Category: stories

Grothendieck’s gribouillis (2)

Published May 31, 2016 by lieven

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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artisanal integers

Published December 31, 2015 by lieven

Summer of 2012. Suddenly several “integer-as-a-service-providers” spring from nowhere. They deliver “artisanal integers”. Integers which (they claim) are “hand-crafted and guaranteed to be unique and hella-beautiful”.

Are you still with me?

Perhaps it helps to have a look at one of the three such services still operational today: Brooklyn Integers, Mission Integers (after San Francisco’s historic Mission District), and London Integers.

Still in the dark?

Here’s my very own, freshly minted, unique integer: $420557015$.

Anyone can check that this is a genuine Brooklyn-integer by looking up its corresponding number-site. In our case the URL would be: http://www.brooklynintegers.com/int/420557015/.

Please pause a moment to admire the hipster-style web-design and the wonderful tagline:

we have infinity on our side (Brooklyn Integers)

Why do we need integer-as-a-service-providers?

This story starts in 2010 when Aaron Straup Cope wants to play with all of the buildings encoded in OpenStreetMap.

Each of these 67 million buildings was added to the database by first creating four or more “nodes” and then grouping them together in a “way”. Aaron wanted to build a catalog of all of these buildings. Because the “ways” had already a numeric ID in the OpenStreetMap-database (using 32-bit integers), he generated a new numeric ID for each and every building, starting at 32-bits plus one, to avoid collisions between the two databases.

He met a similar problem one year later while working on a project called parallel Flickr. Here the idea is to allow individuals to run and maintain their own local copy of Flickr, consisting of their own photos, those of their close friends as well as photos they like.

Assume a fair number of people have set up their local Flickr site and the very worst happens: Flickr itself shuts down. Can we recover the (to us) relevant part of Flickr from all our local copies?

Yes. That is if all of us had the discipline to maintain the original Flickr-IDs and metadata in our parallel Flickrs. A collision between two or more of our databases would only mean we share a copy of the same photo.

If however (and this is infinitely more likely) we all used our own idiosyncratic system to name files, rebuilding the network from our little Flickrs would quickly become hopeless.

Parallel Flickr is an exercise in how individual people can take control and responsibility to archive material they leave on global social media sites such as Facebook, Google+, YouTube, Instagram and the likes. How can we collectively maintain generated content in case the service pulls the plug?

Another example. Consider a large group of people, each of them geotagging and archiving their own tiny part of a larger collective neighbourhood. What’s needed to construct the bigger picture from their individual efforts?

When Aaron asked his pal Mike over coffee in San Francisco’s Mission District, the reply was: “So, are you suggesting that we need something like a centralized ‘integers as a service’ platform?” As a gag Mike also suggested that rather than any old unique integers, there would probably be a demand for hand-crafted artisanal integers.

It’s how Mission Integers was born. And immediately forgotten. There already exist methods to deliver unique IDs, such as the UUID scheme.

The idea was rekindled when Aaron moved to New York and started the Gowanus Heights Neighborhood Project. Here again they needed unique IDs. Because the long strings of alphanumerical characters and dashes provided by UUID are less efficient at the database layer, they needed Mission integers, or perhaps Brooklyn integers, or both.

In order to avoid collisions between these two artisanal integer providers, Mike claimed the even numbers for Mission (as San Francisco is on the ‘left hand’ coast) and Aaron the odd ones for Brooklyn (New York lying on the ‘right hand’ coast).

Remember all of this started in order to empower individuals against the frills of global players, type Facebook.

And now, this new system would depend on a two-party US-based monopoly? Most certainly not!

Along came Dan Catt who created London Integers, using a rather different look-and-feel. “Both Mission and Brooklyn have gone for a hipster boutique type of look, which I wanted to eschew. The London I know is dirty, gritty, beautiful and punk.”

London ‘artisan’ Integers would be multiples of $9$ and in order to avoid collisions with the others he took the maximal integer minted by both Brooklyn or Mission, added a couple of millions to it, and just started distributing integers.

Someone could look at an artisan integer and work out if it was a London Integer by adding up the digits, repeatedly if necessary to get to single digit. If that’s $9$ it’s a London Integer.

If it's not a London Integer then you can tell if it's Brooklyn or Mission on the odd, even front.

Where’s the math in all this?

Ideally, integers-as-a-service-providers will be set up in all major cities, and why not, even in small communities.

Nelson Minar solved two of the most imminent problems arising from having multiple providers of artisanal integers:

all parties producing integers should be aware of one another and honour their respective offsets.
given an artisanal integer one should be able to figure out where it came from

Nelson did this using secret magical powers better known as “maths”. (Aaron Straup Cope)

He used the Cantor pairing to generate a unique integer $z$ corresponding to the $y$-th number produced by the $x$-th foundry:

$z = \frac{(x+y)(x+y+1)}{2} + y$

Conversely, if you’re given the artisanal integer $z$, you can work out its integer-provider by following these rules:

$w=\lfloor \frac{\sqrt{8z+1}-1}{2} \rfloor,~t = \frac{w^2+w}{2},~y=z-t,~x=w-y$

Elegant as this is, there’s a serious flaw. The number of providers will always be significantly smaller than the number of integers they mint. Therefore, most integers will not be used, ever.

Here’s my two-pence worth of advice to build a slightly more economic system:

The $x$-th foundry should only mint multiples of $p(x)$, the $x$-th prime number.
As any time $t$ one should know the product $T$ of all prime numbers of foundries in operation.
In order to decide whether the $x$-th foundry can distribute its $y$-th number $n = y \times p(x)$ one computes $gcd(n,T)$ and look at its largest prime divisor. If this is $p(x)$, the number $n$ can safely be minted. If not it will eventually be distributed by the foundry corresponding to that largest prime factor.
With a little bit of extra work one gets a fairer system. Decompose $gcd(n,T)=p(x_1)^{e_1} p(x_2)^{e_2} \dots p(x_k)^{e_k}$. Then $n$ will be minted by the foundry having the largest exponent. If there are $m$ equal maximal exponents $e$ corresponding to $p(x_{i_1}), p(x_{i_2}) \dots p(x_{i_m})$, then $n$ will be minted by foundry $x_{i_j}$ where $j = e~mod(m)$.
A new foundry will be associated to the next prime number, will advertise its existence to the existing services (changing $T$) and respect as initial offset the smallest prime multiple larger than the maximum integer already minted by the others.

No doubt you will come up with a much cleverer idea! Please leave it in the comments.

Sources:

– H/T Christian Lawson-Perfect via Twitter

– Aaron Straup Cope: “The “Drinking Coffee and Stealing Wifi” 2012 World Tour”

– Rev Dan Catt: “London Artisan Integers; distribution, Hotel Infinity, punk, an excuse & explanation of sorts”

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The Log Lady and the Frobenioid of $\mathbb{Z}$

Published December 14, 2015 by lieven

“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.”

“All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can say it again: some ideas arrive in the form of a dream.”

Here’s such an idea.

It all started when Norma wanted to compactify her twisted-prime-fruit pies. Norma’s pies are legendary in Twin Peaks, but if you never ate them at Double R Diner, here’s the concept.

Start with a long rectangular strip of pastry and decorate it vertically with ribbons of fruit, one fruit per prime, say cherry for 2, huckleberry for 3, and so on.

For elegance, I argued, the $p$-th ribbon should have width $log(p)$.

“That may very well look natural to you,” she said, “but our Geometer disagrees”. It seems that geometers don’t like logs.

Whatever. I won.

That’s Norma’s basic pie, or the $1$-pie as we call it. Next, she performs $n$ strange twists in one direction and $m$ magical operations in another, to get one of her twisted-pies. In this case we would order it as her $\frac{m}{n}$-pie.

Marketing-wise, these pies are problematic. They are infinite in length, so Norma can serve only a finite portion, limiting the number of fruits you can taste.

That’s why Norma wants to compactify her pies, so that you can hold the entire pastry in your hands, and taste the infinite richness of our local fruits.

“Wait!”, our Geometer warned, “You can never close them up with ordinary scheme-dough, the laws of scheme-pastry prohibit this!” He suggested to use a ribbon of marzipan, instead.

“Fine, then… Margaret, before you start complaining again, how much marzipan should I use?”, Norma asked.

“Well,” I replied, “ideally you’d want it to have zero width, but that wouldn’t close anything. So, I’d go for the next best thing, the log being zero. Take a marzipan-ribbon of width $1$.”

The Geometer took a $1$-pie, closed it with marzipan of width $1$, looked at the pastry from every possible angle, and nodded slowly.

“Yes, that’s a perfectly reasonable trivial bundle, or structure sheaf if you want. I’d sell it as $\mathcal{O}_{\overline{\mathbf{Spec}(\mathbb{Z})}}$ if I were you.”

“In your dreams! I’ll simply call this a $1$-pastry, and an $\frac{m}{n}$-pie closed with a $1$-ribbon of marzipan can be ordered from now on as an $\frac{m}{n}$-pastry.”

“I’m afraid this will not suffice,” our Geometer objected, ” you will have to allow pastries having an arbitrary marzipan-width.”

“Huh? You want me to compactify an $\frac{m}{n}$-pie with marzipan of every imaginable width $r$ and produce a whole collection of … what … $(\frac{m}{n},r)$-pastries? What on earth for??”

“Well, take an $\frac{m}{n}$-pastry and try to unravel it.”

Oh, here we go again, I feared.

Whereas Norma’s pies all looked and tasted quite different to most of us, the Geometer claimed they were all the same, or ‘isomorphic’ as he pompously declared.

“Just reverse the operations Norma performed and you’ll end up with a $1$-pie”, he argued.

So Norma took an arbitrary $\frac{m}{n}$-pastry and did perform the reverse operations, which was a lot more difficult that with pies as now the marzipan-bit produced friction. The end-result was a $1$-pie held together with a marzipan-ribbon of width strictly larger or strictly smaller than $1$, but never gave back the $1$-pastry. Strange!

“Besides”, the Geometer added, “if you take two of your pastries, which I prefer to call $\mathcal{L}$ and $\mathcal{M}$, rather than use your numerical system, then their product $\mathcal{L} \otimes \mathcal{M}$ is again a pastry, though with variable marzipan-width.

In the promotional stage it might be nice to give the product for free to anyone ordering two pastries.”

“And how should I produce such a product-pastry?”

“Well, I’m too lazy to compute such things, it must follow trivially from elementary results in Picard-pastry. Surely, our log lady will work out the details in your notation. No doubt it will involve lots of logs…”

And so I did the calculations in my dreams, and I wrote down all formulas in the Double R Diner log-book, for Norma to consult whenever a customer ordered a product, or power of pastries.

A few years ago we had a Japanese tourist visiting Twin Peaks. He set up office in the Double R Diner, consulted my formulas, observed Norma’s pastry production and had endless conversations with our Geometer.

I’m told he categorified Norma’s pastry-bizness, probably to clone the concept to the Japanese market, replacing pastries by sushi-rolls.

When he left, he thanked me for working out the most trivial of examples, that of the Frobenioid of $\mathbb{Z}$…

Added december 2015:

I wrote this little story some time ago.

The last couple of days this blog gets some renewed interest in the aftermath of the IUTT-Mochizuki-Fest in Oxford last week.

I thought it might be fun to include it, if only in order to decrease the bounce rate.

If you are at all interested in the maths, you may want to start with this google+ post, and work your way back using the links curated by David Roberts here.

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