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Category: math

Grothendieck’s gribouillis (2)

Published May 31, 2016 by lievenlb

We left the story of Grothendieck’s Lasserre notes early 2015, uncertain whether they would ever be made public.

Some things have happened since.

Georges Maltsiniotis gave a talk at the Gothendieck conference in Montpellier in june 2015 having as title “Grothendieck’s manuscripts in Lasserre”, raising perhaps even more questions.

Philippe Douroux, a journalist at the French newspaper “Liberation”, had a few months ago his book out “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”. In the first and final couple of chapters he gives details on Grothendieck’s years in Lasserre.


In chapter 46 “Que reste-t-il du tresor de Grothendieck?” (what is left of Grothendieck’s treasure?) he recounts what has happened to the ‘Lasserre gribouillis’ and this allows us to piece together some of the jigsaw-puzzle.

Maltsiniotis’ talk

These days you don’t have to be present at a conference to get the gist of a talk you’re interested in. That is, if at least one of the people present is as helpful as Damien Calaque was in this case. A couple of email exchanges later I was able to get this post out on Google+:

Below is the relevant part of the picture taken by Edouard Balzin, mentioned in the post.

Maltsiniotis blackboard Grothendieck conference

The first three texts are given with plenty of details and add up to say 5000 pages. The fifth text is only given the approximate timing 1993-1998, although they present the bulk of the material (30000 pages).

A few questions come to mind:

– Why didn’t Maltsiniotis give more detail on the largest part of the collection?
– There seem to be at least 15000 pages missing in this roundup (previously, the collection was estimates at about 50000 pages). Were they destroyed?
– What happened to the post-1998 writings? We know from a certain movie that Grothendieck kept on writing until the very end.

Douroux’ book

If you have read Scharlau’s biographical texts on Grothendieck’s life, the middle part of Douroux’ book “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques” will not be too surprising.

However, the first 5 and final 3 chapters contain a lot of unknown information (at least to me) about his life in Lasserre. The story of ‘his last friend Michel’ is particularly relevant.

Michel is a “relieur” (book-binder) and Grothendieck used his services to have carton boxes made, giving precise specifications as to their dimensions in mms, to contain his writings.

In the summer of 2000 there’s a clash between the two, details in chapter 4 “la brouille du relieur”. As a result, all writings from 2000-2014 are not as neatly kept as those before.

Each box is given a number, from 1 to the last one: 41.

In chapter 46 we are told that Georges Maltsiniotis spend two days in Lasserre consulting the content of the first 16 boxes, written between 1992 and 1994. He gives also additional information on the content:

Carton box 1 : “Geometrie elementaire schematique” contains 1100 pages of algebra and algebraic geometry which Maltsiniotis classifies as “assez classique” but which Douroux calls ‘this is solid mathematics on which one has to work hard to understand’ and a bit later (apparently quoting Michel Demasure) ‘we will need 50 years to transform these notes into accessible mathematics’.

Carton boxes 2-4 : “Structure de la psyche” (3700 pages) also being (according to Douroux) ‘a mathematical text in good form’.

Carton boxes 5-16 : Philosophical and mystical reflexions, among which “Psyche et structure” and “Probleme du mal” (7500 pages).

That is, we have an answer to most of the questions raised by Maltsiniotis talk. He only consulted the first 16 boxes, had a quick look at the other boxes and estimated they were ‘more of the same’ and packaged them all together in approximately 30000 pages of ‘Probleme du mal’. Probably he underestimated the number of pages in the 41 boxes containing all writings upto the summer of 2000.

Remains the problem to guess the amount of post 2000 writings. Here’s a picture taken by Leila Schneps days after Grothendieck’s death in Lasserre:

Grothendieck boxes in Lasserre

You will notice the expertly Michel-made carton boxes and a quick count of the middle green and rightmost red metallic box reveals that one could easily pack these 41 carton boxes in 3 metallic cases.

So, a moderate guess on the number of post 2000 pages is : 35000.

Why? Read on.

What does this have to do with the Paris attacks?

Grothendieck boxes in Lasserre

November 13th 2015 is to the French what 9/11 is to Americans (and 22 March 2016 is to Belgians, I’m sad to add).

It is also precisely one year after Grothendieck passed away in Saint-Girons.

On that particular day, the family decided to hand the Grothendieck-collection over to the Bibliotheque Nationale. (G’s last wishes were that everything he ever wrote was to be transferred to the BNF, thereby revoking his infamous letter of 2010, within 7 months after his death, or else had to be destroyed. So, to the letter of his will everything he left should have been destroyed by now. But fortunately none of it is, because 7 months is underestimating the seriousness with which the French ‘notaires’ carry out their trade, I can testify from personal experience).

While the attacks on the Bataclan and elsewhere were going on, a Mercedes break with on board Alexandre Jr. and Jean-Bernard, a librarian specialised in ancient writings, was approaching Paris from Lasserre. On board: 5 metallic cases, 2 red ones, 1 green and 2 blues (so Leila’s picture missed 1 red).

At about 2 into the night they arrived at the ‘commissariat du Police’ of the 6th arrondissement, and delivered the cases. It is said that the cases weighted around 400 kg (that is 80kg/case). As in all things Grothendieck concerned, this seems a bit over-estimated.

Anyway, that’s the last place we know to hold Grothendieck’s Lasserre gribouillis.

There’s this worrying line in Douroux’ book : ‘Who will get hold of them? The BNF? An american university? A math-obsessed billionaire?’

Let’s just hope for the best. That the initial plan to open up the gribouillis to the mathematical community at large will become a reality.

If I counted correctly, there are at least two of these metallic cases full of un-read post 2000 writings. To be continued…

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Map of the Parisian mathematical scene 1933-39

Published January 9, 2015 by lievenlb

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Michele Audin has written a book on the history of the Julia seminar (hat tip +Chandan Dalawat via Google+).

The “Julia Seminar” was organised between 1933 and 1939, on monday afternoons, in the Darboux lecture hall of the Institut Henri Poincare.

After good German tradition, the talks were followed by tea, “aimablement servi par Mmes Dubreil et Chevalley”.

A perhaps surprising discovery Audin made is that the public was expected to pay an attendance fee of 50 Frs. (approx. 32 Euros, today), per year. Fortunately, this included tea…

The annex of the book contains the lists of all people who have paid their dues, together with their home addresses.

The map above contains most of these people, provided they had a Parisian address. For example, Julia himself lived in Versailles, so is not included.

As are several of the first generation Bourbakis: Dieudonne lived in Rennes, Henri Cartan and Andre Weil in Strasbourg, Delsarte in Nancy, etc.

Still, the lists are a treasure trove of addresses of “les vedettes” (the professors and the people in the Bourbaki-circle) which have green markers on the map, and “les figurants” (often PhD students, or foreign visitors of the IHP), the blue markers.

Several PhD-students gave the Ecole Normale Superieure (btw. note the ‘je suis Charlie’-frontpage of the ENS today jan.9th) in the rue d’Ulm as their address, so after a few of them I gave up adding others.

Further, some people changed houses over this period. I will add these addresses later on.

The southern cluster of markers on Boulevard Jourdan follows from the fact that the university had a number of apartment blocks there for professors and visitors (hat tip Liliane Beaulieu).

A Who’s Who at the Julia seminar can be found in Audin’s book (pages 154-167).

Reference:

Michele Audin : “Le seminaire de mathematiques 1933-1939, premiere partie: l’histoire”

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$\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

Published July 31, 2014 by lievenlb

Note to self: check Jack Morava’s arXiv notes on a more regular basis!

It started with the G+-post below by +David Roberts:

Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately.

A couple of years ago I had a brief email exchange with him on the Habiro topology on the roots of unity, and, in the process he send me a 3 page draft with ideas on how this could be relevant to higher dimensional topological QFT (If my memory doesn’t fail me, I can’t find anything remotely related in the arXiv-list).

Being in a number-theory phase lately (yes, I also have to give next year, for the first time, in the second semester, a master-course on Number Theory) the paper A topological group of extensions of $\mathbb{Q}$ by $\mathbb{Z}$ caught my eyes.

The extension group $Ext(\mathbb{Q},\mathbb{Z})$ classifies all short exact sequences of Abelian groups

$0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Q} \rightarrow 0$

upto equivalence, that is commuting sequences with end-maps being identities.

The note by Boardman Some Common Tor and Ext Groups hs a subsection on this group/rational vector space, starting out like this:

“This subsection is strictly optional. The group $Ext(\mathbb{Q}, \mathbb{Z})$ is much more difficult to determine. It is easy to see that it is a rational vector space, simply from the presence of $\mathbb{Q}$, but harder to see what its dimension is. This group is not as mysterious as is sometimes claimed, but is related to adèle groups familiar to number theorists.”

Boardman goes on to show that this extension group can be identified with $\mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}^f_{\mathbb{Q}}$ is the ring of finite adèles, that is, sequence $(x_2,x_3,x_5,…)$ of $p$-adic numbers $x_p \in \widehat{\mathbb{Q}}_p$ with all but finitely many $x_p \in \widehat{\mathbb{Z}}_p$, and $\mathbb{Q}$ is the additive subgroup of constant sequences $(x,x,x,…)$.

Usually though, one considers the full adèle ring $\mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \mathbb{A}^f_{\mathbb{Q}}$ and one might ask for a similar interpretation of the adèle class-group $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$.

This group is known to be isomorphic to the character group (or Pontrtrjagin dual) of the rational numbers, that is, to $\widehat{\mathbb{Q}}$ which are all group-morphisms $\mathbb{Q} \rightarrow S^1$ from the rational numbers to the unit circle. This group is sometimes called the ‘solenoid’ $\Sigma$, it is connected but not path connected and the path-component of the identity $\Sigma_0 = \mathbb{R}$.

A very nice and accessible account of the solenoid is given in the paper The character group of $\mathbb{Q}$ by Keith Conrad.

The point of Morava’s note is that he identifies the solenoid $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ with a larger group of ‘rigidified’ extensions $Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z})$.That is, one starts with a usual extension in $Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$ as above, but in addition, one fixes a splitting of the induced sequence

$0 \rightarrow \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow A \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow 0$

Forgetting the splitting this gives the exact sequence

$0 \rightarrow \mathbb{R} \rightarrow Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z}) \rightarrow Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}) \rightarrow 0$

which is isomorphic to the sequence involving the path-component of the solenoid!

$0 \rightarrow \Sigma_0 = \mathbb{R} \rightarrow \Sigma=\widehat{Q} \rightarrow \mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q} \rightarrow 0$

Morava ends with: “I suppose the proposition above has a natural reformulation
in Arakelov geometry; but I don’t know anything about Arakelov geometry”…

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