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Where are Grothendieck’s writings?

Published September 24, 2016 by lievenlb

You better subscribe to the French newspaper Liberation if you’re interested in the latest whereabouts of Grothendieck’s ‘gribouillis’. And even then it is hard to turn this info into a consistent tale. A futile attempt…

“In the Bibliotheque Nationale de France?”

A year ago it all seemed pretty straightforward. Georges Maltsiniotis gave a talk at the Grothendieck conference on a small part of the 65.000 pages discovered after Grothendieck’s death in Lasserre.

He said that Grothendieck’s family has handed over all non-family related material to the Bibliotheque Nationale de France.

Maltsiniotis insisted that the BNF wants to make these notes available to the academic community, after they made an inventory (which may take some time) and mentioned that the person responsible at the BNF is Isabelle le Masme de Chermont.

A year later there’s still no sign of the Lasserre papers in their database.

Earlier this year, Liberation-jounalist Philippe Douroux, published a book on Grothendieck’s life: “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”.

In this book, and in follow-up articles in Liberation, he follows the trail of Grothendieck’s gribouillis and suggests that we’d better look in stranger places, such as a police station or even a botanical institute…

“In a Parisian police station?”

From chapter 46 of Douroux’ book:

On November 13th 2015, while the terrorist-attacks on the Bataclan and elsewhere were going on, a Mercedes break with on board Alexandre Jr. Grothendieck 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.

At about 2 into the night they arrived at the ‘commissariat du Police’ of the 6th arrondissement. Jean-Bernard pushed open a heavy blue carriage porch, crossed the courtyard opened a second door and then a third one and delivered the cases.

Comparing this description with the image above from google maps, the Lasserre boxes might be in the white building behind the police station.

I have no clue what the function of this building is, or why the boxes were delivered at that place, not at all close to the Bibliotheque Nationale.

As to why they are not at the BNF, this is probably a question of money.

Before the BNF can accept a legacy, French law says they have to agree on its value with the family. Their initial estimate was ridiculously low: 45.000 Euros or less than one Euro a page. In a similar case, the archives of Michel Foucault, a former professor at the College de France, were acquired by the BNF for no less than 4.800.000 Euro.

“At the botanical institute in Montpellier?”

The mysterious white building in Paris is the best guess to hold the 65.000 pages Grothendieck wrote in Lasserre.

However, there are also the 20.000 pages of the Mormoiron gribouillis, consisting of 5 boxes (Pamper-boxes it is said) rescued by Malgoire in 1991 from Grothendieck’s bonfire.

In 2010, after Grothendieck’s letter that his work should be destroyed, Malgoire donated the Pamper-boxes to the university of Montpellier. The university put them in solid archive boxes and placed them in the Botanical Institute.

As Grothendieck donated these writings to Malgoire, who donated them in turn to his university, the University of Montpellier claimed to own the Mormoiron-gribouillis, and started a silly legal battle with Grothendieck’s children.

On May 3rd the children won, and the documents should have been handed over to the family by mid July 2016. The intention was that they would join the Lasserre notes in Paris.

Mid June, however, the region of Languedoc-Roussillon gave the University of Montpellier 57.000 Euro so that the Grothendieck-notes could be scanned and archived. Probably, a delaying tactic.

So, my best guess is that the Mormoiron gribouillis are still in Montpellier.

Topology and the symmetries of roots

Published September 21, 2016 by lievenlb

We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.

In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots are again roots and by a symmetry we mean a map on all roots, sending sums to sums and products to products and leaving all the green dots (the rational numbers) fixed.

John Baez has an excellent post on the beauty of roots, including a picture of all polynomials of degree $\leq 5$ with integer coefficients between $-4$ and $4$ and, this time, colour-coded by: grey=2, cyan=3, red=4 and black=5.

beauty of roots

In both pictures there’s a hint of the unit circle, black in the title picture and spanning the ‘white gaps’ in the picture above.

If we’d only consider the sub-picture of all (sums and products of) roots including the rational numbers on the horizontal axis and the roots of unity on the unit circle we’d get the cyclotomic field $\mathbb{Q}_{cyc} = \mathbb{Q}(\mu_{\infty})$. Here we know all symmetries: they are generated by taking powers of the roots of unity. That is, we know all about the Galois group $Gal(\mathbb{Q}_{cyc}/\mathbb{Q})$.

The ‘missing’ symmetries, that is the Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$ remained a deep mystery, until last week…

[section_title text=”The oracle speaks”]

On september 15th, Robert Kucharczyk and Peter Scholze (aka the “oracle of arithmetic” according to Quanta-magazine) arXived their paper Topological realisations of absolute Galois groups.

Peter Scholze

They discovered a concrete compact connected Hausdorff space $M_{cyc}$ such that Galois extensions of $\mathbb{Q}_{cyc}$ correspond to connected etale covers of $M_{cyc}$.

Let’s look at a finite field $\mathbb{F}_p$. Here, Galois extensions of $\mathbb{F}_p$ (and there is just one such extension of degree $n$, upto isomorphism) correspond to connected etale covers of the circle $S^1$.

An etale map $X \rightarrow S^1$ is such that every circle point has exactly $n$ pre-images. Here again, up to homeomorphism, there is a unique such $n$-fold cover of $S^1$ (the picture on the left gives the cover for $n=2$).

If we replace $\mathbb{F}_p$ by the cyclotomic field $\mathbb{Q}_{cyc}$ then the compact space $M_{cyc}$ replaces the circle $S^1$. So, if we take a splitting polynomial of degree $n$ with coefficients in $\mathbb{Q}_{cyc}$, then there is a corresponding etale $n$-fold cover $X \rightarrow M_{cyc}$ such that for a specific point $p$ in $M_{cyc}$ its pre-images correspond to the roots of the polynomial. Nice!

Sadly, there’s a catch. Even though we have a concrete description of $M_{cyc}$ it turns out to be a horrible infinite dimensional space, it is connected but not path-connected, and so on.

Even Peter Scholze says it’s unclear whether new results can be proved from this result (see around 39.15 in his Next Generation Outreach Lecture).

Btw. if your German is ok, this talk is a rather good introduction to classical Galois theory and etale fundamental groups, including the primes=knots analogy.

[section_title text=”the imaginary field with one element”]

Of course there’s no mention of it in the Kucharczyk-Scholze paper, but this result is excellent news for those trying to develop a geometry over the imaginary field with one element $\mathbb{F}_1$ and hope to apply this theory to problems in number theory.

As a side remark, some of these people have just published a book with the EMS Publishing House: Absolute arithmetic and $\mathbb{F}_1$-geometry

The basic idea is that the collection of all prime numbers, $\mathbf{Spec}(\mathbb{Z})$ is far too large an object to be a terminal object (as it is in schemes). One should therefore extend the setting of schemes to so called $\mathbb{F}_1$-schemes, in which $\mathbf{Spec}(\mathbb{Z})$ is some higher dimensional object.

Initially, one hoped that $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ might look like a curve, so that one could try to mimick Weil’s proof of the Riemann hypothesis for curves to prove the genuine Riemann hypothesis.

But, over the last decade it became clear that $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ looks like an infinite dimensional space, a bit like the space $M_{cyc}$ above.

I’ll spare this to a couple of follow-up posts, but for now I’ll leave you with the punchline:

The compact connected Hausdorff space $M_{cyc}$ of Kucharczyk and Scholze is nothing but the space of complex points of $\mathbf{Spec}(\mathbb{Q}_{cyc})/\mathbb{F}_1$!

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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…

neverendingbooks Posts

Where are Grothendieck’s writings?

Topology and the symmetries of roots

Grothendieck’s gribouillis (2)