A,Hom_S(B,C))=Hom_S(A \otimes_R B,C)$

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

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

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

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

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

Miniskirts and spectral sequences

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

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

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Who was ‘le P. Adique’?

Published July 8, 2010 by lievenlb

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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Olivier Messiaen & Mathieu 12

Published December 31, 2009 by lievenlb

To mark the end of 2009 and 6 years of blogging, two musical compositions with a mathematical touch to them. I wish you all a better 2010!

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 $M_{12} $ 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 $M_{12} $. This can be seen by realizing the two permutations as the rotational permutations

and identifying them with the Mongean shuffles generating $M_{12} $. 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)=e*a
c=(1,12,11,9,5,4,6,2,10,7)(3,8)=e*d
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)

Here’s the piece performed on organ :

Considering the permutations $X=d.a^{-1} $ and $Y=(a.d^2.a.d^3)^{-1} $ one obtains canonical generators of $M_{12} $, that is, generators satisfying the defining equations of this sporadic group

$X^2=Y^3=(XY)^{11}=[X,Y]^6=(XYXYXY^{-1})^6=1 $

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!

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The artist and the mathematician

Published October 19, 2009 by lievenlb

Over the week-end I read The artist and the mathematician (subtitle : The story of Nicolas Bourbaki, the genius mathematician who never existed) by Amir D. Aczel.

Whereas the central character of the book should be Bourbaki, it focusses more on two of Bourbaki’s most colorful members, André Weil and Alexander Grothendieck, and the many stories and myths surrounding them.

The opening chapter (‘The Disappearance’) describes the Grothendieck’s early years (based on the excellent paper by Allyn Jackson Comme Appelé du Néant ) and his disappearance in the Pyrenees in the final years of last century. The next chapter (‘An Arrest in Finland’) recount the pre-WW2 years of Weil and the myth of his arrest in Finland and his near escape from execution (based on Weil’s memoires The Apprenticeship of a Mathematician). Chapter seven (‘The Café’) describes the first 10 proto-Bourbaki meetings following closely the study ‘A Parisian Café and Ten Proto-Bourbaki Meetings (1934-1935)‘ by Liliane Beaulieu. Etc. etc.

All the good ‘Bourbaki’-stories get a place in this book, not always historically correct. For example, on page 90 it is suggested that all of the following jokes were pulled at the Besse-conference, July 1935 : the baptizing of Nicolas, the writing of the Comptes-Rendus paper, the invention of the Bourbaki-daughter Betti and the printing of the wedding invitation card. In reality, all of these date from much later, the first two from the autumn of 1935, the final two no sooner than april 1939…

One thing I like about this book is the connection it makes with other disciplines, showing the influence of Bourbaki’s insistence on ‘structuralism’ in fields as different as philosophy, linguistics, anthropology and literary criticism. One example being Weil’s group-theoretic solution to the marriage-rules problem in tribes of Australian aborigines studied by Claude Lévi-Strauss, another the literary group Oulipo copying Bourbaki’s work-method.

Another interesting part is Aczel’s analysis of Bourbaki’s end. In the late 50ties, Grothendieck tried to convince his fellow Bourbakis to redo their work on the foundations of mathematics, changing these from set theory to category theory. He failed as others felt that the foundations had already been laid and there was no going back. Grothendieck left, and Bourbaki would gradually decline following its refusal to accept new methods. In Grothendieck’s own words (in “Promenade” 63, n. 78, as translated by Aczel) :

“Additionally, since the 1950s, the idea of structure has become passé, superseded by the influx of new ‘categorical’ methods in certain of the most dynamical areas of mathematics, such as topology or algebraic geometry. (Thus, the notion of ‘topos’ refuses to enter into the ‘Bourbaki sack’ os structures, decidedly already too full!) In making this decision, in full cognizance, not to engage in this revision, Bourbaki has itself renounced its initial ambition, which has been to furnish both the foundations and the basic language for all of modern mathematics.”

Finally, it is interesting to watch Aczel’s own transformation throughout the book, from slavishly copying the existing Weil-myths and pranks at the beginning of the book, to the following harsh criticism on Weil, towards the end (p. 209) :

“From other information in his autobiography, one gets the distinct impression that Weil was infatuated with the childish pranks of ‘inventing’ a person who never existed, creating for him false papers and a false identity, complete with a daughter, Betti, who even gets married, parents and relatives, and membership in a nonexistent Academy of Sciences of the nonexistent nation of Polvedia (sic).
Weil was so taken with these activities that he even listed, as his only honor by the time of his death ‘Member, Poldevian Academy of Sciences’. It seems that Weil could simply not go beyond these games: he could not grasp the deep significance and power of the organization he helped found. He was too close, and thus unable to see the great achievements Bourbaki was producing and to acknowledge and promote these achievements. Bourbaki changed the way we do mathematics, but Weil really saw only the pranks and the creation of a nonexistent person.”

Judging from my own reluctance to continue with the series on the Bourbaki code, an overdose reading about Weil’s life appears to have this effect on people…

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Where is the Royal Poldavian Academy?

Published October 11, 2009 by lievenlb

Among the items found on Andre Weil at the time of his arrest was “a packet of calling cards belonging
to Nicolas Bourbaki, member of the Royal Academy of Poldavia”.

But then, where is the Royal Poldavian Academy situated? Well, surely in the Kingdom of Poldavia, which is a very strange country indeed, its currency unit being the bourbaki and there exist only two types of coins: gold ones (worth n bourbakis) and silver ones (worth m bourbakis). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. Prove that any payment above mn-2 bourbakis can be made without the need to receive change.

However, the Kingdom of Poldavia isn’t another Bourbaki concoction. The name goes back at least to a joke pulled by the right-wingers of the Action Francaise in may 1929. Here’s the TIME article of May 20th 1929 :

“When 28 French Republican deputies sat down to their breakfast coffee and croissants early last week, each found a large crinkly letter from Geneva in his morning’s mail. Innocent and refreshed after a sound night’s sleep, not one Republican deputy saw anything untoward in the fact that the large crinkly letters were embossed on the stationery of “Foreign Minister Lamidaeff, of the Kingdom of Poldavia.” They saw nothing strange in the fact that Poldavians were in financial difficulties, and they found Minister Lamidaeff most thoughtful in not asking for money, but merely for an expression of “moral support” from the Deputies in his campaign to aid Poldavian sufferers. “We believe that our interests were betrayed at the Peace Conference,” wrote Poldavian Lamidaeff. “and we appeal to you as a member of the French Parliament to do your utmost to help us in this our hour of need. The whole nation of Poldavia and its noble monarch who disregarded personal safety in 1916, and joined France in her War for justice and righteousness, pray you to remember our sacrifices.”

What could be fairer than that? Legislators all over the world are always ready to write enthusiastic platitudes in favor of anything that sounds like a good cause. The wronged Poldavians seemed a very good cause. Each of the 28 deputies sat down at his desk and pledged his moral support to “Foreign Minister Lamidaeff of Poldavia.”

None of the 28 deputies noticed that the old Poldavian name of Lamidaeff might read “I’Ami d’A. F.”—”the friend of A. F.,” “the friend of L’Action Française” famed royalist newspaper of which the editor is Leon Daudet, bon vivant, practical jokester, son of famed Author Alphonse Daudet (Tartarin de Tarascon), exile from the republic he has so consistently lampooned (TIME, June 13, 1927, et seq.). Three days after the 28 gullible deputies replied to the “Poldavian Minister,” a special edition of L’Action Française appeared.

“People of France,” wrote exiled Editor Daudet, who once escaped from La Sante prison through a hoaxed release order telephoned from the office of the Minister of the Interior, “—People of France, how much longer will you permit such ignorant deputies to represent you before the world? Here are 28 of your elected representatives, and they actually believe there is a Kingdom of Poldavia, and that Lamidaeff is its Foreign Minister. Lamidaeff, c’est moi!””

The consul of Poldavia also appears in the 1936 Tintin-story The Blue Lotus by Hergé. In view of the above AF-connection, it should’t come as a surprise that Hergé is often accused of extreme-right sympathies and racism.

To some, Poldavia is a small country in the Balkans, to others it lies in the Caucasus, but has disappeared from the map of Europe. All accounts do agree on one point, namely that Poldavia is a mountainous region.

Today we are pleased to disclose the exact location of the Royal Poldavian Academy, and, thanks to the wonders of Google Earth you can explore the Kingdom of Poldavia at your leisure if you give it the coordinates 45.521082N,2.935495E. Or, you can use the Google-map below :

View Larger Map

The evidence is based on a letter sent by Andre Weil to Elie Cartan when the Bourbakis wanted to submit a note for the Comptes Rendus des Séances Hebdomadaires de l’Académie des Sciences under the pseudonyme Nicolas Bourbaki. As the academy requires a biographical note on the author, Weil provided the following information about Bourbaki’s life :

“Cher Monsieur,
Je vous envoie ci-joint, pour les C.R., une note que M.Bourbaki m’a chargé de vous transmettre. Vous n’ignorez pas que M.Bourbaki est cet ancien professeur à l’Université Royale de Besse-en-Poldévie, dont j’ai fait la connaissance il y a quelque temps dans un café de Clichy où il passe la plus grande partie de la journée et même de la nuit ; ayant perdu, non seulement sa situation, mais presque toute sa fortune dans les troubles qui firent disparaître de la carte d’Europe la malheureuse nation poldève, il gagne maintenant sa vie en donnant, dans ce café, des leçons de belote, jeu où il est de première force.
Il fait profession de ne plus s’occuper de mathématiques, mais il a bien voulu cependant s’entretenir avec moi de quelques questions importantes et même [ajout manuscrit : me laisser] jeter un coup d’œil sur une partie de ses papiers ; et j’ai réussi à le persuader de publier, pour commencer, la note ci-jointe, qui contient un résultat fort utile pour la théorie moderne de l’intégration, je pense que vous ne verrez pas de difficulté à l’accueillir pour les Comptes-Rendus ; si même les renseignements que je vous donne au sujet de M.Bourbaki ne paraissaient pas suffisamment clairs, j’imagine qu’il n’appartient à l’Académie, et en particulier à celui qui présente la note, que de s’assurer de la valeur scientifique de celle-ci, et non de faire une enquête au sujet de l’auteur. Or j’ai examiné soigneusement le résultat de M.Bourbaki, et son exactitude est hors de doute.
Veuillez recevoir, je vous prie, les remerciements de M.Bourbaki et les miens, et croyez toujours à mes sentiments bien affectueusement et respectueusement dévoués.
A.Weil”

That is, ‘Besse-en-Poldevie’, or simply ‘Besse’ as in this line from the wedding announcement “Mademoiselle Betti Bourbaki, a former student of the Well-Ordereds of Besse” must be the capital of Poldavia where the Academy is housed.

You may have never heard of Poldavia, but if you are a skiing or cycling enthusiast, the name of its capital sure does ring a bell, or rather so does the name of its sub-part Super Besse. The winter sports resort of Super Besse is located in the commune of Besse-et-Saint-Anastaise in the Parc naturel régional des volcans d’Auvergne in the department of Puy de Dôme, in Auvergne. Situated approximately 50 km from Clermont-Ferrand, it is located at an altitude of 1350 m on the slopes of Puy de Sancy, Puy de la Perdrix and Puy Ferrand. Surely a mountainous region …

Besse-et-Saint-Anastaise, or rather Besse-en-Chandesse as it was formerly called, was the venue of the very first Bourbaki Congres 1935. Surely, they used the ‘Royal Poldavian Academy’ as their meeting place. But, where is it?

At the Besse meeting were present : Claude Chevalley, Jean Dieudonné, René de Possel, Henri Cartan, Szolem Mandelbrojt, Jean Delsarte, André Weil, the physicist Jean Coulomb, Charles Ehresmann and a ‘cobaye’ called Mirles.

Of these men three held a position at the University Blaise Pascal of Clermont-Ferrand : Mandelbrojt, de Possel and Coulomb and they arranged that the Bourbaki-group could use the universities’ biology-outpost in Besse-en-Chandesse. Photographic evidence for this is provided by the man standing apart in the right hand-picture above : the biologist Luc Olivier.

Concluding : the Royal Poldavian Academy is located at the ‘Station Biologique de l’Université Blaise Pascal’, Rue du Lavoir, Besse-et-Saint-Anastaise.

On July 12th 2003 a ceremony was held at the Biology-station commemorating the birth of Nicolas Bourbaki (the group), supposedly born July 12th 1935. A plate at an exterior wall of the Biology-station was unveiled.

More information about the mysterious country of Poldavia can be found in the article La verité sur la Poldévie by Michele Audin.

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The wedding invitation that nearly killed Andre Weil

Published October 6, 2009 by lievenlb

Andre Weil wrote about his arrest as a Russian spy in november 1939 :

“The manuscripts they found appeared
suspicious – like those of Sophus Lie, arrested on charges of spying
in Paris, in 1870. They also found several rolls of stenotypewritten
paper at the bottom of a closet. When I said these were the text of a
Balzac novel, the explanation must have seemed far-fetched. There was
also a letter in Russian, from Pontrjagin, I believe, in response to
a letter I had written at the beginning of the summer regarding a
possible visit to Leningrad; and a packet of calling cards belonging
to Nicolas Bourbaki, member of the Royal Academy of Poldavia, and even
some copies of his daughter Betti Bourbaki’s wedding invitation, which
I had composed and had printed in Cambridge several months earlier in
collaboration with Chabauty and my wife.”

I’ve always wondered how on earth the Finnish police could interpret mathematical texts as coded messages. Reading the ‘faire-part’ (attempted translation below) it is hard to view it as anything but a coded message…

Here it is : a copy of the ‘faire-part’ of Betti Bourbaki’s wedding to Hector Petard, that nearly did cost Andre Weil’s life.

Monsieur Nicolas Bourbaki, Canonical Member of the Royal Academy of Poldavia, Grand Master of the Order of Compacts, Conserver of Uniforms, Lord Protector of Filters, and Madame nee One-to-One, have the honor of announcing the mariage of their daughter Betti with Monsieur Hector Petard, Delegate Administrator of the Society of Induced Structures, Member of the Institute of Classified Archeologists, Secretary of the Work of the Lion Hunt.

Monsieur Ersatz Stanislas Pondiczery, retired First Class Covering Complex, President of the Reeducation Home for Weak Convergents, Chevalier of the Four U’s, Grand Operator of the Hyperbolic Group, Knight of the Total Order of the Golden Mean, L.U.B., C.C., H.L.C., and Madame nee Compact-in-itself, have the honor of announcing the marriage of their ward Hector Petard with Mademoiselle Betti Bourbaki, a former student of the Well-Ordereds of Besse.

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.

The organ will be played by Monsieur Modulo, Assistant Simplex of the Grassmannian (Lemmas will be sung by Scholia Cartanorum). The result of the collection will be given to the House of Retirement foor Poor Abstracts, Convergence is assured.

After the congruence, Monsieur and Madame Bourbaki will receive guests in their Fundamental Domain; there will be dancing with music by the Fanfare of the VIIth Quotient Field.

Canonical Tuxedos (ideals left of the buttonhole). QED.

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seen this quiver?

Published March 25, 2005 by lievenlb

The above quiver on 10 vertices is not symmetric, but has the
interesting property that every vertex has three incoming and three
outgoing arrows. If you have ever seen this quiver in another context,
please drop me a line. My own interest for it is that it is the ‘one
quiver’ for a non-commutative compactification of $GL_2(\mathbb{Z}) $. If
you like to know what I mean by this, you might consult the
Granada-notes which I hope to post over the weekend.

On a
different matter, if you want to know what all this hype on derived
categories and the classification project is about but got lost in the
pile of preprints, you might have a look at the Bourbaki talk by Raphael
Rouquier Categories
derivees et geometrie birationnelle posted today on the arXiv.

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