Grothendieck’s gallery No. 154

Since mid May the Montpellier part of Grothendieck’s gribouillis are online and for everyone available at the Archives Grothendieck.

The story is well-known.

End of June 1990, Grothendieck phoned Jean Malgoire warning him to come asap if he wanted to safeguard the better part of G’s mathematical archive, for he was making a bonfire…

A second handover apparently took place on July 28th 1995.

Malgoire kept these notes (in huge Pampers boxes!) until 2010 when he got cold feet as a result of Grothendieck’s letter. He then donated the boxes to the University of Montpellier in 2012.

After Grothendieck’s death in 2014, Montpellier started a project to scan each and every page and put them online, with the backing of Grothendieck’s children (and generous financial support from the local authorities).

So here we are now, and… nobody seems to care.

I’m aware only of this post on MathOverflow by someone who wants to LaTex some of the material on motives.

Perhaps this is due to the less than optimal presentation of the material, or more likely, Grothendieck’s terrible handwriting. Perhaps the University of Montpellier should partner up with the (older generation of) French pharmacists.

But then, there’s this artistic gem in the archive: cote 154 systemes the pseudo-droites written in 1983-84.

It is written on ancient matrix-plotter page. Here’s a typical example

Which mathematical department wouldn’t want to acquire a framed version of one of these original pages?

That’s the point I wanted to make early may in this G+-post, hoping to raise money to safeguard the Lasserre part of Grothendieck’s gribouillis.

When in need for a header picture, I’ll use a page of Grothendieck’s gallery No 154 from now on.

Here’s evidence that Grothendieck was working on GUTS! (literally).

The subway singularity

The Boston subway is a complex system, spreading out from a focus at Park Street.

On March 3rd, the Boylston shuttle went into service, tying together the seven principal lines, on four different levels.

A day later, train 86 went missing on the Cambridge-Dorchester line.

The Harvard algebraist R. Tupelo suggested the train might have hit a node, a singularity. By adding the Boylston shuttle, the connectivity of the subway system had become infinite…

Never heard of this tragic incident?

Time to read up on A.J. Deutsch’s classic ‘A subway named Moebius’ from 1950. A 12 page pdf of this short story is available via the Rio Rancho Math Camp.

The ‘explanation’ given in the story is that the Moebius strip has a singularity. Before you yell that this is impossible, have a look at this or that.

A ‘non spatial network’ where ‘an exclusion principle operates’, Deutsch’s story says.

Here’s another take.

The train took the exceptional fiber branch, instead of remaining on the desingularisation?

Whatever really happened, it’s a fun read, mathematics clashing with bureaucracy.

In 1996 Gustavo Mosquera directed the film ‘Moebius’, set in Buenos Aires, loosely based on Deutsch’s story.

Here’s the full version (90 min.), with subtitles. Have fun!

MOEBIUS dirigido por Gustavo Mosquera from Universidad del Cine on Vimeo.

Where are Grothendieck’s writings? (2)

A couple of days ago, there was yet another article by Philippe Douroux on Grothendieck’s Lasserre writings “Inestimables mathématiques, avez-vous donc un prix?” in the French newspaper Liberation.

Not that there is much news to report.

I’ve posted on this before: Grothendieck’s gribouillis, Grothendieck’s gribouillis (2), and more recently Where are Grothendieck’s writings?

In that last post I claimed that the five metallic cases containing Grothendieck’s Lasserre notes were in a white building behind the police station of the sixth arrondissement of Paris.

I was wrong.

There’s a detail in Douroux’ articles I forgot to follow-up before.

Here’s the correct location:

[section_title text=”What went wrong?”]

Here’s my ‘translation’ of part of chapter 46 of Douroux’ book “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”:

“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.”

It all seemed to fit together: the ‘commissariat’ has a courtyard (but then, so do most buildings in the neighborhood) and has a blue carriage porch:

portepolicejpg

What went wrong?

I should have trusted Google-translate instead.

It translates the original text “…il garait sa voiture pres du commissariat…” more correctly into “…he parked his car near the police station…”. ‘Near’ as apposed to ‘at’…

We should have looked for a location close to the police station.

And, I should have looked up “Jean-Bernard, a librarian specialised in ancient writings”.

[section_title text=”Who is Jean-Bernard?”]

In Douroux’ latest article there’s this sentence:

“Dès lors, on comprend mieux le travail de Jean-Bernard Gillot, libraire à Paris et expert en livres anciens et manuscrits scientifiques pour lequel les cinq malles contenant les écrits de Lasserre représentent l’affaire d’une vie.”

I’m not even going to make an attempt at translation, you know which tool to use if needed. Suffice it to say that the mysterious Jean-Bernard is no other than Jean-Bernard Gillot.

jbgillotjpg

In 2005, Jean-Bernard Gillot took over the Librairie Alain Brieux, specialising in ancient scientific books and objects. Here’s a brief history of this antiques shop.

Relevant to our quest is that it is located 48, rue Jacob in Paris, just around the corner of the Police Station of the 6th arrondissement.

And, there is a beautiful heavy blue carriage porch, leading to an interior courtyard…

portelibrairiejpg

A quick look at the vast amount of scientific objects (such as these Napier’s bones) indicates that there must be adequate and ample storage space in the buildings behind the shop.

This is where the five metallic cases containing the Lasserre writings are at this moment.

[section_title text=”What’s next?”]

We’re lightyears removed from Maltsiniotis’ optimistic vision, broadcast at the Grothendieck conference in Montpellier last year, that the BNF would acquire the totality of the writings and make them available to the mathematical community at large.

Apart from Maltsiniotis’ cursory inventory of (part of) the 93.000 pages, nobody knows what’s inside these five boxes, making it impossible to put a price tag on them.

Perhaps, the family should grant some bloggers access to the cases, in return for a series of (live)posts on what they’ll find inside…?!

NaNoWriMo (3)

In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote:

“Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can be joined together. Add creativity, imagination and a bit of ingenuity. Build anything.

Mathematics is exactly the same. We start with some basic building blocks and ways of joining them together. And then we use creativity, and, yes, imagination and certainly ingenuity, and try to build anything.”

She then goes on to explain category theory, higher dimensional topology, and the process of generalisation in mathematics, whole the time using lego as an analogy. But, she doesn’t get into the mathematics of lego, perhaps because the talk was aimed at students and researchers of all levels and all disciplines.

There are plenty of sites promoting lego in the teaching of elementary mathematics, here’s just one link-list-page: “27 Fantastic LEGO Math Learning Activities for All Ages”. I’m afraid ‘all ages’ here means: under 10…

lego-math-teaching-children-alycia-zimmerman-fb__700-png

Can one do better?

Everyone knows how to play with lego, which shapes you can build, and which shapes are simply impossible.

Can one tap into this subconscious geometric understanding to explain more advanced ideas such as symmetry, topological spaces, sheaves, categories, perhaps even topos theory… ?

Let’s continue our

[section_title text=”imaginary iterview”]

Question: What will be the opening scene of your book?

Alice posts a question on Lego-stackexchenge. She wants help to get hold of all imaginary lego shapes, including shapes impossible to construct in three-dimensional space, such as gluing two shapes over some internal common sub-shape, or Escher like constructions, and so on.

escherlego

Question: And does she get help?

At first she only gets snide remarks, style: “brush off your French and wade through SGA4”.

Then, she’s advised to buy a large notebook and jot down whatever she can tell about shapes that one can construct.

If you think about this, you’ll soon figure out that you can only add new bricks along the upper or lower bricks of the shape. You may call these the boundary of the shape, and soon you’ll be doing topology, and forming coproducts.

These ‘legal’ lego shapes form what some of us would call a category, with a morphism from $A$ to $B$ for each different way one can embed shape $A$ into $B$.

Of course, one shouldn’t use this terminology, but rather speak of different instruction-manuals to get $B$ out of $A$ (the morphisms), stapling two sets of instructions together (the compositions), and the empty instruction-sheet (the identity morphism).

Question: But can one get to the essence of categorical results in this way?

Take Yoneda’s lemma. In the case of lego shapes it says that you know a shape once you know all morphisms into it from whatever shape.

For any coloured brick you’re given the number of ways this brick sits in that shape, so you know all the shape’s bricks. Then you may try for combination of two bricks, and so on. It sure looks like you’re going to be able to reconstruct the shape from all this info, but this quickly get rather messy.

But then, someone tells you the key argument in Yoneda’s proof: you only have to look for the shape to which the identity morphism is assigned. Bingo!

Question: Wasn’t your Alice interested in the ‘illegal’ or imaginary shapes?

Once you get to Yoneda, the rest follows routinely. You define presheaves on this category, figure out that you get a whole bunch of undesirable things, bring in Grothendieck topologies to be the policing agency weeding out that mess, and keep only the sheaves, which are exactly the desired imaginary shapes.

Question: Your book’s title is ‘Primes and other imaginary shapes’. How do you get from Lego shapes to prime numbers?

By the standard Gödelian trick: assign a prime number to each primitive coloured brick, and to a shape the product of the brick-primes.

That number is a sort of code of the shape. Shapes sharing the same code are made up from the same set of bricks.

Take the set of all strictly positive natural numbers partially ordered by divisibility, then this code is a functor from Lego shapes to numbers. If we extend this to imaginary shapes, we’ll rapidly end up at Connes’ arithmetic site, supernatural numbers, adeles and the recent realisation that the set of all prime numbers does have a geometric shape, but one with infinitely many dimensions.

primenumbers

Not sure yet how to include all of this, but hey, early days.

Question: So, shall we continue this interview at a later date?

No way, I’d better start writing.