Category: stories

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.

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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…

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artisanal integers

Published December 31, 2015 by lievenlb

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 lievenlb

“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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Map of the Parisian mathematical scene 1933-39

Published January 9, 2015 by lievenlb

[put_wpgm id=1]

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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Children have always loved colimits

Published January 7, 2015 by lievenlb

If Chad Orzel is able to teach quantum theory to his dog, surely it must be possible to explain schemes, stacks, toposes and motives to hipsters?

Perhaps an idea for a series of posts?

It’s early days yet. So far, I’ve only added the tag sga4hipsters (pun intended) and googled around for ‘real-life’ applications of sheaves, cohomology, and worse.

Sooner or later one ends up at David Spivak’s MIT-webpage.

David has written a book “category theory for scientists” and has several papers on applications of category theory to databases.

There’s also this hilarious abstract, reproduced below, of a talk he gave in 2007 at many cheerful facts.

If this guy ever decides to write a novel, I’ll pre-order it on the spot.

Presheaf, the cobbler.
by David Spivak

Children have always loved colimits.

Whether it be sorting their blocks according to color, gluing a pair of googly eyes and a pipe-cleaner onto a piece of yellow construction paper, or simply eating a peanut butter sandwich, colimits play a huge role in their lives.

But what happens when their category doesn’t have enough colimits?

In today’s ”ownership” society, what usually happens is that the parents upgrade their child’s category to a Presheaf category. Then the child can cobble together crazy constructions to his heart’s content.

Sometimes, a kid comes up to you with an FM radio she built out of tinkertoys, and says
”look what I made! I call it ’182 transisters, 11 diodes, 6 plastic walls, 3 knobs,…’”

They seem to go on about the damn thing forever.

Luckily, Grothendieck put a stop to this madness.

He used to say to them, ever so gently, ”I’m sorry, kid. I’m really proud of you for making this ’182 transistors’ thing, but I’m afraid it already has a name. It’s called a radio.

And thus Grothendieck apologies were born.

Two years later, Grothendieck topologies were born of the same concept.

In this talk, I will teach you to build a radio (that really works!) using only a category of presheaves, and then I will tell you about the patent-police, known as Grothendieck topologies.

God willing, I will get through SGA 4 and Lurie’s book on Higher Topos Theory.”

Further reading:

David Spivak’s book (old version, but freely available) Category theory for scientists.

The published version, available from Amazon.

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Can one explain schemes to hipsters?

Published January 5, 2015 by lievenlb

Nature (the journal) asked David Mumford and John Tate (of Fields and Abel fame) to write an obituary for Alexander Grothendieck.

Probably, it was their first experience ever to get a paper… rejected!

What was their plan?

How did they carry it out?

What went wrong?

And, can we learn from this?

the plan

Mumford and Tate set themselves an ambitious goal. Although Nature would have been happiest with a purely biographical note, they seized the opportunity to explain three ‘simple’ things to a wider audience: (1) schemes, (2) category theory, and, (3) cohomology…

“Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology.”

1. Schemes

Here, the basic stumbling block, as Mumford acknowledged afterwards, is of course that most people don’t know what a commutative ring is. If you’ve never encountered a scheme before in broad daylight, I’m not certain this paragraph tells you how to recognise one:

“… In simplest terms, he proposed attaching to any commutative ring (any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients) a geometric object, called the Spec of the ring (short for spectrum) or an affine scheme, and patching or gluing together these objects to form the scheme. …”

2. Categories

Here they do a pretty good job, I think. They want to explain Grothendieck’s ‘functor of points’ and the analogy they used with several measuring experiments is neat:

“… Grothendieck used the web of associated maps — called morphisms — from a variable scheme to a fixed one to describe schemes as functors and noted that many functors that were not obviously schemes at all arose in algebraic geometry.

This is similar in science to having many experiments measuring some object from which the unknown real thing is pieced together or even finding something unexpected from its influence on known things….”

3. Cohomology

Here, Mumford “hoped that the inclusion of the unit 3-sphere in $\mathbb{C}^2- \{ (0,0) \}$ would be fairly clear to most scientists and so could be used to explain the Mike Artin’s breakthrough that $H^3_{et}(\mathbb{A}^2 – \{ (0,0) \}) \not= 0$.”

I’d love to know the fractional odds an experienced bookmaker would set in case someone (not me!) wants to bet on them successfully getting this message across.

“… Using complex coordinates (z,w), a plane has four real dimensions and taking out a point, what’s left is topologically a three dimensional sphere. Following the inspired suggestions of Grothendieck, Artin was able to show how with algebra alone that a suitably defined third cohomology group of this space has one generator, that is the sphere lives algebraically too. Together they developed what is called étale cohomology at a famous IHES seminar. …”

the aftermath

The good news is that Nature will still publish the Tate-Mumford obit, is some form or another, next week, on januari 15th. According to Mumford they managed to sneak in three examples of commutative rings in passing: polynomials, dual numbers and finite fields.

what went wrong?

The usual?

We mathematicians are obsessed with getting definitions right. We truly believe that no-one can begin to understand the implications of an idea if we don’t teach them the nitty gritty details of our treasured definitions first.

It appears that we are alone on this.

Did physicists smack us in the face with the full standard-model Lagrangian, demanding us to digest the minute details of it first, before they could tell us they had discovered the Higgs boson?

No, most scientists know how to get a message across. You need 3 things:

– a catchy name (the ‘God Particle’)

– good graphics (machines at CERN, collision pictures)

– a killer analogy (the most popular, in relation to the Higgs particle, seems to be “like Maggie Tatcher walking into a room”…)

can we learn from this?

Of course we can.

And frankly, I’m somewhat surprised Mumford missed this chance.

After all, he dreamed up the graphics and the killer analogy…

Further reading

– Mumford’s original rant : Can one explain schemes to biologists?

– John Baez’ follow-up post : Can one explain schemes to biologists?

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Grothendieck’s gribouillis

Published January 3, 2015 by lievenlb

A math-story well worth following in 2015.

What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)?

Here’s the little I know about this:

1. The Mormoiron scribbles

During the 80ties Grothendieck lived in ‘Les Aumettes’ in Mormoiron

In 1991, just before he moved to the Pyrenees he burned almost all of his personal notes in the garden. He phoned Jean Malgoire:

“Si tu ne viens pas chercher mon bordel mathématique, il va brûler avec le reste.”

Malgoire sped to Mormoiron and rescued 5 boxes containing about 20.000 pages. The next 20 years he kept them in his office, not exactly knowing what to do with them.

On january 3rd 2010, Grothendieck wrote his (in)famous letter forbidding others to share or publish any of his writings. (Picture via the SecretBloggingSeminar)

Malgoire feared that Grothendieck would soon ask him to destroy the Mormoiron-gribouillis and decided to donate them to the University of Montpellier.

They are kept somewhere in their archives, the exact location known only to Jean Malgoire, Luc Gomel (who is in charge of the patrimonium of the University of Montpellier) and the person who put the boxes away.

After Grothendieck’s death on november 13th, FranceTV3 did broadcast a short news-item.

If Grothendieck’s children agree, the University of Montpellier intends to make an inventory of the 5 boxes and will make them available, at least to researchers.

2. The Lasserre scribbles

The final 23 years of his life, Grothendieck lived in the small village of Lasserre in the French Pyrenees.

Here he could be glimpsed blurrily through the window as he wrote for hours during the night.(Picture via the GrohendieckCircle)

Leila Schneps and her husband Pierre Lochak did visit the house and met with Grothendieck’s family the week after his death.

Before she went, she was optimistic about the outcome as she emailed:

“I have already started modifying the Grothendieck circle website and it will of course eventually return completely. Plus many things will be added, as we will now have access to Grothendieck’s correspondence and many other papers.”

Her latest comment, from december 16th, left on the Grothendieck-circle bulletin board, is more pessimistic:

“Il a ecrit a Lasserre sans cesse pendant plus de 20 ans. Je n’ai pu que jeter un rapide coup d’oeil sur tout ce qu’il a laisse. Il y a de tout: des maths, des reflexions sur lui-meme, et des reflexions sur la nature humaine et sur l’univers. Rien n’est disponible pour le moment. Ces manuscrits finiront dans une bibliotheque et seront peut-etre un jour consultables.”

The good news is that there appears to be some mathematics among the Lassere-gribouillis. The sad news being that none of this is available at the moment, and perhaps never will.

So, what happened? Here’s my best guess:

Grothendieck’s children were pretty upset a private letter from one of them turned up in the French press, a couple of years ago.

Perhaps, they first want to make sure family related material is recovered, before they’ll consider donating the rest (hopefully to the University of Montpellier to be reunited with Grothendieck’s Mormoiron-notes).

This may take some time.

Further reading (in French):

Grothendieck, mon tresor (nationale)

Un génie mystérieux, un secret bien gardé

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On categories, go and the book $\in$

Published July 24, 2014 by lievenlb

A nice interview with Jacques Roubaud (the guy responsible for Bourbaki’s death announcement) in the courtyard of the ENS. He talks about go, categories, the composition of his book $\in$ and, of course, Grothendieck and Bourbaki.

Clearly there are pop-math books like dedicated to $\pi$ or $e$, but I don’t know just one novel having as its title a single mathematical symbol : $\in$ by Jacques Roubaud, which appeared in 1967.

The book consists of 361 small texts, 180 for the white stones and 181 for the black stones in a game of go, between Masami Shinohara (8th dan) and Mitsuo Takei (2nd Kyu). Here’s the game:

In the interview, Roubaud tells that go became quite popular in the mid sixties among French mathematicians, or at least those in the circle of Chevalley, who discovered the game in Japan and became a go-envangelist on his return to Paris.

In the preface to $\in$, the reader is invited to read it in a variety of possible ways. Either by paying attention to certain groupings of stones on the board, the corresponding texts sharing a common theme. Or, by reading them in order of how the go-game evolved (the numbering of white and black stones is not the same as the texts appearing in the book, fortunately there’s a conversion table on pages 153-155).

Or you can read them by paragraph, and each paragraph has as its title a mathematical symbol. We have $\in$, $\supset$, $\Box$, Hilbert’s $\tau$ and an imagined symbol ‘Symbole de la réflexion’, which are two mirrored and overlapping $\in$’s. For more information, thereader should consult the “Dictionnaire de la langue mathématique” by Lachatre and … Grothendieck.

According to the ‘bibliographie’ below it is number 17 in the ‘Publications of the L.I.T’.

Other ‘odd’ books in the list are: Bourbaki’s book on set theory, the thesis of Jean Benabou (who is responsible for Roubaud’s conversion from solving the exercises in Bourbaki to doing work in category theory. Roubaud also claims in the interview that category theory inspired him in the composition of the book $\in$) and there’s also Guillaume d’Ockham’s ‘Summa logicae’…

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Oulipo’s use of the Tohoku paper

Published July 17, 2014 by lievenlb

Many identify the ‘Tohoku Mathematical Journal’ with just one paper published in it, affectionately called the Tohoku paper: “Sur quelques points d’algèbre homologique” by Alexander Grothendieck.

In this paper, Grothendieck reshaped homological algebra for Abelian categories, extending the setting of Cartan-Eilenberg (their book and the paper both appeared in 1957). While working on the Tohoku paper in Kansas, Grothendieck did not have access to the manuscript of the 1956 book of Cartan-Eilenberg, about which he heard from his correspondence with Serre.

Concerning the title, an interesting suggestion was made by Mathieu Bélanger in his thesis “Grothendieck et les topos: rupture et continuité dans les modes d’analyse du concept d’espace topologique”, (footnote 18 on page 164):

“There is a striking resemblance between the title of the Grothendieck’s article “Sur quelques points d’algèbre homologique”, and that of Fréchet‘s thesis “Sur quelques points d’analyse fonctionelle”. Why? Grothendieck remains silent about it. Perhaps he saw a methodological similarity between the introduction, by Fréchet, of abstract spaces in order to develop the foundations of functional calculus and that of the Abelian categories he needed to clarify the homological theory. Compared with categories of sets, groups, topological spaces, etc. that were used until then, Abelian categories are in effect abstract categories.”

But, what does this have to do with the literary group OuLiPo (ouvroir de littérature potentielle, ‘workshop of potential literature’)?

Oulipo was founded in 1960 by Raymond Queneau and François Le Lionnais. Other notable members have included novelists Georges Perec and Italo Calvino, poets Oskar Pastior, Jean Lescure and poet/mathematician Jacques Roubaud.

Several members of Oulipo were either active mathematicians or at least had an interest in mathematics. Sometimes, Oulipo is said to be the literary answer to Bourbaki. The group explored new ways to create literature, often with methods coming from mathematics or programming.

One such method is described in “Chimères” by Le Lionnais:

One takes a source text A. One ’empties’ it, that is, one deletes all nouns, adjectives and verbs, but marks where they were in the text. In this way we have ‘prepared’ the text.

Next we take three target texts and make lists of words from them, K the list of nouns of the first, L the list of adjectives of the second and M the list of verbs of the third. Finally, we fill the empty spaces in the source text by words from the target lists, in the order that they appeared in the target texts.

In the example Le Lionnais gives, the liste M is the list of all verbs appearing in the Tohoku paper.

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