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Le Guide Bourbaki : Pelvoux

Pelvoux is a former commune (now merged into Vallouise-Pelvoux) in the Hautes-Alpes department in the Provence-Alpes-Côte d’Azur region in southeastern France. No less than five summer-Bourbaki congresses took place in Pelvoux:

  • La Tribu 25 : Congres oecumenique de Pelvoux (June 25th – July 8th 1951)
  • La Tribu 28 : Congres de la motorisation de l’ane qui trotte (June 25th – July 8th 1952)
  • La Tribu 45 : Congres des hyperplans (June 25th – July 7th 1958)
  • La Tribu 48 : Congres de cerceau (June 25th – July 8th 1959)
  • La Tribu 51 : Congres un peu sec (June 25th – July 7th 1960)

Bourbaki’s Diktat of the 1951-meeting tells us:

“The 1951 Ecumenical Congress will be held at the Hotel d’Ailefroide, Pelvoux-le-Poet (Hautes Alpes), from June 25 at 10 a.m. to July 8 at 6 p.m. The recommended means of communication are:
A) The train, Briancon line, get off at Argentiers la Bessee then the bus, direct to the hotel.
B) plane, boat, helicopter
Please do not confuse the Hotel d’Ailefroide in Pelvoux-le-Poet with the locality of Ailefroide which is elsewhere.”

You can still book a stay at Le Chalet Hotel d’Ailefroide in Pelvoux, but we will see that this is not the place we are looking for.

From the history of the Rolland family in Ailefroide:

“In 1896, Jean, the older brother of our grandfather Pierre, built two hotels simultaneously: the Hôtel d’Ailefroide in the hamlet of Poêt, very close to our family home, and the Chalet Hôtel d’Ailefroide, very close to our chalet ‘le Saint Pierre’.”

The ‘Hôtel d’Ailefroide’ in Poêt no longer functions as hotel, but some old postcards of its circulate on the web:

To convince ourselves that this is really the place of venue of the Bourbaki-congresses, compare the balustrade of the terras, and the main entrance door of the Hotel to these two pictures taken at the 1951-conference:

(From left to right: Jacques Dixmier, Jean Dieudonne, Pierre Samuel, Andre Weil, Jean Delsarte, and partially hidden, Laurent Schwartz.)

(Laurent Schwartz before Bourbaki’s famous portable blackboard.)

We’ve seen that in Amboise, Bourbaki made pilgrimages to Chancay. When in Pelvoux, He made a pilgrimage towards Les Bans, where Herbrand fell to his death.

Jacques Herbrand was considered one of the greatest younger logicians and number-theorists when he fell to his death on july 27th 1931, only 23 years old, while mountain-climbing in the Massif des Ecrins.

He was awarded a Rockefeller fellowship that enabled him to study in Germany in 1931, first with John von Neumann in Berlin, then during June with Emil Artin in Hamburg, and finally with Emmy Noether in Göttingen.

Herbrand was a close friend of Andre Weil and, in particular, of Claude Chevalley. From Chevalley is the quote: “Jacques Herbrand would have hated Bourbaki”.

In the summer-vacation of 1931 he went mountain-climbing with a couple of friends in the French Alps. They set off from Le refuge de la Pilatte

and took the normal route to the south summit of “Les Bans” (the blue, followed by green track in the map below). A more detailed description of the route and its difficulties can be found here.

They did reach the summit, as illustrated by this classic picture of Herbrand (in the mddle) but the accident happened in the descent.

The French mathematical society has donated a commemorative plaque to the chapel of Notre Dame des Neiges in La Bérarde.

For much more information, see this excellent article by Mathouriste.

From La Tribu 25 (translated by Maurice Mashaal in Bourbaki, a secret society of mathematicians, page 108):

“In addition to the regime imposed by the High Commisions, a terrible schism threatens Bourbaki, that between the mountaineers and the couch potatoes. Faced with an alpine valley, one person is afraid of snow and makes a dash for the Tropics, another rebels against ‘these horrible mountains, enormous masses lacking formality and structure’, a third, motorized, is surprised by the insistence of the mountaineers to be driven to the bottom of each and every valley and abandons them to their sad fate. On the other hand, a delegation representing all ages and ranks sets off to survey glaciers and neves, defy crevasses and mountain sickness, and plant Bourbaki’s flag above Refuge Caron, at 3160 meters.”

Mashaal’s book also contains a picture (copyright Archives Association de N. Bourbaki) of the delegation of mountaineers, taken on Wednesday July 4th 1951, with the Barre des Ecrins in the background:

From left to right: Laurent Schwartz, Andre Weil, Pierre Cartier, Pierre Samuel, Jean-Pierre Serre, and their guide. Presumably, Terry Mirkil took the photograph.

Present at the congress were: Cartan, Delsarte, Dieudonne, Dixmier, Godement, Sammy, Samuel, Schwartz, Serre, Weil; the Foreign visitors : Hochschild, Borel,and Guinea pigs : Cartier and Mirkil.

I’ll let you figure out who Bourbaki’s couch potatoes were.

As we are on a mission to find all places of Bourbaki congresses in the 50ties, does the building of the ‘Hôtel d’Ailefroide’ in Pelvoux-le-Poêt still exists, and what is its exact location?

Coordinates: 44.853904, 6.492673.

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Le Guide Bourbaki : Amboise

Between 1955 and 1960 four Bourbaki congresses were held in Amboise, a small market town on the river Loire, and once home of the French royal court.

  • La Tribu 38, from March 11th-17th 1956, ‘Congres des trois angles plats’
  • La Tribu 40, fromOctober 7th-14th 1956, ‘Congres de l’intelligence peu commune’
  • La Tribu 41, from March 17th-24th 1957, ‘Congres du foncteur inflexible’
  • La Tribu 47, from March 7thh-14th 1959, ‘Congres “Chez mon cousin”‘

Seldom a congress-location was described in such detail. On page 1 of La Tribu 38 one reads:

‘The congress was held in Amboise from March 11th till March 17th 1956, in the salons de l’Hotel de la Breche, situated in the rue de Pocé, between the railway station and the bridge.’

Hotel de la Breche, Amboise in 1956 (Photo from Bourbaki et la Touraine by Jacques Borowczyk)

Today, there is no rue de Pocé in Amboise, but the Hotel de la Breche still exists, the restaurant run by a father-daughter combo as chefs. Its address is 26, Rue Jules Ferry, Rive Droite, 37400 Amboise. The Rue Jules Ferry goes from the centre of Amboise in the direction of nearby Pocé-sur-Cisse so it may have been named Rue de Pocé in the 50ties. It definitely is the same Hotel.

In this period several of the Bourbaki-members obtained prestigious positions at Institutes and Universities, resulting in some banter in La Tribu.

In La Tribu 38 page 2 the expulsion is threatened of all members which are not ‘Professors of the first rank’.

“In the meantime, the regulations have been supplemented by articles making it compulsory to wear a broken collar and a tie, the use of the word ‘Monsieur’ when speaking of the undisputed leaders of La Sorbonne and the College, formal address will be compulsory between members, and the guinea pigs will use the third person to address their elders.”

Recall that Jean-Pierre Serre received the Fields medal at age 27 in 1954, and was nominated in 1956 as the youngest Professor of the Collège de France (chair of algebra and geometry).

Claude Chevalley had a difficult time after WW2 to get a position at a French university as he stayed in the US when war broke out. Eventually his friends managed to create a chair for him at La Sorbonne in 1957 (chair of analytic geometry and group theory). (see here for a list of all chairs in mathematics over the years).

From La Tribu 47 page 2:

“Inspired by his writings on Logic, Bourbaki wondered if the system of axioms formed by the Motchane Institute, the Princeton Institute, the College, Polytechnique and the little Sorbonne is compatible; it seems that we are on the way to an affirmative answer thanks to the work of various congressmen whom La Tribu does not want to name.”

Here, ‘l’Institut Motchane’ if of course the IHES, which was founded in 1958 by businessman and mathematical physicist Léon Motchane, with the help of Robert Oppenheimer and Jean Dieudonné, who would become the first permanent professor. Dieudonne accepted the position only after Grothendieck was also offered a position.

L’Institut de Princeton is the Institute for Advanced Studies where Andre Weil obtained a permanent position in 1958. We saw already that ‘College’ means Serre, and ‘Sorbonne’ Chevalley.

Amboise is not far from Chancay where the second and third pre-WW2 Bourbaki-conferences were held, at the estate of the parents of Chevalley in La Massotterie, where this iconic picture was taken.

During at least three of the four meetings in Amboise a pilgrimage to Chancay was organised.

In La Tribu 38 on page 2:

“A pilgrimage to Chancy gives rise to a great sponging session. Some will regret that there was no cellar visit session.”

In La Tribu 40 on page 2:

“We find all the same the strength and the courage to go to Chancay to taste white wine, and meditate on the sheaves of germs of carrots.”

Finally, in La Tribu 47 on page 2:

“Accompanied by a plumber, the Congress made a pilgrimage to Chancay; he finds that the pipes were not leaking excessively, and that the tap at Vouvray was even working very well.”

Note that Vouvray is an ‘appellation d’origine contrôlée’ of white wines produced around the village Vouvray, so all white wines from Chancay are Vouvray-wines.

The first few pages of most La Tribu-issues are full of these tiny tidbits of French knowledge. Perhaps I should start another series ‘La Tribu Trivia’?

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Le Guide Bourbaki : Sallieres-les-bains

For three summers in a row, Bourbaki held its congres in ‘Sallieres-les-bains’, located near Die, in the Drôme.

  • La Tribu 36, from June 27th till July 9th 1955
  • La Tribu 39, from June 24th till July 7th 1956, ‘Congres des Tapis’
  • La Tribu 42, from June 23rd till July 7th 1957, ‘Congres oecumenique du diabolo’

There are several ways to determine the exact location of a Bourbaki congres.

The quickest one is to get hold of the corresponding Diktat which often not only gives the address, but also travel instructions on how to get there. But strangely, no Diktats for congresses after 1953 are cleared by the ACNB.

Next, one can look in the previous La Tribu issue, as it contains a section on the next congres. In La Tribu 35 we find on page 2 : “Next congres : from June 25th till July 6th, in a location to be determined”. La Tribu 38 on page 5 reads (rough translation)

Next congres : will be held around the usual dates (June 23rd till July 7th, with a margin of 2 or 3 days). To facilitate things for Borel and Weil, Koko (=Koszul) will quickly look for a pleasant place in the Vosges or l’Alsace region. If this fails, he’ll immediately warn Cartan who will then take care of Die.

La Tribu 41 mentions on page 6: next congress will be held in Die from June 23rd till July 7th.

Finally, it may be that La Tribu itself gives more details. Strangely, La Tribu 36 is not among the issues recently cleared by the ACNB.

We know of its existence from Kromer’s paper La Machine de Grothendieck, and from a letter from Serre to Grothendieck from July 13th 1955 in which he writes that the Bourbaki congres in Sallieres-les-bains went well and that Grothendieck’s paper on Homological algebra (now known as the Tohoku-paper) was carefully read and converted everyone (‘even Dieudonne, who seems completely functorised’).

In La Tribu 39 we immediately strike gold, the heading tells us that the congres was held in the ‘Etablissement Thermo-resineux de Sallieres les bains’.

But, if you google for this, all you get are some pretty old postcards, such as this one

with one exception, a site set up to save the chapel of the Thermes de Sallieres-les-bains, which gives some historical information (google-translated):

“In Die, in the middle of the 19th century, the thermo-resinous establishments of Salières-les-bains opened. Until 1972, i.e. for 120 years, spa guests came there every summer to treat their bronchial tubes and rheumatism with the vapours of mugho pine. The center of Die and its cathedral being 4km away, it is in this 51m² chapel that the curists gathered. Mass was even sometimes said there because a priest was regularly among the spa guests. But after the closure, the small family farm can no longer maintain all the large buildings of the inn and their chapel, whose roof has now collapsed…”

And, there is the book Des bains de vapeurs térébenthinés aux pastilles de Pin mugho by Cécile Raynal, containing a short paragraph on the installation in Sallieres: (G-translate)

“Located a short distance from Martouret, this hydro-mineral establishment was created by a breeder, owner of the Sallieres estate, Mr. Taillotte. He equipped himself with facilities for resinous baths and also used hydrotherapy. More especially frequented by the patients of the surroundings, under the supervision of Dr. Magnan, a doctor from Die, the establishment charged moderate prices and functioned only in the summer. The installations would have lasted until the 1970s.”

So, it is perfectly possible that the Bourbakis stayed here in the mid 50ties. But, how did they know of this place and what’s the link with Cartan?

If you look at the map (Sallieres is the red marker) you’ll find in the immediate neighborhood the former Abbey of Valcroissant (for the Dome du Glandasse read La Tribu 42, page2)

“The abbey was bought in the 1950s by the mathematician and philosopher Marcel Légaut and his wife, who chose to restore it while maintaining agricultural activity, particularly livestock. The restoration led in particular to the classification of the abbey in the inventory of historical monuments, a classification which took place on October 25, 1971. The restoration continued in the 21st century, led by Rémy Légaut, son of Marcel, his wife Martine, and the association of “Friends of Valcroissant” created by André Pitte and Serge Durand.”

Marcel Legaut was a very interesting person, who did a Grothendieck avant-la-lettre. From wikipedia

“Marcel Légaut was born in Paris, where he received his Ph.D. in Mathematics from the École Normale Supérieure in 1925. He taught in various faculties (among them Rennes and Lyon) until 1943. Under the impact of the Second World War and the rapid French defeat in 1940, Légaut acknowledged the lack of certain fundamental aspects in his life as well as in the lives of other university professors and civil servants. That is why he tried to alternate teaching with farm work. After three years his project was no longer accepted and he left the University to live as a shepherd in the Pré-Alpes (Haut-Diois).”

Legaut also wrote about twenty books on catholic faith. Again from wikipedia (and compare to Grothendieck’s later years):

“Légaut offers, in his books, his meditation, his testimony and his prayer, resulting from the intimate conversation he holds with himself, with his friends and with God. Meditation, testimony and prayer are, in every human being, the three categories corresponding to the different destinataries of intimate “conversation”, which is, in short, the sort of communication that every spiritual life aims to achieve according to its deep instinct.”

Marcel Legaut is also one of the 24 ‘mutants’ in Grothendieck’s Clef des songes. Is it possible the two met during the Bourbaki congres in Sallieres-les-bains?

In this article on Legaut there’s this recollection by Pierre Cartier:

“Pierre Cartier believes that Grothendieck and Légaut had already met in the fifties, on the occasion of a Bourbaki meeting which took place in the Alps in Pelvoux-le Poët. Légaut, who lived at no great distance, was acquainted with Henri Cartan, André Weil and other members of Bourbaki. Cartier remembers that he himself visited Légaut at the time, and recalls Légaut actually attending the Bourbaki meeting.”

I beg to differ on the place of the Bourbaki meeting, I’m convinced it was during a congres in Sallieres-les-bains. We now also see the link with Cartan. Probably it was Legaut who mentioned the nearby wellness-center to Cartan.

Do the buildings of the ‘Etablissement Thermo-resineux de Sallieres les bains’ still exist, and what is their exact location?

If you intend to go on a little pelgrimage, point your GPS to 44.737347, 5.398835. Perhaps you can stay for a few days in the renovated Abbaye de Valcroissant, they offer courses in herbal medicine, aromatherapy and natural cosmetics, which are organised from March to November.

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Le Guide Bourbaki : Marlotte

During the 1950ties, the Bourbakistas usually scheduled three meetings in the countryside. In the spring and autumn at places not too far from Paris (Royaumont, Celles-sur-plaines, Marlotte, Amboise…), in the summer they often went to the mountains (Pelvoux, Murols, Sallieres-les-bains,…).

Being a bit autistic, they preferred to return to the same places, rather than to explore new ones: Royaumont (6 times), Pelvoux (5 times), Celles-sur-plaine (4 times), Marlotte (3 times), Amboise (3 times),…

In the past, we’ve tried to pinpoint the exact locations of the pre-WW2 Bourbaki-conferences: in 1935 at le Station Biologique de l’Université Blaise Pascal’, Rue du Lavoir, Besse-et-Saint-Anastaise, in 1936 and 1937 at La Massotterie in Chancay, and in 1938 at l’ecole de Beauvallon (often mistakingly referred to as the ‘Dieulefit-meeting’).

Let’s try to do the same for their conferences in the 1950ties. Making use of the recent La Tribu releases for he period 1953-1960, let’s start arbitrarily with the 1955 fall meeting in Marlotte.

Three conferences were organised in Marlotte during that period:

  • La Tribu 37 : ‘Congres de la lune’, October 23-29 1955
  • La Tribu 43 : ‘Congres de la deuxieme lune’, October 6-11 1957
  • La Tribu 44 : ‘Congres des minutes de silence’, March 16-22 1958

Grothendieck was present at all three meetings, Weil at the last two. But let us return to the fight between these two (‘congres des minutes de silence’) regarding algebraic geometry/category theory in another post.

Today we’ll just focus on the location of these meetings. At first, this looks an easy enough task as on the opening page of La Tribu we read:

“The conference was held at the Hotel de la mare aux canards’ (‘Hotel of the duck pond’) in Marlotte, near Fontainebleau, from October 23rd till 29th, 1955”.

Just one little problem, I can’t find any reference to a ‘Hotel de la Mare aux Canards’ in Marlotte, neither at present nor in the past.

Nowadays, Bourron-Marlotte is mainly a residential village with no great need for lodgings, apart from a few ‘gites’ and a plush hotel in the local ‘chateau’.

At the end of the 19th century though, there was an influx of painters, attracted by the artistic ‘colonie’ in the village, and they needed a place to sleep, and gradually several ‘Auberges’ and Hotels opened their doors.

Over the years, most of these hotels were demolished, or converted to family houses. The best list of former hotels in Marlotte, and their subsequent fate, I could find is L’essor hôtelier de Bourron et de Marlotte.

There’s no mention of any ‘Hotel de la mare aux canards’, but there was a ‘Hotel de la mare aux fées’ (Hotel of the fairy pond), which sadly was demolished in the 1970ties.

There’s little doubt that this is indeed the location of Bourbaki’s Marlotte-meetings, as the text on page one of La Tribu 37 above continues as (translation by Maurice Mashaal in ‘Bourbaki a secret society of mathematicians’, page 109):

“Modest and subdued sunlight, lustrous bronze leaves fluttering in the wind, a pond without fairies, modules without end, indigestible stones, and pierced barrels: everything contributes to the drowsiness of these blasé believers. ‘Yet they are serious’, says the hotel-keeper, ‘I don’t know what they are doing with all those stones, but they’re working hard. Maybe they’re preparing for a journey to the moon’.”

Bourbaki didn’t see any fairies in the pond, only ducks, so for Him it was the Hotel of the duck pond.

In fact La mare aux fées is one of the best known spots in the forest of Fontainebleau, and has been an inspiration for many painters, including Pierre-August Renoir:

Here’s the al fresco restaurant of the Hotel de la mare aux fées:

Both photographs are from the beginning of the 20th century, but also in the 50ties it was a Hotel of some renown as celebreties, including the actor Jean Gabin, stayed there.

The exact location of the former Hotel de la mare aux fées is 83, Rue Murger in Bourron-Marlotte.

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The (somewhat less) Secret Bourbaki Archive

It has been many, many years since I’ve last visited the Bourbaki Archives.

The underground repository of the Bourbaki Secret Archives is a storage facility built beneath the cave of the former Capoulade Cafe. Given its sporadic use by staff and scholars, the entire space – including the Gallery of all intermediate versions of every damned Bourbaki book, the section reserved to Bourbaki’s internal notes, such as his Diktats, and all numbers of La Tribu, and the Miscellania, containing personal notes and other prullaria once belonging to its members – is illuminated by amber lighting activated only when movement is detected by strategically placed sensors, and is guarded by a private security firm, hired by the ACNB.

This description (based on that of the Vatican Secret Archives in the book The Magdalene Reliquary by Gary McAvoy) is far from the actual situation. The Bourbaki Archive has been pieced together from legates donated by some of its former members (including Delsarte, Weil, de Possel, Cartan, Samuel, and others), and consist of well over a hundredth labeled carton and plastic cases, fitting easily in a few standard white Billy Ikea bookcases.

The publicly available Bourbaki Archive is even much smaller. The Association des collaborateurs de Nicolas Bourbaki has strong opinions on which items can be put online. For years the available issues of La Tribu were restricted to those before 1953. I was once told that one of the second generation Bourbaki-members vetoed further releases.

As a result, we only had the fading (and often coloured) memories of Bourbaki-members to rely on if we wanted to reconstruct key events, for example, Bourbaki’s reluctance to include category theory in its works. Rather than to work on source material, we had to content ourselves with interviews, such as this one, the relevant part starts at 51.40 into the clip. See here for some other interesting time-slots.

On a recent visit to the Bourbaki Archives I was happy to see that all volumes of “La Tribu” (the internal newsletter of Bourbaki) are now online from 1940 until 1960.

Okay, it’s not the entire story yet but, for all you Grothendieck aficionados out there, it should be enough as G resigned from Bourbaki in 1960 with this letter (see here for a translation).

Grothendieck was present at just twelve Bourbaki congresses in the period between 1955 and 1960 (he was also present as a ‘cobaye’ at a 1951 congress in Nancy).

The period 1955-60 was crucial in the modern development of algebraic geometry. Serre’s ‘FAC’ was published, as was Grothendieck’s ‘Tohoku-paper’, there was the influential Chevalley seminar, and the internal Bourbaki-fight about categories and the functorial view.

Perhaps the definite paper on the later issue is Ralf Kromer’s La ‘Machine de Grothendieck’ se fonde-t-elle seulement sur les vocables metamathematiques? Bourbaki et les categories au cours des annees cinquante.

Kromer had access to most issues of La Tribu until 1962 (from the Delsarte archive in Nancy), but still felt the need to justify his use of these sources to the ACNB (footnote 9 of his paper):

“L’autorisation que j’ai obtenue par le Comité scientifique des Archives de la création des mathématiques, unité du CNRS qui fut chargée jusqu’en 2003 de la mise à disposition de ces archives, me donne également le droit d’utiliser les sources datant des années postérieures à l’année 1953, que j’avais consultées auparavant aux Archives Jean Delsarte, soit avant que l’ACNB (Association des Collaborateurs de Nicolas Bourbaki) ne rende publique sa décision d’ouvrir ses archives et ne décide des parties qui seraient consultables.

J’ai ainsi bénéficié d’une occasion qui ne se présenterait sans doute plus aujourd’hui, mais c’est en toute légitimité que je puis m’appuyer sur cette riche documentation. Toutefois, la collection des Archives Jean Delsarte étant à son tour limitée aux années antérieures à 1963, je n’ai pu étudier la discussion ultérieure.”

The Association des Collaborateurs de Nicolas Bourbaki made retirement from active B-membership mandatory at the age of 50. One might expect of it to open up all documents in its archives which are older than fifty years.

Meanwhile, we’ll have a go at the 1940-1960 issues of La Tribu.

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the topos of unconsciousness

Since wednesday, as mentioned last time, the book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient” is available in the better bookshops.

There’s no need to introduce Alain Connes on this blog. Patrick Gauthier-Lafaye is a French psychiatrist and psycho-analyst, working in Strassbourg.

The book is a lengthy dialogue in which the authors try to find a use for topos theory in Jaques Lacan’s psycho-analytical view of the unconscious.

If you are a complete Lacanian virgin, it may be helpful to browse through “Lacan, a beginners guide” (by Lionel Bailly) first.

If this left you bewildered, for example by Lacan’s strange (ab)use of mathematics, rest assured, you’re not alone.

It is no coincidence that Lacan’s works are the first case-study in the book “Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science” by Alan Sokal (the one of the hoax) and Jean Bricmont. You can download the book from this link.

If now you feel that Sokal and Bricmont are way too harsh on Lacan, I urge you to have a go at the book “Writing the structures of the subject, Lacan and topology” by Will Greenshields.

If you don’t have the time or energy for this, let me give you one illustrative example: the topological explanation of Lacan’s formula of fantasy:

\$~\diamond~a \]

Loosely speaking this formula says “the barred subject stands within a circular relationship to the objet petit a (the object of desire), one part of which is determined by alienation, the other by separation”.

Lacan was obsessed with the immersion of the projective plane $\mathbb{P}^2(\mathbb{R})$ into $\mathbb{R}^3$ as the cross-cap. Here’s an image of it from his 1966-67 seminar on ‘Logique du fantasme’ (213 pages).

This image includes the position of the objet petit $a$ as the end point of the self-intersection curve, which itself is referred to as the ‘castration’, or the ‘phallus’, or whatever.

Brace yourself for the ‘explanation’ of $\$~\diamond~a$: if you walk twice around $a$ this divides the cross-cap into a disk and a Mobius-strip!

The mathematics is correct but I fail to see how this helps the psycho-analyst in her therapy. But hey, everyone will tell you I have absolutely no therapeutic talent.

Let’s return to the brand new book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient”.

It was to be expected that they would defend Lacan’s exploitation of (surface) topology by saying that he was just unfortunate not to have the more general notion of toposes available, as well as their much subtler logic. Perhaps someone should write a fictional parody on Greenshields book: “Lacan and the topos”…

Connes’ first attempt to construct the topos of unconsciousness was also not much of a surprise. According to Lacan the unconscious is ‘structured like a language’.

So, a natural approach might be to start with a ‘dictionary’-category (words and relations between them) or any other known use of a category in linguistics. A good starting point to read up on this is the blog post A new application of category theory in linguistics.

Eventually they settled for a much more ambitious project. To Connes and Gauthier-Lafaye every individual has her own topos and corresponding logic.

They don’t specify how to construct these individual toposes, but postulate that they are all connected to a classifying topos, which is their incarnation of the world of ‘myths’ and ‘fantasies’.

Surely an idea Lacan would have liked. Underlying the unconscious must be, according to Connes and Gauthier-Lafaye, a geometric theory! That is, it can be fully described by first order sentences.

Lacan himself used already some first order sequences in his teachings, such as in his logic of sexuation:

\forall x~(\Phi~x)~\quad \text{but also} \quad \exists x~\neg~(\Phi~x) \]

where $\Phi~x$ is the phallic function. Quoting from Greenshield’s book:

“While all (the sons) are subject to ($\forall x$) the law of castration ($\Phi~x$), we also learn that this law nevertheless resides upon an exception: there exists a subject ($\exists x$) that is not subject to this law ($\neg \Phi~x$). This exception is embodied by the despotic father who, not being subject to the phallic function, experiences an impossible mode of totalised jouissance (he enjoys all the women). He is, quite simply, the exception that proves the law a necessary beyond that enables the law’s geometric bounds to be defined.”

It will be quite hard (but probably great fun for psycho-analysts) to turn the whole of Lacanian theory on the unconscious into a coherent geometric theory, construct its classifying topos, and apply the Joyal-Reyes theorem to get at the individual cases/toposes.

I’m sure there are much deeper insights to be gained from Connes’ and Gauthier-Lafaye’s book, but this is what i got from a first, fast, cursory reading of it.

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Grothendieck meets Lacan

Next month, a weekend-meeting is organised in Paris on Lacan et Grothendieck, l’impossible rencontre?.

Photo from Remembering my father, Jacques Lacan

Jacques Lacan was a French psychoanalyst and psychiatrist who has been called “the most controversial psycho-analyst since Freud”.

What’s the connection between Lacan and Grothendieck? Here’s Stephane Dugowson‘s take (G-translated):

“As we know, Lacan was passionate about certain mathematics, notably temporal logic and the theory of knots, where he thought he found material for advancing the theory of psychoanalysis. For his part, Grothendieck testifies in his non-strictly mathematical writings to his passion for the psyche, as shown by many pages of his Récoltes et Semailles just published by Gallimard (in January 2022), or even, among the tens of thousands of pages discovered at his death and of which we know almost nothing, the 3700 pages of mathematics grouped under the title ‘Structure of the Psyche’.

One might therefore be surprised that the two geniuses never met. In fact, a lunch did take place in the early 1970s organized by the mathematician and psychoanalyst Daniel Sibony. But a lunch does not necessarily make a meeting, and it seems that this one unfortunately did not happen.”

As it is ‘bon ton’ these days in Parisian circles to utter the word ‘topos’, several titles of the talks given at the meeting contain that word.

There’s Stephane Dugowson‘s talk on “Logique du topos borroméen et autres logiques à trois points”.

Lacan used the Borromean link to illustrate his concepts of the Real, Symbolic, and Imaginary (RSI). For more on this, please read chapter 6 of Lionel Baily’s excellent introduction to Lacan’s work Lacan, A Beginner’s Guide.

The Borromean topos is an example of Dugowson’s toposes associated to his ‘connectivity spaces’. From his paper Définition du topos d’un espace connectif I gather that the objects in the Borromean topos consist of a triple of set-maps from a set $A$ (the global sections) to sets $A_x,A_y$ and $A_z$ (the restrictions to three disconnected ‘opens’).

\xymatrix{& A \ar[rd] \ar[d] \ar[ld] & \\ A_x & A_y & A_z} \]

This seems to be a topos with a Boolean logic, but perhaps there are other 3-point connectivity spaces with a non-Boolean Heyting subobject classifier.

There’s Daniel Sibony‘s talk on “Mathématiques et inconscient”. Sibony is a French mathematician, turned philosopher and psychoanalyst, l’inconscient is an important concept in Lacan’s work.

Here’s a nice conversation between Daniel Sibony and Alain Connes on the notions of ‘time’ and ‘truth’.

In the second part (starting around 57.30) Connes brings up toposes whose underlying logic is much subtler than brute ‘true’ or ‘false’ statements. He discusses the presheaf topos on the additive monoid $\mathbb{N}_+$ which leads to statements which are ‘one step from the truth’, ‘two steps from the truth’ and so on. It is also the example Connes used in his talk Un topo sur les topos.

Alain Connes himself will also give a talk at the meeting, together with Patrick Gauthier-Lafaye, on “Un topos sur l’inconscient”.

It appears that Connes and Gauthier-Lafaye have written a book on the subject, A l’ombre de Grothendieck et de Lacan : un topos sur l’inconscient. Here’s the summary (G-translated):

“The authors present the relevance of the mathematical concept of topos, introduced by A. Grothendieck at the end of the 1950s, in the exploration of the structure of the unconscious.”

The book will be released on May 11th.


The monster prime graph

Here’s a nice, symmetric, labeled graph:

The prime numbers labelling the vertices are exactly the prime divisors of the order of the largest sporadic group: the monster group $\mathbb{M}$.
\# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^ \]

Looking (for example) at the character table of the monster you can check that there is an edge between two primes $p$ and $q$ exactly when the monster has an element of order $p.q$.

Now the fun part: this graph characterises the Monster!

There is no other group $G$ having only elements of these prime orders, and only these edges for its elements of order $p.q$.

This was proved by Melissa Lee and Tomasz Popiel in $\mathbb{M}, \mathbb{B}$, and $\mathbf{Co}_1$ are recognisable by their prime graphs, by using modular character theory.

The proof for the Monster takes less than one page, so it’s clear that it builds on lots of previous results.

There’s the work of Mina Hagie The prime graph of a sporadic simple group, who used the classification of all finite simple groups to put heavy restrictions on possible groups $G$ having the same prime graph as a sporadic simple group.

For the Monster, she proved that if the prime graph of $G$ is that of the monster, then the Fitting subgroup $F(G)$ must be a $3$-group, and $G/F(G) \simeq \mathbb{M}$.

Her result, in turn, builds on the Gruenberg-Kegel theorem, after Karl Gruenberg and Otto Kegel.

The Gruenberg-Kegel theorem, which they never published (a write-up is in the paper Prime graph components of finite groups by Williams), shows the wealth of information contained in the prime graph of a finite group. For this reason, the prime graph is often called the Gruenberg-Kegel graph.

The pictures above are taken from a talk by Peter Cameron, The Gruenberg-Kegel graph. Peter Cameron’s blog is an excellent source of information for all things relating groups and graphs.

The full proof of the Gruenberg-Kegel theorem is way too involved for a blogpost, but I should give you at least an idea of it, and of one of the recurrent tools involved, the structural results on Frobenius groups by John Thompson.

Here’s lemma 1.1 of the paper On connection between the structure of a finite group and the properties of its prime graph by A.V. Vasil’ev.

Lemma: If $1 \triangleleft K \triangleleft H \triangleleft G$ is a series of normal subgroups, and if we have primes $p$ dividing the order of $K$, $q$ dividing the order of $H/K$, and $r$ dividing the order of $G/H$, then there is at least one edge among these three vertices in the prime graph of $G$.

Okay, let’s suppose there’s a counterexample $G$, and take one of minimal order. Let $P$ be a Sylow $p$-subgroup of $K$, and $N$ its normaliser in $G$. By the Frattini argument $G=K.N$ and so $G/K \simeq N/(N \cap K)$.

Then there’s a normal series $1 \triangleleft P \triangleleft (N \cap H)=N_H(P) \triangleleft N$, and by Frattini $H=K.(N \cap H)$. But then, $N/(N \cap H)=H.N/H = G/H$ and $(N \cap H)/P$ maps onto $(N \cap H)/(N \cap H \cap K) = H/K$, so this series satisfies the conditions for the three primes $p,q$ and $r$.

But as there is no edge among $p,q$ and $r$ in the prime graph of $G$, there can be no such edge in the prime graph of $N$, and $N$ would be a counterexample of smaller order, unless $N=G$.

Oh, I should have said this before: if there is an edge between two primes in the prime graph of a subgroup (or a quotient) of $G$, then such as edge exists also in the prime group of $G$ (trivial for subgroups, use lifts of elements for quotients).

The only way out is that $N=G$, or that $P$ is a normal subgroup of $G$. Look at quotients $\overline{G}=G/P$ and $\overline{H}=H/P$, take a Sylow $q$-subgroup of $\overline{H}$ and $\overline{M}$ its normaliser in $\overline{G}$.

Frattini again gives $\overline{M}/(\overline{M} \cap \overline{H}) = \overline{G}/\overline{H}$, and $r$ is a prime divisor of the order of $\overline{M}/\overline{Q}$.

Lift the whole schmuck to the lift of $\overline{M}$ in $G$ and get a series of normal subgroups
1 \triangleleft P \triangleleft Q \triangleleft M \]
satisfying the three primes condition, so would give a smaller counter-example unless $M=G$ and $Q$ (the lift of $\overline{Q}$ to $G$) is a normal subgroup of $G$.

Sooner or later, in almost all proofs around the Gruenberg-Kegel result, a Frobenius group enters the picture.

Here, we take an element $x$ in $G$ of order $r$, and consider the subgroup $F$ generated by $Q$ and $x$. The action of $x$ on $Q$ by conjugation is fixed-point free (if not, $G$ would have elements of order $p.r$ or $q.r$ and there is no edge between these prime vertices by assumption).

But then, $F$ is a semi-direct product $Q \rtimes \langle x \rangle$, and again because $G$ has no elements of order $p.r$ nor $q.r$ we have:

  • the centraliser-subgroup in $F$ of any non-identity element in $\langle x \rangle$ is contained in $\langle x \rangle$
  • the centraliser-subgroup in $F$ of any non-identity element in $Q$ is contained in $Q$

So, $F$ is a Frobenius group with ‘Frobenius kernel’ $Q$. Thompson proved that the Frobenius kernel is a nilpotent group, so a product of its Sylow-subgroups. But then, $Q$ (and therefore $G$) contains an element of order $p.q$, done.

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Mamuth to Elephant (3)

Until now, we’ve looked at actions of groups (such as the $T/I$ or $PLR$-group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords.

Elephant-lovers recognise such settings as objects in the presheaf topos on the one-object category $\mathbf{M}$ corresponding to the group or monoid. That is, we look at contravariant functors $\mathbf{M} \rightarrow \mathbf{Sets}$.

Last time we’ve encountered the ‘Cube Dance Grap’ which depicts a particular relation among the major, minor, and augmented chords.

Recall that the twelve major chords (numbered for $1$ to $12$) are the ordered triples of tones in $\mathbb{Z}_{12}$ of the form $(n,n+4,n+7)$ (such as the triangle on the left). The twelve minor chords (numbered from $13$ to $24$) are the ordered triples $(n,n+3,n+7)$ (such as the middle triangle). The four augmented chords (numbered from $25$ to $28$) are the triples of the form $(n,n+4,n+8)$ (such as the rightmost triangle).

The Cube Dance Graph relates two of these chords when they share two tones (pitch classes) whereas the remaining tones differ by a halftone.

Picture modified from this post.

We can separate this symmetric binary relation into three sub-relations: the extension of the $P$ and $L$-operations on major and minor chords to the augmented ones (these are transformations), and the remaining relation $U$ which connects the major and minor chords to the augmented chords (and which is not a transformation).

Binary relations on the same set can be composed, so we get a monoid $\mathbf{M}$ generated by the three relations $P,L$ and $U$. The action of $\mathbf{M}$ on the $28$ chords no longer gives us an ordinary presheaf (because $U$ is not a transformation), but a relational presheaf as in the paper On the use of relational presheaves in transformational music theory by Alexandre Popoff.

That is, the action defines a contravariant functor $\mathbf{M} \rightarrow \mathbf{Rel}$ where $\mathbf{Rel}$ is the category (actually a $2$-category) of sets, but with binary relations as morphisms (that is, $Hom(X,Y)$ is all subsets of $X \times Y$), and the natural notion of composition of such relations. The $2$-morphism between relations is that of inclusion.

To compute with monoids generated by binary relations in GAP one needs to download, compile and load the package semigroups, and to represent the binary relations as partitioned binary relations as in the paper by Martin and Mazorchuk.

This is a bit more complicated than working with ordinary transformations:


But then, GAP quickly tells us that $\mathbf{M}$ is a monoid consisting of $40$ elements.

gap> M:=Semigroup([P,L,U]);
gap> Size(M);

The Semigroups-package can also compute Green’s relations and tells us that there are seven such $R$-classes, four consisting of $6$ elements, two of four, and one of eight elements. These are also visible in the Cayley graph, exactly as last time.

Or, if you prefer the cleaner picture of the Cayley graph from the paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis by Popoff, Andreatta and Ehresmann.

This then allows us to compute the Heyting algebra of the subobject classifier, and all the Grothendieck topologies, at least for the ordinary presheaf topos of $\mathbf{M}$-sets, not for the relational presheaves we need here.

We can consider the same binary relation on the larger set of triads when we add the suspended triads. These are the ordered triples in $\mathbb{Z}_{12}$ of the form $(n,n+5,n+7)$, as in the rightmost triangle below.

There are twelve suspended chords (numbered from $29$ to $40$), so we now have a binary relation $T$ on a set of $40$ triads.

The relation $T$ is too coarse, and the art is to subdivide $T$ is disjoint sub-relations which are musically significant, between major and minor triads, between major/minor and augmented triads, and so on.

For each such partition we can then consider the monoids generated by these sub-relations.

In his paper, Popoff suggest relevant sub-relations $P,L,T_U,T_V$ and $T_U \cup T_V$ of $T$ which in our numbering of the $40$ chords can be represented by these PBR’s (assuming I made no mistakes…ADDED march 24th: I did make a mistake in the definition of L, see comment by Alexandre Popoff, below the corect L):

L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[ -10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28],[-29], [-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40]],[[17], [18], [19], [ 20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5], [6], [7],[8],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40] ]);

The resulting monoids are huge:

gap> G:=Semigroup([P,L,TU,TV]);
gap> Size(G);
gap> H:=Semigroup([P,L,TUV]);
gap> Size(H);

In Popoff’s paper these monoids have sizes respectively $473,293$ and $994,624$. Strangely, the offset is in both cases $144=12^2$. (Added march 24: with the correct L I get the same sizes as in Popoff’s paper).

Perhaps we should try to transform such relational presheaves to ordinary presheaves.

One approach is to use the Grothendieck construction and associate to a set with such a relational monoid action a directed graph, coloured by the elements of the monoid. That is, an object in the presheaf topos of the category
\xymatrix{C & E \ar[l]^c \ar@/^2ex/[r]^s \ar@/_2ex/[r]_t & V} \]
and then we should consider the slice topos over the one-vertex bouquet graph with one loop for each element in the monoid.

If you want to have more details on the musical side of things, for example if you want to know what the opening twelve chords of “Take a Bow” by Muse have to do with the Cube Dance graph, here are some more papers:

A categorical generalization of Klumpenhouwer networks, A. Popoff, M. Andreatta and A. Ehresmann.

From K-nets to PK-nets: a categorical approach, A. Popoff, M. Andreatta and A. Ehresmann.

From a Categorical Point of View: K-Nets as Limit Denotators, G. Mazzola and M. Andreatta.


Mamuth to Elephant (2)

Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon.

If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the major $0$-chord) or $2,3$ and $4$ vertices (for a minor chord, here on the right the minor $0$-chord).

The symmetries of the $12$-gon, the dihedral group $D_{12}$, act on the $24$ major- and minor-chords transitively, preserving the type for rotations, and interchanging majors with minors for reflections.

Mathematical Music Theoreticians (MaMuTh-ers for short) call this the $T/I$-group, and view the rotations of the $12$-gon as transpositions $T_k : x \mapsto x+k~\text{mod}~12$, and the reflections as involutions $I_k : x \mapsto -x+k~\text{mod}~12$.

Note that the elements of the $T/I$-group act on the vertices of the $12$-gon, from which the action on the chord-triangles follows.

There is another action on the $24$ major and minor chords, mapping a chord-triangle to its image under a reflection in one of its three sides.

Note that in this case the reflection $I_k$ used will depend on the root of the chord, so this action on the chords does not come from an action on the vertices of the $12$-gon.

There are three such operations: (pictures are taken from Alexandre Popoff’s blog, with the ‘funny names’ removed)

The $P$-operation is reflection in the longest side of the chord-triangle. As the longest side is preserved, $P$ interchanges the major and minor chord with the same root.

The $L$-operation is refection in the shortest side. This operation interchanges a major $k$-chord with a minor $k+4~\text{mod}~12$-chord.

Finally, the $R$-operation is reflection in the middle side. This operation interchanges a major $k$-chord with a minor $k+9~\text{mod}~12$-chord.

From this it is already clear that the group generated by $P$, $L$ and $R$ acts transitively on the $24$ major and minor chords, but what is this $PLR$-group?

If we label the major chords by their root-vertex $1,2,\dots,12$ (GAP doesn’t like zeroes), and the corresponding minor chords $13,14,\dots,24$, then these operations give these permutations on the $24$ chords:


Then GAP gives us that the $PLR$-group is again isomorphic to $D_{12}$:

gap> G:=Group(P,L,R);;
gap> Size(G);
gap> IsDihedralGroup(G);

In fact, if we view both the $T/I$-group and the $PLR$-group as subgroups of the symmetric group $Sym(24)$ via their actions on the $24$ major and minor chords, these groups are each other centralizers! That is, the $T/I$-group and $PLR$-group are dual to each other.

For more on this, there’s a beautiful paper by Alissa Crans, Thomas Fiore and Ramon Satyendra: Musical Actions of Dihedral Groups.

What does this new MaMuTh info learns us more about our Elephant, the Topos of Triads, studied by Thomas Noll?

Last time we’ve seen the eight element triadic monoid $T$ of all affine maps preserving the three tones $\{ 0,4,7 \}$ of the major $0$-chord, computed the subobject classified $\Omega$ of the corresponding topos of presheaves, and determined all its six Grothendieck topologies, among which were these three:

Why did we label these Grothendieck topologies (and corresponding elements of $\Omega$) by $P$, $L$ and $R$?

We’ve seen that the sheafification of the presheaf $\{ 0,4,7 \}$ in the triadic topos under the Grothendieck topology $j_P$ gave us the sheaf $\{ 0,3,4,7 \}$, and these are the tones of the major $0$-chord together with those of the minor $0$-chord, that is the two chords in the $\langle P \rangle$-orbit of the major $0$-chord. The group $\langle P \rangle$ is the cyclic group $C_2$.

For the sheafication with respect to $j_L$ we found the $T$-set $\{ 0,3,4,7,8,11 \}$ which are the tones of the major and minor $0$-,$4$-, and $8$-chords. Again, these are exactly the six chords in the $\langle P,L \rangle$-orbit of the major $0$-chord. The group $\langle P,L \rangle$ is isomorphic to $Sym(3)$.

The $j_R$-topology gave us the $T$-set $\{ 0,1,3,4,6,7,9,10 \}$ which are the tones of the major and minor $0$-,$3$-, $6$-, and $9$-chords, and lo and behold, these are the eight chords in the $\langle P,R \rangle$-orbit of the major $0$-chord. The group $\langle P,R \rangle$ is the dihedral group $D_4$.

More on this can be found in the paper Commuting Groups and the Topos of Triads by Thomas Fiore and Thomas Noll.

The operations $P$, $L$ and $R$ on major and minor chords are reflexions in one side of the chord-triangle, so they preserve two of the three tones. There’s a distinction between the $P$ and $L$ operations and $R$ when it comes to how the third tone changes.

Under $P$ and $L$ the third tone changes by one halftone (because the corresponding sides skip an even number of vertices), whereas under $R$ the third tone changes by two halftones (a full tone), see the pictures above.

The $\langle P,L \rangle = Sym(3)$ subgroup divides the $24$ chords in four orbits of six chords each, three major chords and their corresponding minor chords. These orbits consist of the

  • $0$-, $4$-, and $8$-chords (see before)
  • $1$-, $5$-, and $9$-chords
  • $2$-, $6$-, and $10$-chords
  • $3$-, $7$-, and $11$-chords

and we can view each of these orbits as a cycle tracing six of the eight vertices of a cube with one pair of antipodal points removed.

These four ‘almost’ cubes are the NE-, SE-, SW-, and NW-regions of the Cube Dance Graph, from the paper Parsimonious Graphs by Jack Douthett and Peter Steinbach.

To translate the funny names to our numbers, use this dictionary (major chords are given by a capital letter):

The four extra chords (at the N, E, S, and P places) are augmented triads. They correspond to the triads $(0,4,8),~(1,5,9),~(2,6,10)$ and $(3,7,11)$.

That is, two triads are connected by an edge in the Cube Dance graph if they share two tones and differ by an halftone in the third tone.

This graph screams for a group or monoid acting on it. Some of the edges we’ve already identified as the action of $P$ and $L$ on the $24$ major and minor triads. Because the triangle of an augmented triad is equilateral, we see that they are preserved under $P$ and $L$.

But what about the edges connecting the regular triads to the augmented ones? If we view each edge as two directed arrows assigned to the same operation, we cannot do this with a transformation because the operation sends each augmented triad to six regular triads.

Alexandre Popoff, Moreno Andreatta and Andree Ehresmann suggest in their paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis that one might use a monoid generated by relations, and they show that there is such a monoid with $40$ elements acting on the Cube Dance graph.

Popoff claims that usual presheaf toposes, that is contravariant functors to $\mathbf{Sets}$ are not enough to study transformational music theory. He suggest to use instead functors to $\mathbf{Rel}$, that is Sets with as the morphisms binary relations, and their compositions.

Another Elephant enters the room…

(to be continued)

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