the bourbaki code

Surely there must be a more elegant formulation than that of Mochizuki of an arithemetic Frobenioid, some commenters on google+ pointed out this week. Here, a feeble attempt on my part.

An arithmetic Frobenioid is a category associated to a Galois extension $L$ of the rational numbers $\mathbb{Q}$ and consists of two parts. First, one associates a 'layer of objects' $Frob(K)$ to every subfield $\mathbb{Q} \subset K \subset L$ and next one links these different layers by 'pull-back' or Galois morphisms.

Let $R$ be the ring of integers in $K$ (that is, $R$ is the integral closure of $\mathbb{Z}$ in $K$), then $R$ is a Dedekind domain implying that all non-zero ideals of $R$ are maximal and that any non-zero ideal $I$ of $R$ can be written uniquely as a product of prime-ideals

$$I=P_{i_1}^{e_1}.P_{i_2}^{e_2}. \cdots . P_{i_k}^{e_k}$$

and therefore the non-zero ideals of $R$ form an abelian monoid

$$Div(R) = \bigoplus_{P \in \mathbf{Spec}(R)} \mathbb{N}$$

the ideal $I$ above corresponding the the element having $e_l$ in the factors determined by $P_{i_l}$ and zeroes elsewhere. The group generated by it is the divisor group $Div(K)$ and consists of all fractional ideals of $K$. Such a fractional ideal has again a unique decomposition in powers of prime ideals, this time allowing for negative exponents where one defines for a prime ideal $P \in \mathbf{Spec}(R)$

$$P^{-1} = \{ k \in K~|~P.k \subset R \}$$

There's a natural equivalence relation on fractional ideals by declaring $I \sim J$ if and only if there is a $k \in K$ such that $I.k=J$.

To some category theorists, sets with an equivalence relation are easy examples of groupoids which are very special categories in which every morphism is an isomorphism. The objects are the elements of the set (in our case the fractional ideals) and there is exactly one isomorphism between any pair of elements which are equivalent to each other.

These isomorphisms are precisely the isomorphisms in the layer $Frob(K)$. From number theory one gets that the number of isomorphism classes (or connected components in the groupoid) is finite and that the classes are labeled by the elements of a finite Abelian group $Cl(R)$ which is called the class group of $R$. This group is also the quotient monoid

$$Cl(R) \simeq Div(R)/Prin(R)$$

where $Prin(R)$ is the sub-monoid of all principal ideals $Rr$ of $R$.

The remaining morphisms in $Frob(R)$ are given by the action-map of a certain noncommutative monoid on $Div(K)$ (or on its groupoid). This monoid is generated by the operations

$[p](I) = I^p$ for any prime number $p$, and

$P(I) = P.I$ for any non-zero prime ideal $P$ of $R$.

One easily verifies that the only relations between these are (for prime numbers $p$ and $q$ and prime ideals $P$ and $Q$)

$[p] \circ [q] = [q] \circ [p]$, $P \circ Q = Q \circ P$ and $[p] \circ Q = \underbrace{Q \circ Q \circ \cdots \circ Q}_p \circ [p]$

and so the acting monoid is the twisted monoid $\mathbb{N}^{\times}_{>0} \ast Div(R)$. These morphisms turn $Frob(K)$ into a category and it is not that difficult to see that the endomorphism-monoid of any element $I \in Div(K)$ is isomorphic (as abstract monoid) to the twisted monoid $\mathbb{N}^{\times}_{>0} \ast Prin(R)$.

The remaining part in the definition of the arithmetic Frobenioid are the morphisms between the different layers. So, assume $K \subset K' \subset L$ and let $\sigma$ be a $\mathbb{Q}$-automorphism of $K'$ fixing $K$. then there is a ringmorphism between the rings of integers $\sigma~:~R \rightarrow R'$.

If $I \in Div(K)$ there is for each $\sigma$ a pull-back morphism

$\sigma~:~R'.\phi(I) \rightarrow I$

and one verifies that composition of pull-backs are again a pull-back morphism, so that we have indeed associated a category to any Galois extension $L/\mathbb{Q}$ : the arithmetic Frobenioid of $L$.

Next time we will try to reconstruct $L$ from the category...

Welcome back to the second episode in the stills-YouTube-channel MinuteMochizuki. Today we will dismantle not just one scheme, but a simple cover of arithmetic schemes and replace it by a

Quadratic Arithmetic Frobenioid

Let $m$ be a square-free number and for simplicity we assume it not to be of the form $m=4l+1$. The ring of integers in the field $\mathbb{Q}(\sqrt{m})$ is then $\mathbb{Z}[\sqrt{m}] = \mathbb{Z}.1 \oplus \mathbb{Z}\sqrt{m}$ and we want to consider the degree two cover of arithmetic schemes

$$\mathbf{Spec}(\mathbb{Z}[\sqrt{m}]) \rightarrow \mathbf{Spec}(\mathbb{Z})$$

So we have to describe all prime ideals of $\mathbb{Z}[\sqrt{m}]$ and see how they lie over the prime numbers $p$ in $\mathbb{Z}$. This all depends on whether or not $\overline{m}$ is a square in $\mathbb{Z}/p\mathbb{Z}$. If $p \not\mid 4m$ then

- if $\overline{m}$ is a square modulo $p$ then there are two prime ideals $P_1$ and $P_2$ of $\mathbb{Z}[\sqrt{m}]$ lying over $(p)$ and $P_1.P_2=\mathbb{Z}[\sqrt{m}]p$. Some say $p$ splits in $\mathbb{Z}[\sqrt{m}]$ and there are infinitely many $p$ for which this happens.
- if $\overline{m}$ is not a square modulo $p$ then there is just one prime $P=\mathbb{Z}[\sqrt{m}]p$ lying over $(p)$. Some say $(p)$ remains prime, or is inert in $\mathbb{Z}[\sqrt{m}]$ and there are infinitely many $p$ for which this happens.

In the remaining (finite number of) cases, when $p \mid 4m$, there is one prime $P$ over $(p)$ but it ramifies meaning that $P^2=\mathbb{Z}[\sqrt{m}]p$. For example, when $m=3$ we get the following picture

and we see that $\mathbf{Spec}(\mathbb{Z}[\sqrt{3}])$ is a two-sheeted cover (some say etale cover) away from the 'bad points', the ramified primes $2$ and $3$.

In the inert cases we have that $\mathbb{Z}[\sqrt{m}]/p\mathbb{Z}[\sqrt{m}] \simeq \mathbb{F}_{p^2}$ and the Frobenius map $\overline{x} \rightarrow \overline{x}^p$ is a non-trivial automorphism of the quotient. We would like to be able to lift all these Frobenius automorphisms from quotients to ring-maps of $\mathbb{Z}[\sqrt{m}]$, but clearly the map $x \rightarrow x^p$ does not behave well for addition in $\mathbb{Z}$.

(Smart-asses will object at this point and argue that as $\mathbb{Q}(\sqrt{m})$ is an Abelian extension we may do all of this, somehow, using $\lambda$-rings. Point being, one would also like to do this in non-Abelian situations and preferable even for $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.)

In order to include information of these power-maps we will have no choice but to dismantle the two prime-spectra as well as their cover map and to replace this classical setting by the

Frobenioid $C(m)$

We'll start with the recipe of last time, replacing each point (prime ideal) in the two prime spectra by a component $\mathbb{Z}$ into the poset which is this time the disjoint union of two posets, one for each scheme. That is the objects of $C(m)$ are the elements of

$$\bigoplus_{P \in \mathbf{Spec}(\mathbb{Z}[\sqrt{m}])} \mathbb{Z} \bigsqcup \bigoplus_{p~\text{prime}} \mathbb{Z}$$

Last time we saw that we could identify elements of the second factor with positive rational numbers (that is, fractions). We can do something similar with elements of the first factor, but then we have to replace 'fractions' by 'fractional ideals' which are just ideals of $\mathbb{z}[\sqrt{m}]$ divided by an integer.

The poset-structure can again be defined in terms of divisibility if we (again) replace integers by ideals, that is, two fractional ideals $I$ and $J$ satisfy $I \mid J$ if there is an ideal $N \triangleleft \mathbb{Z}[\sqrt{m}]$ such that $I.N = J$.

Morphisms in $C(m)$ come in two flavours (or, as the inimitable Mochizuki would say: "the fundamental dichotomy between types of combinatorial structures, between 'etale-like' structures which are 'indifferent to order' and 'Frobenius-like' structures which are 'order-conscious'." (GeomFrob1, p.9)).

First flavour comes, as last time, from the action-maps of a monoid on each of the components. We already described this action on the component corresponding to $\mathbf{Spec}(\mathbb{Z})$. The action on the other component is similar, apart from having to trow in an extra factor, the Galois group $Gal(\mathbb{Q}(\sqrt{m})/\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}$. So, the action monoid is $\mathbb{N}_{> 0} \times \mathbb{Q}(\sqrt{m})^{\ast} \times Gal$ with product-rule

$$(n,q_1,\sigma_1).(m,q_2,\sigma_2) = (n.m,\sigma_1(q_2)^n.q_1,\sigma_1 \circ \sigma_2)$$

and action $(n,q,\sigma).I = \sigma(I)^n.q^{-1}$. Again, morphisms in $C(m)$ are induced by combining action with poset-maps in both layers.

The second flavour of maps (the 'etale-like' in M-parlance) are substitutes for the cover-maps between the two layers. For every integer $n$ and every $q \in \mathbb{Q}(\sqrt{m})$ we have a map between a fractional ideal $I \in Div(\mathbb{Z}[\sqrt{m}])$ (the upper layer) and a rational number $r \in \mathbb{Q}_{>0}^*$ (the bottom layer) whenever

$$q.(\mathbb{Z}[\sqrt{m}].r).I^{-n}~\text{is an ideal of}~\mathbb{Z}[\sqrt{m}]$$

Again, one shows as last time that this actually is a category, meaning that compositions of maps are well-defined.

Problem for light insomniacs : show that every morphism in $C(m)$ is an epimorphism (that is, if the compositions with two maps are equal, then these two maps have to be equal).

Problem for incurable insomniacs : show that if there is an equivalence of categories between $C(m)$ and $C(n)$, then $\mathbb{Q}(\sqrt{m}) = \mathbb{Q}(\sqrt{n})$.

This week on G+ i said i'd better set-up a YouTube channel MinuteMochizuki (mimicking MinutePhysics) but it's way quicker to doodle a few things, instant-upload a snap-shot and add some explanation. But, there's only that much math one can cram into a G+ post, so i will try to cross-post here.

A year ago Mochizuki announced a proof of the ABC-conjecture, but apart from a wonderful entry on MathOverflow there's not much helpful information to be found on the web. I'd imagined lots of people to set-up work-seminars going through the IUTeich-papers and blogging about the progress they made, but it appears that the whole world is anxiously waiting for anonymous experts to deliver their verdict.

As long as the weather in Antwerp and the south of France remains lousy i force myself to work through Mochizuki's paper The Geometry of Frobenioids 1 : the general theory and post the bits and pieces i hope to understand on G+. If this project suddenly stops, check the weather-charts before drawing conclusions.

the plan

I'd love to understand the information contained in 'arithmetic frobenioids', which are very special categories that should contain enough information to reconstruct from the category-maps all arithmetic scheme covers $\mathbf{Spec}(S) \rightarrow \mathbf{Spec}(R)$ where $R$ and $S$ are rings of integers in finite extensions of $\mathbb{Q}$, together with 'lifts' of all frobenius-maps from quotients. The point being that one cannot lift these frobenius-maps to ring maps and hence they cannot be lifted within the framework of scheme theory.

That's why Mochizuki decided to 'partially dismantle scheme theory', that is to replace the prime spectra and the cover-maps between them by a huge poset, roughly replacing each point (prime ideal) by a factor $\mathbb{Z}$ into a union of free Abelian groups with the natural poset-structure and let certain non-commutative monoids act on these posets, part of the monoid structure encoding power-maps $x \rightarrow x^n$ for all $n$, which should then be considered the lifts of the frobenius-info.

the baby version

Before bringing in Galois information, let us see what the 'dismantled' $\mathbf{Spec}(\mathbb{Z})$ looks like. So, for each prime number $p$ we take a component $\mathbb{Z}$ (with natural ordering) and look at the direct sum group (or poset)

$$Div(\mathbb{Z}) = \bigoplus_{p~\text{prime}} \mathbb{Z}$$

Perhaps it is better to identify elements of this poset with strictly positive rational numbers $a \in \mathbb{Q}_{> 0}$, the correspondence given by factoring $a$ into prime factors

$$a = \frac{p_{i_1}^{n_1} \cdots p_{i_k}^{n_k}}{q_{j_1}^{m_1} \cdots q_{j_l}^{m_l}}$$

and so $a$ corresponds to the element of $Div(\mathbb{Z})$ having $n_z$ at component $p_{i_z}$ and $-m_z$ at component $q_{j_z}$ and zeroes elsewhere. Under this correspondence the poset structure on $Div(\mathbb{Z})$ translates into the poset structure on $\mathbb{Q}_{> 0}$ given by divisibility, that is

$$a \mid b~\text{if and only if there is a natural number $m \in \mathbb{N}$ such that}~a.m=b$$

Now consider the category $\mathcal{C}_{\mathbb{Z}}$ having as its objects the $a \in \mathbb{Q}_{>0}$ and arrows

$$(n,q)~:~a \rightarrow b~\text{if and only if}~a^n \mid q.b$$

for all $n \in \mathbb{N}_0$ and all $q \in \mathbb{Q}_{> 0}$. If we want this to be a category we'd better check that compositions are well-defined. So, assume we have a map $(n,q)~:~a \rightarrow b$ and a map $(m,r)~:~b \rightarrow c$ then we have the conditions

$$a^n \mid q.b~\text{and}~b^m \mid r.c$$

and hence $a^{n.m} \mid q^m.b^m \mid q^m.r.c$ so we have indeed a composed map $(n.m,q^m.r)~:~a \rightarrow c$.

Alternatively, one can make $\mathbb{N}_0 \times \mathbb{Q}_{>0}$ into a noncommutative monoid by defining a product

$$(n,q).(m,r) = (n.m,q^m.r)$$

and then there's an action of this monoid on the poset $Div(\mathbb{Z}) = \mathbb{Q}_{>0}$ given by $(n,q).a = a^n.q^{-1}$ and define the arrows in $\mathcal{C}_{\mathbb{Z}}$ to be induced by this action-map and the poset structure.

For insomniacs : get all prime numbers back using only the category structure of $\mathcal{C}_{\mathbb{Z}}$.

For me this quest is over. All i did was following breadcrumbs left by others.

Fellow-travelers arrived there before. What did they do next?

The people from the esoteric site L'Astrée, write literary texts on Grothendieck, mixing strange details (such as the kiosque de la place Pinel, the village of Fougax-et-Barrineuf and even 'Winnie' or 'Fred le Belge, notre indic vers Grothendieck') with genuine finds, such as this 'petite annonce' in the journal for this le 09

which reads:

"RETRAITE (PROFESSEUR UNIVERSITE) CHERCHE -eau de vie de pays pour mes préparations de plantes. Ecrire à M. Grothendieck."

Caterine Aira makes a movie

Most of you will be perfectly happy to know Grothendieck lives in a tiny village close to the market-town of Saint-Girons. A few may click through the map below to satisfy their need to know the name of 'le petit village de l'Ariège'.

To do what exactly, i wonder.

You can write a letter, but it will be returned unopened.

You can email 'la Mairie' (btw. it's the 'orange'-address rather than the 'wanadoo' ones), but i doubt they'll update their Wikipedia-page to acknowledge Grothendieck among the 'Personnalités liées à la commune'.

You can go there in person to hear the villagers out, but, until you're a 'résident permanent', you will be considered an outsider, and treated as one.

If it's knowledge you're after, Grothendieck made it plain he no longer wants to be part of the mathematical society.

His mathematical brain is scattered in the 20.000 pages, kept in 5 boxes at the university of Montpellier. This is the genuine treasure, and should be made public without further delay.

I trust you'll proceed wisely.

To 'Monsieur Alexandre', on his 85th birthday:

happier days!

Previous in this series:

- Vendargues

- Mormoiron

- Massy

- Olmet-et-Villecun

- un petit village de l'Ariège

- Saint-Girons

'

Roy Lisker (remember him from the Mormoiron post?) has written up his Grothendieck-quest(s), available for just 23$, and with this strange blurb-text:

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"The author organized a committee to search for him that led to his discovery, in good health and busily at work, in September, 1996. This committee has since become the Grothendieck Biography Project. All of this is recorded in a 300 page account in 3 parts."

Probably he refers to the trip made by Leila Schneps and Pierre Lochak, nicely described in Sam Leith's The Einstein of maths:

"One of the last members of the mathematical establishment to come into contact with him was Leila Schneps. Through a series of coincidences, she and her future husband, Pierre Lochak, learned from a market trader in the town he left in 1991 that ‘the crazy mathematician’ had turned up in another town in the Pyrenees. Schneps and Lochak in due course staked out the marketplace of the town, carrying an out-of-date photograph of Grothendieck, and waited for the greatest mathematician of the 20th century to show up in search of beansprouts.

‘We spent all morning there in the market. And then there he was.’ Were they not worried he’d run away? ‘We were scared. We didn’t know what would happen. But he was really, really nice. He said he didn’t want to be found, but he was friendly. We told him that one of his conjectures had been proved. He had no idea. He’d stopped being interested in maths at that stage. He thought his unpublished work would all have been long forgotten.’"

To city-cats this may seem an improbable coincidence, but if you live in the French mountains for some time, you learn to group your shoppings, and do them on market-days. The nearest market-town, where you can find a decent 'boulangerie' or supermarket, may be just 20 kms down the road, but it'll take you close to an hour to get there.

If you sit near the town-fountain on market-days, for some weeks, you will have seen most of the people living in the vast neighborhood.

So, we'd better try to find Leila's market-town.

One of the nicer talks on the life of Grothendieck was given by Winfried Scharlau (who also has two books on offer on Grothendieck's life, seems to become an emerging bisiness ...) at the IHES Grothendieck colloque.

This video is stuffed with unknown (at least to me) pictures of Grothendieck, his places at Mormoiron and Villecun and of his four children still living in France. Highly recommended!

But, the lecture has a very, very strange ending.

At 1hr 06.51 into the video he shows the slide reproduced on the left below and says: "Okay and here's a picture on which I will not further comment. That's the last thing I want to show you. I thank you very much for your patience."

Leila Schneps has a page with pictures on her website, including 3 pictures of her house, and then the one on the right above, merely described as 'Another house'.

And then there's this paragraph from Roy Lisker's (him again) Travelogue-France (March 8-April 5, 2005) part 2

"I left the IHP around 11 to return to the CNRS research center at 175 rue du Chevaleret. Pierre Lochak and I discussed the possibility of my going to the town of St. Giron outside of Toulouse to make another impromptu visit to La Maison d'Alexandre Grothendieck."

So, here we have three founding members of the Grothendieck circle linking publicly to the same picture of that one place they want to keep secret at all cost?

Dream on!

If you followed this series at all and have looked at the pictures of Grothendieck's houses in Mormoiron or Villecun it is hard to imagine him living in a bourgeois-house, dating from the end of the 19th century, in a medium-sized market-town.

Still, it is quite likely that the picture is indeed taken in Saint-Girons, on some saturday in 1996 when Leila and Pierre bumped into Grothendieck on the market in Saint-Girons.

After all, Saint-Girons is the market-town closest to the final Grothendieck-spot...

Previous in this series:

- Vendargues

- Mormoiron

- Massy

- Olmet-et-Villecun

- un petit village de l'Ariège

We would love to conclude this series by finding the location of the "final" Grothendieck-spot, before his 85th birthday, this thursday.

But, the road ahead will be treacherous, with imaginary villages along the way and some other traps planted by the nice people of the Grothendieck Fan Club

It is well-known that some members (if not all) of the GFC know the exact location of Grothendieck's hideout in the Pyrenees. Trying to pry this information from them, pledging to keep the name secret, is described as 'solving an equation in n unknowns' in the article Le trésor oublié du génie des maths (h/t +David Roberts):

"Cela fait aujourd’hui vingt-deux ans qu’il vit reclus au pied des Pyrénées, dans un village où personne ne va par hasard et dont le nom doit rester secret. Il le souhaite et ceux qui, de loin, le protègent le souhaitent également. Obtenir l’adresse contre l’assurance de ne pas le déranger prend le temps de résoudre une équation à «n» inconnues. Se poster devant chez lui permet de constater qu’il est bien vivant au milieu d’un village qui le regarde comme «le savant» sans chercher à en savoir plus. A 84 ans, il vient se chauffer au soleil devant son portail puis rentre dans sa maison où nul ne pénètre."

As we don't want to take this vow of secrecy, we will have to rely on the few hints they left in the literature. Presumably, the most trustworthy information is to be found in Pierre Cartier's paper A country of which nothing is known but the name, Grothendieck and “motives”:

"As I already said, he retired in 1988, and has lived since then in self-imposed exile. At first he lived near the Fontaine de Vaucluse, in the middle of a little vineyard that he cultivated, and near to his daughter Johanna and his grandchildren. But later he broke off every family relation. He didn’t seem to mind that the place where he lived was located so near to the infamous Camp du Vernet which played a sad role in his childhood. He lived for years without any contact with the outside world and only a few people even knew where he was. He chose to live alone, considered by his neighbors as a “retired mathematics professor who’s a bit mad”."

There is a small (but for our purposes important) addition to the first sentence in the French version:

"... il a pris sa retraite en 1988, et vit depuis un exil intérieur dans un petit village de l'Ariège."

This addition makes our quest a bit more 'doable'. The department of l'Ariège is one of the lesser populated ones in France (having less that 150.000 inhabitants), and has 'only' 332 villages.

One can divide this number roughly by 2, leaving out the larger villages and towns and those situated in the higher mountains, where living must be extremely difficult for an 85 year old.

An alternative reason for leaving out the more southern villages is Cartier's claim that 'le petit village' is close to the Camp du Vernet, which is the place from which Grothendieck's father was deported to Auschwitz.

This former concentration camp is located in Le Vernet, close to the town of Pamiers (central upper part of the map).

So, one can safely assume that the final G-spot must lie on the map below (click on it to navigate and explore).

Previous in this series:

- Vendargues

- Mormoiron

- Massy

- Olmet-et-Villecun

Before we start the quest for the final G-spot, hopefully in time for Grothendieck's 85th birthday, one more post on Alexandre's 'hippy-days'.

In the second part of Allyn Jackson's "The Life of Alexandre Grothendieck" she tells the story that AG, while touring the US to spread the gospel of the eco-mouvement "Survivre et Vivre" (the deal was that he gave 1 math-talk if he was allowed to give another one on ecology/politics), met a graduate student of Daniel Gorenstein, Justine Skalba, who quickly became a G-groupie and returned with him after the US-trip to France, where she lived with him for two years (and had one child with him, John, who later also became a mathematician).

Allyn Jackson writes:

"In early 1973 he (AG) and Skalba moved to Olmet-le-sec (probably she means: Olmet sec, so without any additions), a rural village in the south of France. This area was at the time a magnet for hippies and others in the counterculture movement who wanted to return to a simpler lifestyle close at hand (I would have added: and, it still is). Here Grothendieck again attempted (he did this once before in his Parisian period, setting up a commune in Chatenay-Malabry) to start up a commune, but personality conflicts led to its collapse. At various times three of Grothendieck's children came to live in the Paris commune and in the one in Olmet (probably this being: Johanna, Mathieu and Alexandre who even today maintain an alternative lifestyle). After the commune disolved, he moved with Skalba and his children to Villecum, a short distance away."

As Yves Ladegaillerie tells Jackson, Grothendieck lived an ascetic, unconventional life in an old house without electricity in Villecun, about thirty-five miles outside of Montpellier. Ladegaillerie remembered seeing Justine Skalba and her baby there. Many friends, acquantances and students went to visit Grothendieck there, including people from the ecology movement.

Here's the (in)famous house in Villecun (h/t Winfried Scharlau)

And, if you are a bit like me, wanting to see everything with G-earth or maps, here's the scenery (click on the image to be there).

Again, if someone at the Mairie d'Olmet-et-Villecun reads this, please consider adding to your list of 'Personnalités liées à la commune'

- Michel Chevalier

- Paul Dardé

this one:

- Alexandre Grothendieck

Merci infiniment!

Previous in this series:

- Vendargues

- Mormoiron

- Massy

One week from now, Alexandre Grothendieck will turn 85. Today, we'll have a glance at his 'wilder years', the early 70ties, when he resigned from the IHES and became one of the leading figures in the French eco-movement. This iconic picture is from those days

The text reads:

"Schurik entre les "frères ennemis" Gaston Galan et Dyama, rue Polonceau.

Derrière, Chantal et Motito (femme et fille de Gaston)."

Schurik (that is, AG) between the 'hostile brothers' Gaston Galan and Dyama in the 'rue Polonceau'. Behind, Chantal and Motito (wife and daughter of Gaston).

However, if you stroll down the Rue Polonceau via StreetView (note to self: high time to revisit Paris IRL) it is unclear where this picture might have been taken. One notable exception perhaps, at 38, Rue Polonceau.

Today, this address houses the feminist group Ruptures with the noble goal to establish a society based on a genuine equality between women and men.

"L’association se donne pour objectif de substituer à la société patriarcale une société fondée sur une égalité réelle et pas seulement formelle entre les femmes et les hommes dans le domaine économique, social, politique et culturel. Elle est basée sur la laïcité et la parité."

Besides, they want to encourage cooperation with other movements striving for a better world:

"Convergences des luttes féministes, altermondialistes, écologistes, antiracistes"

It is thus very well possible that this address was already used in the 70ties by similar social groups, such as the ecological movement "Survivre et Vivre" (Survive and Live), a movement founded in 1970 by three renowned mathematicians: Grothendieck, Claude Chevalley and Pierre Samuel. The origins and evolution of Survivre et Vivre are nicely described on this page at Science et Société.

So, whoever wrote that text beneath the photograph is probably right, though I'd love to hear more details. Still, this picture was the first thing on my mind when i found the place where Grothendieck lived in his IHES-years (and shortly afterwards).

The first issue of the Bulletin of Survivre et Vivre (btw. most issues are available from the Grothendieck circle and are fun reading material if you are, like me, in constant need to brush up your French) concludes with a list of the names, professions and addresses of the group's members (25 at the time, including AG's mother-in-law (Julienne Dufour, mother of his wife Mireille Dufour) and his son (from another mother) Serge):

So, here we are, Grothendieck lived with his wife, children (apart from Serge who was at the time based in Nice) and mother-in-law at 2, Avenue de Verrières, Massy, France

If you click on the picture, you can walk around this G-spot, located just across the Gare de Massy.

If some of you have better info on this or other Grothendieck-spots, please fill me in.

I'm bound to travel south, possibly in search for more information, end of next week...

Previous in this series:

- Vendargues

- Mormoiron

With Grothendieck's 85th brithday coming up, march 28th, we continue our rather erratic quest to locate the spots that once meant a lot to him.

Ever wondered what Grothendieck's last-known hideout looked like? Well, here's the answer:

(h/t gruppe eM)

And, here's the story.

One of the stranger stories to be found on the web is the Grothendieck quest by Roy Lisker. In 1988, after AG declined the Crafoord Prize, Roy convinced an editor of Le Nouvel Observateur to hire him to uncover the whereabouts of Alexandre Grothendieck and, if possible, to interview him.

The 'quest' is an hilarious account of Roy's attempts to prise AG's address out of the people from the Montpellier maths department, his subsequent travels and stay at Grothendieck's place.

He put the text online in 2008 and made it intentionally opaque wrt. AG's phone number and address:

"His phone, if in fact this notorious hermit bothered with such contrivances, was unlisted."

"...of his adopted village of Lessmoiron (after a 20 year silence it is permissible to reveal its name) , in the department of the Vacluse, a region of France long habituated to the herbergement of exiled or alienated Popes."

By that time the Grothendieck-Serre correspondence had been published for over 4 years, including a letter dated 2 september 1984, giving away this 'secret information':

So, not only do we have a phone number (today it would be 0033(0)4 90 61 88 30), but also that AG lived in the hamlet "Les Aumettes" in the village of Mormoiron (and not 'Less'moiron, duh), close to the famous (to any bicyclist) Mont Ventoux.

From Roy's quest we learn that it is about 3 kms from the center of Mormoiron and that

"Grothendieck's cottage was built up against a hillside, it's conical shape hugging the hill like the helmet of a medieval knight. The lower entrance was graced by a pair of sturdy French windows. Above these, at the level of the attic, two tiny rectangular windows filtered light into the bedroom."

If you want to explore the immediate neighborhood of Les Aumettes, click on the picture below (bonus points for anyone who is able to pinpoint the exact location on the map).

If someone at the Mairie de Mormoiron reads this, please consider adding to your list of 'Personnalités liées à la commune'

- Raymond Guilhem de Budos (? - 1363), neveu de Clément V, seigneur de Clermont, Lodève, Budos,Beaumes-de-Venise, Bédoin, Caromb, Entraigues, Loriol et Mormoiron, gouverneur de Bénévent, Maréchal de la Cour pontificale et Recteur du Comtat Venaissin de 1310 à 1317.

- Guillaume-Emmanuel-Joseph, baron Guilhem de Sainte-Croix, (1746-1809), membre de l'Institut, auteur d'un essai Examen critique des anciens historiens d'Alexandrie, couronné par l'Académie des Inscriptions et Belles-Lettres en 177224.

- Paul Vialis, ancien maire de Moirmoiron ou il est né en 1848, et député de Vaucluse.

- Albert Schou (da), (27 mars 1849 – 4 février 1900), photographe danois

this one:

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

Merci!

Previous in this series:

- Vendargues

In a couple of days, on march 28th, Alexandre Grothendieck will turn 85.

To mark the occasion we'll run a little series, tracking down places where he used to live, hoping to entice some of these villages in the south of France to update their Wikipedia-page by adding under 'Personnalités liées à la commune' the line

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

as did the village of Le Chambon-sur-Lignon, where Grothendieck was kept safe from 1942-1945, separated from his mother who was send to an internment camp (his father was deported by the French authorities in august 1942 and killed by the Nazis in Auschwitz).

After the war, Alexandre was reunited with his mother and, according to Allyn Jackson's As If Summoned from the Void: The Life of Alexandre Grothendieck, they "went to live in Maisargues, a village in the wine-growing region outside of Montpellier".

Amir Aczel adds to this in his book The artist and the mathematician, the story of Nicolas Bourbaki: "From 1945 until 1948, mother and son lived in the small hamlet of Mairargues, virtually hidden among the vineyards, a dozen kilometers from Montpellier. They had a marvelous small garden: they never had to work at gardening and yet the earth was so fertile, and the rains so abundant, that the garden produced a plentiful harvest of figs, spinach, and tomatoes. Their garden was at the verge of splendid poppies. Grothendieck remembers his time there with his mother as "la belle vie"."

But, there is no Maisargues nor Mairargues to be found in France.

There is the village of Caissargues, close to Nimes, about 50 kms from Montpellier, and, there is the village of Meyrargues, close to Pertuis, more than 170 kms from Montpellier.

So, where is the hamlet of "la belle vie"?

Jackson's and Aczel's info is based on a footnote in Grothendieck's Recoltes et semailles (in fact, Aczel's text is a mere translation of it):

"Entre 1945 et 1948, je vivais avec ma mère dans un petit hameau à une dizaine de kilomètres de Montpellier, Mairargues (par Vendargues), perdu au milieu des vignes. (Mon père avait disparu à Auschwitz, en 1942.) On vivait chichement sur ma maigre bourse d’étudiant. Pour arriver à joindre les deux bouts, je faisais les vendanges chaque année, et après les vendanges, du vin de grapillage, que j’arrivais à écouler tant bien que mal (en contravention, paraît-il, de la législation en vigueur. . . ) De plus il y avait un jardin qui, sans avoir à le travailler jamais, nous fournissait en abondance figues, épinards et même (vers la fin) des tomates, plantées par un voisin complaisant au beau milieu d’une mer de splendides pavots. C’était la belle vie."

Although Grothendieck misspells Mayrargues, he points to the village of Vendargues which is situated 12 kms east of Montpellier and has a hamlet called Mayrargues (foto above). Via Google Maps you can visit "l'hameau de la belle vie" by yourself (it even has streetview).

If someone at the Mairie de Vendargues comes across this post, please consider adding to your list of famous (former) inhabitants:

- Marcelin Albert (1851-1921), séjourne au mazet de Montmaris, leader de la révolte viticole, est le parrain de Marcellin Guille né en 1907 et oncle d'Archiguille.

- Sabri Allouani (1978-), raseteur (Septuple Vainqueur du Championnat de France de la Course Camarguaise au As 2000-2007)

- Archiguille (Augustin François Guille, peintre contemporain "Transfigurations") vivant en Suisse.

- Laurent Ballesta (1974-), Biologiste marin, plongeur, photographe, collaborateur de Nicolas Hulot)

- Le général Pierre Berthezène (1775-1847), baron d'Empire, pair de France (1775-1847)

- Jerôme Bonnisel (joueur de football professionnel)

- le baron Pierre Le Roy de Boiseaumarié, (1890-1967), fondateur des appellations d'origine contrôlées, vigneron à Châteauneuf-du-Pape.

this one:

- Alexandre Grothendieck (né en 1928), mathématicien français ayant reçu la Médaille Fields.

Thanks!