\font\bg=cmbx12 at 14pt
\magnification=\magstep1
\font\tiny=cmr8
\ \vskip .5cm
\centerline{\bg Chapter 3.  From student to celebrity: 1949-1952}
\vskip .5cm
\noindent {\bf An amazing breakthrough: 1949-1950}
\vskip .5cm
An anecdote survives about Grothendieck's arrival in Nancy: the story of his
rude reception at the hands of Dieudonn\'e when, on their very first contact, 
he showed him the work on ``generalized integrals'' about which Dieudonn\'e had 
written to him so warmly just months earlier, praising his ``ardor'' for mathematics. 
Those who recall this incident (or rather, who recall Dieudonn\'e's telling them
about it) claim that Dieudonn\'e gave Grothendieck a rather sharp dressing down, 
finding that the work displayed a reprehensible tendency to gratuitous generality.  
Whether or not his assessment is justifiable, it is clear in any case that he immediately
seized the essence of Grothendieck's approach -- and was quick to discourage 
it.  As Schwartz recounts it in his autobiography:
\vskip .3cm
{\it Il pr\'esenta d'abord \`a Dieudonn\'e un article d'une cinquantaine
de pages, sur ``L'int\'e\-gration \`a valeurs dans un groupe topologique.''
C'\'etait exact, mais rigoureusement sans aucun int\'er\^et.  Dieudonn\'e,
avec l'agressivit\'e (toujours passag\`ere) dont il \'etait capable, lui passa
un savon m\'emorable, arguant qu'on ne devait pas travailler de cette mani\`ere,
en g\'en\'eralisant pour le plaisir de g\'en\'eraliser.  Il fallait que le
probl\`eme trait\'e soit difficile, et susceptible d'applications dans le
reste des math\'ematiques (ou d'autres sciences); ses r\'esultats ne
serviraient jamais \`a rien ni \`a personne.} [Sch 292]$^{1}$
\vskip .3cm
The irrepressible Grothendieck does not seem to have conceived the slightest
grudge towards Dieudonn\'e on this account; perhaps his own long experience with
aggressivity and anger made him dismiss such scenes easily, or perhaps the scolding 
tone was not enough to suppress the underlying warm generosity and benevolence that 
Dieudonn\'e was noted for as much or more than for his brusque manner and frequent,
rather loud explosions of temper.  Quite probably, Grothendieck simply accepted the rebuke 
as valid, cast aside his past work without a thought as he had already done twice before, 
and rolled up his sleeves to have a new stab at proving his worth, by working for the first 
time on problems set to him by others.

In order to direct the young student to more concrete problems, Dieudonn\'e and 
Schwartz handed him a copy of their most recent paper, {\it La dualit\'e dans les
espaces ${\cal F}$ et ${\cal LF}$}.  The main novelty of this article was the systematic 
consideration of (LF)-spaces, which are built from the (F)-spaces by a process of inductive 
sequential limit.  At that time, the ideas of projective and inductive limits were
relatively new; although they existed, they were used in specific situations for
specific constructions, without recourse to a general definition.  But it would
be wrong to consider that Grothendieck's interest in functional analysis at this
time was an entirely separate thing from the style of his later work.  The fundamental
ideas that struck him at this time are not only recognizable, but are even in a
sense the underpinnings of his later work.  

The article ended with fourteen questions that the authors had not been
able to solve.  Dieudonn\'e handed the article to Grothendieck with the 
injunction to choose some of them and think about them.  As Schwartz
recounts it, Grothendieck disappeared totally for some weeks, after which he 
returned with profound and difficult solutions to several of them,
Jacques Dixmier recalls Dieudonn\'e bounding into a Bourbaki meeting with his 
usual energy and telling everyone how a new student had shown up in Nancy with
some useless manuscript about integrals and had earned what
Dieudonn\'e felt was a well-deserved rebuke -- but then had gone off and
solved the bulk of the problems at the end of his recent paper with Schwartz.

Just weeks after Grothendieck's arrival in Nancy, Dieudonn\'e was considering
him as a suitable recruit for Bourbaki.  Grothendieck felt surprised and delighted by 
Dieudonn\'e's enthusiasm: ``Si j'ai vu un math\'ematicien faire usage d'un 
puissant et \'el\'ementaire ``pouvoir d'encouragement'', c'est bien lui!
Je n'y ai jamais re-song\'e avant cet instant, mais je me souviens que c'est
dans ces dispositions aussi qu'il avait accueilli d\'ej\`a mes tout
premiers r\'esultats \`a Nancy, r\'esolvant des questions qu'il avait
pos\'ees avec Schwartz (sur les espaces ${\cal F}$ et ${\cal LF}$).
C'\'etaient des r\'esultats tout modestes, rien de g\'enial ni d'extraordinaire
certes, on pourrait dire qu'il n'y avait pas de quoi s'\'emerveiller.''

In spite of this modest assessment of his own work, Schwartz and Dieudonn\'e
remained enthusiastic. Grothendieck's solutions to their 
problems gave rise to his first three publications, three notes submitted to 
the {\it Comptes Rendus de l'Acad\'emie des Sciences} in 1950. MORE

Discussion of publications 1950-53. C.R. notes etc.

\vskip .5cm
\noindent {\bf  A thesis topic: 1950-51}
\vskip .3cm
In the meantime, Schwartz set about to find a proper -- deeper -- thesis topic for 
Grothendieck from within the questions arising from his own work.  Generaling from 
concerns with particular
types of spaces, Schwartz asked Grothendieck the following question:
{\it put a good topology on the tensor product $E\otimes F$ of two
locally convex spaces $E$ and $F$.}  Having set the problem, Schwartz went
off to Brazil for the summer.
\vskip .3cm
{\it At the end of July in Brazil, I received a very disappointed letter from 
him: there were two locally convex topologies on $E\otimes F$,
both equally natural, but different!  So there was nothing interesting to
say.  I didn't know what to answer him.  It was clear that on
${\cal D'}\otimes F$ there was only one obvious topology.  But difficulty
and defeat can often be a source of victory.  Two weeks later, I received
a new and triumphant letter: the two topologies coincided in the case of
${\cal D'}\otimes F$.  There are certain locally convex spaces, which he
called ``nuclear'', such that for every $F$, the two topologies on
$E\otimes F$ coincide.  Everything became clear.}
\vskip .3cm
Schwartz gives the year of Grothendieck's arrival in Nancy as 1951 and
claims he worked out the theory of nuclear spaces in the summer of 1952.
It would seem, however, that these dates are erroneous.  Grothendieck
arrived in Nancy in October 1949, and over the course of the year 1950 he
published three notes in the {\it Comptes Rendus} on the Dieudonn\'e-Schwartz
problems.  In the fall of 1950, he was living at 33 rue du Mar\'echal
G\'erard and writing to Dixmier.  By the spring of 1951 he had moved to
3 chemin du Grand Moulin.  The tone of his letters gains in both
authority and familiarity over the course of the years 1950-1952:  the first 
letter, dated November 30, 1950, begins: ``Je me permets de vous \'ecrire \`a 
propos d'un probl\`eme sur lequel je m'acharne pendant un certain temps sans 
obtenir de r\'esultat d\'efinitif,'' the fourth, dated May 2, 1952, starts:
``Je n'ai jamais prouv\'e ni pr\'etendu avoir prouv\'e le th\'eor\`eme
dont tu parles, et qui d'ailleurs est faux.''

In 1951 Grothendieck published two further notes, now discussing the topic of
topologies on tensor products of topological vector spaces, and in 1952,
apart from an article on compactness criteria, he published a summary article
in the {\it Annales de l'Institut Fourier} containing all the essentials of
his work on nuclear spaces.  So it seems probable that Schwartz gave him the
thesis topic in the summer of 1950, and that the essential work of the 
thesis was finished and published by 1952.  

Although he worked so intensively during his thesis years in Nancy, 
Grothendieck's personal life was by no means uneventful.  Soon after his
arrival in Nancy, his mother Hanka came to join him, and her encouragement
and support must have been intense as always, since he dedicated his
thesis to her ``mit Verehrung''.  [Quote thesis]

While not seeking friends, he accepted the friendship of some of the other
students at the university.  Paulo Ribenboim has many memories of their
friendship:
\vskip .3cm
{\it 1950: When Ribenboim was a student in Rio, he came across the notes of
Dieudonn\'e's course in Sao Paulo.  He got a French scholarship to go to
France, almost by chance, one of his professors said ``Go to France''.
Ribenboim was in Nancy from 1950-1952.  He often went to Schwartz's house,
from the very beginning.  Schwartz at that time had just received the
Fields Medal.  One day Schwartz told him ``Toi, Paulo, t'as l'air  normal,
t'es pas fou.  J'ai un \'etudiant, il est tr\`es fort mais il travaille
trop.  Il va venir tout \`a l'heure, vous deviendrez peut-\^etre amis.''
It was Grothendieck.  He came by bike; he was very strong.  They talked
about biking which they loved, about Brazil.  They decided to go to some
concerts together.

Ribenboim, bourgeois family, was a pretty good pianist.  He liked to play
Debussy.  One day Grothendieck said to him, ``I'm going to learn to play
the piano.''  ``Well, go rent a piano.''  ``But look how I'm dressed, I can't
go like this, you go for me.'' So Ribenboim went and rented a piano for
Grothendieck.  Grothendieck worked on it very hard. He used to play until
very late at night.  His landladies all hated this, and  he ended up having
to move as many as 23 times around Nancy.

Dieudonn\'e to Ribenboim: ``What do you know?''  Ribenboim: ''Lattices...''
``Forget all that!  Take Bourbaki, and read it.''  Every Friday, Ribenboim
would go to Dieudonn\'e's house to ask questions and work out exercises
from Bourbaki.  Once he was amazed to see even Dieudonn\'e stop and think.
The question was too difficult...  (He also remembers, and his wife remembers,
how rude and unkind Dieudonn\'e was with his wife, and once with his
teenaged daughter and her Latin homework.)

Grothendieck came once to see Ribenboim in his little room and stared at a
huge pile of math books.  ``What's that?  I bet you'll never read that in your
whole life.''}
\vskip .3cm
Grothendieck was also made very welcome in the Schwartz household, where
Schwartz and his wife received him warmly while observing his peculiarities
at close range.  Claudine Schwartz, who was a very little girl at the time,
remembers him vaguely, but her parents spoke of him often.
\vskip .3cm
{\it C'est le seul qui \'etait re\c cu chez mes parents comme un fils.  Ma
m\`ere disait toujours: il va faire un enfant \`a sa logeuse -- et il
l'a fait.  Il aimait beaucoup notre domestique, Alice.  Alice \'etait \^ag\'ee,
plus \^ag\'ee que ma m\`ere.  Elle mangeait \`a table avec nous, et
Grothendieck \'etait toujours d'une grande gentillesse avec elle.  (Nous
habitions cours L\'eopold.)  On a l'impression que Grothendieck avait une
envie volontaire (mais aussi une capacit\'e naturelle) d'\^etre bien avec
des gens de toutes les classes sociales.

Plus tard, mon p\`ere essayait de lui parler, de le sermonner; il ne donnait
pas assez d'argent \`a Mireille pour nourrir correctement ses enfants.  Pendant
ces ann\'ees-\`a, \`a Paris (1957-60), pour l'inviter chez nous, c'\'etait
tr\`es compliqu\'e, il fallait chercher une heure o\`u le repas tombait
juste, parce que Grothendieck avait un rythme biologique qui n'\'etait pas
sur 24 heures, ce qui faisait qu'il veillait souvent une partie de la nuit
et dormait le jour.  Mon p\`ere sentait une vraie responsabilit\'e pour
lui, et essay\'e de lui parler quand il faisait ``n'importe quoi''.  Selon
les \'epoques, il ne mangeait pas de viande ou alors il en mangeait.
Mon p\`ere discutait longuement avec lui sur cette histoire de rythme.  Ca
l'int\'eressait beaucoup au niveau biologique, mais il s'inqui\'etait pour
sa vie familiale.  Il trouvait qu'il ne s'occupait pas bien de ses enfants.
Ma m\`ere \'etait aussi tr\`es maternelle pour lui.   Ils essayaient (en
vain) de lui faire comprendre qu'il fallait tenir compte des autres.}
\vskip .3cm
Roger Godement shared his memories of Grothendieck during the years 1949-1952
in the following terms:
\vskip .3cm
{\it Grothendieck venait souvent chez nous pendant son sejour a Nancy.  Mais
c'etait un vrai sauvage!

Il avait des manieres parfaitement agreables, il etait souriant, on voyait
qu'il avait ete bien eleve.  Il avait une voix douce et ne disait jamais
une chose mechante.  [Entre parentheses, Serre, qui venait a Nancy seulement
en etoile filante, vivant a Paris, etait tres sympathique aussi a cette
epoque-la, et jamais cassant et mechant avec les moins forts que lui,
comme il l'est devenu plus tard.] Si on avait su a cette epoque la comment son
enfance s'etait passee, on lui aurait pose des questions a ce sujet.  Mais
on ne savait rien.  On s'imaginait que tout s'etait passe normalement.

J'ai rencontre sa mere a Nancy une fois ou deux, mais je n'ai aucun souvenir
particulier, sauf qu'elle etait venue habiter avec lui.

Non, mais quand il arrivait chez nous, pour dejeuner, par exemple, il s'etonnait
de voir que nous mangions bien, sans nous priver.  "Pourquoi vous mangez
comme ca?" criait-il.  "On n'a pas besoin de toute cette nourriture, toute
cette viande.  Moi, je ne me nourris que de lait et de fromage, et de pain."
Et la femme de Godement devait sortir du lait et du fromage expres pour
Grothendieck, il ne mangeait pas la meme chose que les autres.

D'ailleurs il avait du merite, parce que le lait a cette-epoque la n'etait
ni sterilise, ni meme pasteurise.  Tout ca nous est venu des Etats-Unis,
un peu plus tard.  Chaque annee a Nancy, un gosse ou deux mourait en bas
age a cause du lait.  J'ai un souvenir de Bob McKee qui est venu passer
un moment a Nancy, et a remarque un "Milk-Bar" dans la rue.  (Takahashi --
Reiji? etait a Nancy a cette epoque aussi.)  Il s'y est
precipite et a engouffre un enorme verre de lait avec un gros morceau de
Camembert, combinaison d'ailleurs qui m'avait fait bien rire.  Eh bien, il
a paye ca de 48 heures de maux de ventre epouvantables.

Oui, puis Grothendieck avait des idees politiques tres frustes.  Il etait
anarchiste, en gros, mais ca se voyait que ses idees n'etaient pas digerees,
qu'il les avait heritees toutes crues de ses parents.  (Plus tard, en 1970,
c'est moi qui ai du le convaincre que les centrales nucleaires etaient
infiniment moins dangereux pour l'espece humaine que les armes nucleaires,
idee qu'il a fini par comprendre tout a fait, car il a cesse de parler de
centrale nucleaire tres rapidement.  Mais il faut dire qu'il ne s'est jamais
donne la peine de lire le moindre livre americain sur le sujet, pour se
renseigner, et il y en avait des excellents.)
Il etait tres perfectionniste, une fois je me souviens qu'il avait decide
d'entreprendre de traduire un roman, je ne sais plus si c'etait du francais
vers l'allemand ou le contraire, pour se perfectionner.  (Note: Clef p. 124
Grothendieck parle, mais c'est en 1957, de son envie de traduire la Cornette
de Rilke d'allemand en fran\c cais.)

Il vivait dans un quartier un peu loin du centre de Nancy.  Il avait parfois
des ennuis dans son quartier.  Une fois, il engueulait un gosse de trop
chahuter, et le gar\c con lui a dit "Sale boche!"  Il a pris une belle
paire de claques ou plus, et Grothendieck s'est retrouve au commissariat
de police.  Il pretendait qu'il etait bon boxeur d'ailleurs.  Je ne sais pas
si c'etait vrai.

Grothendieck semblait content a Nancy, mais il se rendait bien compte comme
tout le monde, que le niveau intellectuel de la population n'etait pas
brillante.  Et meme le departement de maths n'etait ni Harvard ni Princeton.
Paris n'etait pas mieux que Nancy a ce moment-la.  Comme type qui faisait
des maths il n'y avait que Cartan.

La medaille Fields de Schwartz ne l'impressionnait pas plus que ca.  Nous
savions tous que Schwartz etait un bon mathematicien, nous n'avions que
faire de sa medaille.  De toute facon, Grothendieck aurait rencontre
Hilbert, il lui aurait dit "How are you?"  Il n'avait aucun sentiment
d'inferiorite meme devant le plus grand des mathematicien.  C'etait evident
qu'il se considerait deja comme un tres grand mathematicien, il etait tres
sur de lui.

Il s'habillait tres rudimentairement: chemise, pantalon, sandales.  S'il y
avait eu des jeans, il en aurait peut-etre porte, mais il n'y en avait pas
encore.  Mais tout etait toujours extremement propre chez lui.  Il se souciait
de sa, comme de sa nourriture, dans une idee de purete.

J'etais vaguement au courant qu'il avait fait en enfant a une femme a cette
epoque-la.  Je crois qu'il m'en a parle.}
\vskip .4cm
Reiji Takahashi, who arrived in Nancy in 1953 after Grothendieck had left,
was struck by the strong traces he had left behind.  He even recalls
going into a post office in Nancy and being suddenly asked, perhaps by
someone who recognized him as a mathematician -- ``Do you know Grothendieck?''
Surprised, Takahashi responded that he did not.  ``He was very strange,''
murmured the employee before returning to his other tasks.  Takahashi
also recalls sitting out a dance during a party at the home of Delsarte, 
the father to two charming daughters, and finding himself in a corner with 
Serre and Delsarte, who were heatedly discussing nothing other than 
the merits of Grothendieck's mathematical style, with Delsarte criticizing
it in the face of Serre's eager support.


RS (Terry Mirkil)


Aline Dricquert, Serge.
Talk to Serge, find out more about who Marcelle was (5 children from different
fathers?  Note what Yolande said about her.  Contact these children?  Do
they remember Schurik?  Can they tell about their mother, and about her
subsequent relation with him?  About the fight over Serge coming for vacations?)
\vskip .5cm
SchwartzX
First letters to Dixmier
Pragacz
Saphar



\vfill\eject
\noindent {\bf Translation of the French quotations}
\vskip .4cm
$^{1}$ He first gave Dieudonn\'e an article of some 50 pages on ``Integration
with values in a a topological group''.  It was exact, but totally
uninteresting.  Dieudonn\'e with the (always temporary) aggressivity of 
which he was sometimes capable, gave him a memorable dressing-down, arguing
that one couldn't work that way, generalizing for the pleasure of generalizing.
The problem under consideration had to be difficult, and to have applications
in the rest of mathematics (or other sciences); his results would never
be of any use to anyone for anything.


\bye
