Comme Appelé du Néant
—
As If Summoned from the Void:
The Life of Alexandre
Grothendieck
Allyn Jackson
1196
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
A Different Way of Thinking
Dans le travail de découverte, cette at-
tention intense, cette sollicitude ardente
sont une force essentielle, tout comme
la chaleur du soleil pour l’obscure ges-
tation des semences enfouies dans la
terre nourricière, et pour leur humble
et miraculeuse éclosion à la lumière du
jour.
In the work of discovery, this intense at-
tention, this ardent solicitude, are an es-
sential force, just like the warmth of
the sun for the obscure gestation of
seeds covered in nourishing soil, and for
their humble and miraculous blossom-
ing in the light of day.
—
Récoltes et Semailles
, page P49
Grothendieck had a mathematical style all his
own. As Michael Artin of the Massachusetts Insti-
tute of Technology commented, in the late 1950s
and 1960s “the world needed to get used to him,
to his power of abstraction.” Nowadays Grothen-
dieck’s point of view has been so thoroughly ab-
sorbed into algebraic geometry that it is standard
fare for graduate students starting in the field,
many of whom do not realize that things were
once quite different. Nicholas Katz of Princeton Uni-
versity said that when as a young mathematician
he first encountered Grothendieck’s way of think-
ing, it seemed completely different and new. But
it is hard to articulate what the difference was. As
Katz put it, the change in point of view was so fun-
damental and profound and, once adopted, so
completely natural “that it’s sort of hard to imag-
ine the time before you thought that way.”
Although Grothendieck approached problems
from a very general point of view, he did so not for
generality’s sake but because he was able to use gen-
erality in a very fruitful way. “It’s a kind of approach
that in less gifted hands just leads to what most
people would say are sterile generalities,” Katz
commented. “He somehow knew what general
things to think about.” Grothendieck always sought
the precise level of generality that would provide
precisely the right leverage to gain insight into a
problem. “He seemed to have the knack, time after
time, of stripping away just enough so that it wasn’t
a special case, but it wasn’t a vacuum either,” com-
mented John Tate of the University of Texas at
Austin. “It’s streamlined; there is no baggage. It’s
just right.”
One striking characteristic of Grothendieck’s
mode of thinking is that it seemed to rely so little
on examples. This can be seen in the legend of the
so-called “Grothendieck prime”. In a mathematical
conversation, someone suggested to Grothendieck
that they should consider a particular prime num-
ber. “You mean an actual number?” Grothendieck
asked. The other person replied, yes, an actual
prime number. Grothendieck suggested, “All right,
take 57.”
Allyn Jackson is senior writer and deputy editor of the
Notices
. Her email address is
axj@ams.org
.
This is the second part of a two-part article
about the life of Alexandre Grothendieck. The
first part of the article appeared in the October
2004 issue of the
Notices
.
N
OVEMBER
2004
N
OTICES OF THE
AMS
1197
But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.”
Consider by contrast the Indian mathematician
Ramanujan, who was intimately familiar with prop-
erties of many numbers, some of them huge. That
way of thinking represents a world antipodal to that
of Grothendieck. “He really never worked on exam-
ples,” Mumford observed. “I only understand things
through examples and then gradually make them
more abstract. I don’t think it helped Grothendieck
in the least to look at an example. He really got con-
trol of the situation by thinking of it in absolutely
the most abstract possible way. It’s just very strange.
That’s the way his mind worked.” Norbert A’Campo
of the University of Basel once asked Grothendieck
about something related to the Platonic solids.
Grothendieck advised caution. The Platonic solids
are so beautiful and so exceptional, he said, that one
cannot assume such exceptional beauty will hold in
more general situations.
One thing Grothendieck said was that one should
never try to prove anything that is not almost ob-
vious. This does not mean that one should not be
ambitious in choosing things to work on. Rather,
“if you don’t see that what you are working on is
almost obvious, then you are not ready to work on
that yet,” explained Arthur Ogus of the University
of California at Berkeley. “Prepare the way. And that
was his approach to mathematics, that everything
should be so natural that it just seems completely
straightforward.” Many mathematicians will choose
a well-formulated problem and knock away at it,
an approach that Grothendieck disliked. In a
well-known passage of
Récoltes et Semailles
, he
describes this approach as being comparable to
cracking a nut with a hammer and chisel. What he
prefers to do is to soften the shell slowly in water,
or to leave it in the sun and the rain, and wait for
the right moment when the nut opens naturally
(pages 552–553). “So a lot of what Grothendieck did
looks like the natural landscape of things, because
it looks like it grew, as if on its own,” Ogus noted.
Grothendieck had a flair for choosing striking,
evocative names for new concepts; indeed, he saw
the act of naming mathematical objects as an in-
tegral part of their discovery, as a way to grasp them
even before they have been entirely understood
(
R&S
, page P24). One such term is
étale
, which in
French is used to describe the sea at slack tide, that
is, when the tide is neither going in nor out. At slack
tide, the surface of the sea looks like a sheet, which
evokes the notion of a covering space. As Grothen-
dieck explained in
Récoltes et Semailles
, he chose
the word
topos
, which means “place” in Greek, to
suggest the idea of “the ‘object
par excellence
’ to
which topological intuition applies” (pages 40–41).
Matching the concept, the word
topos
suggests the
most fundamental, primordial notion of space. The
term
motif
(“motive” in English) is intended to
evoke both meanings of the word: a recurrent
theme and something that causes action.
Grothendieck’s attention to choosing names
meant that he loathed terminology that seemed un-
suitable: In
Récoltes et Semailles
, he said he felt an
“internal recoiling” upon hearing for the first time
the term
perverse sheaf
. “What an idea to give such
a name to a mathematical thing!” he wrote. “Or to
any other thing or living being, except in sternness
towards a person—for it is evident that of all the
‘things’ in the universe, we humans are the only
ones to whom this term could ever apply” (page
293).
Although Grothendieck possessed great tech-
nical power, it was always secondary; it was a
means for carrying out his larger vision. He is
known for certain results and for developing cer-
tain tools, but it is his creation of a new viewpoint
on mathematics that is his greatest legacy. In this
regard, Grothendieck resembles Evariste Galois;
indeed, in various places in
Récoltes et Semailles
Grothendieck wrote that he strongly identified with
Galois. He also mentioned that as a young man he
read a biography of Galois by Leopold Infeld [In-
feld] (page P63).
Ultimately, the wellspring of Grothendieck’s
achievement in mathematics is something quite
humble: his love for the mathematical objects he
studied.
A Spirit in Stagnation
[P]endant vingt-cinq ans, entre 1945
(quand j’avais dix-sept ans) et 1969
(quand j’allais sur les quarante-deux),
j’ai investi pratiquement la totalité de
Grothendieck lecturing at the IHÉS.
1198
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
mon énergie dans la recherche mathé-
matique. Investissement démesuré,
certes. Je l’ai payé par une longue stag-
nation spirituelle, par un “épaississe-
ment” progressif, que j’aurai plus d’une
fois l’occasion d’évoquer dans les pages
de
Récoltes et Semailles
.
[F]or twenty-five years, between 1945
(when I was seventeen years old) and
1969 (when I reached forty-two), I in-
vested practically my entire energy into
mathematical research. An excessive in-
vestment, certainly. I paid for it with a
long spiritual stagnation, with a pro-
gressive “dulling”, that I have more than
once found occasion to evoke in the
pages of
Récoltes et Semailles
.
—
Récoltes et Semailles
, page P17
During the 1960s, Barry Mazur of Harvard Uni-
versity visited the Institut des Hautes Études Sci-
entifiques (IHÉS) with his wife. Although by that
time Grothendieck had a family and a house of his
own, he also kept an apartment in the same build-
ing where the Mazurs were living and frequently
worked there late into the night. Because the apart-
ment keys did not open the outside doors, which
were locked at 11:00 p.m., one might have trouble
getting into the building after an evening in Paris.
But “I remember we never had any problems,”
Mazur recalled. “We would take the last train back,
absolutely certain that there would be Grothen-
dieck working, his desk by the window. We would
throw some gravel at his window and he would
open the outside door for us.” Grothendieck’s
apartment was sparely furnished; Mazur remem-
bered a wire sculpture in the outline of a goat and
an urn filled with Spanish olives.
This somewhat lonely image of Grothendieck
working away into the night in a spartan apartment
captures one aspect of his life during the 1960s.
At this time he did mathematics nonstop. He was
talking to colleagues, advising students, lecturing,
carrying on extensive correspondence with math-
ematicians outside of France, and writing the seem-
ingly endless volumes of
EGA
and
SGA
. It is no
exaggeration to say that he was single-handedly
leading a large and thriving segment of worldwide
research in algebraic geometry. He seemed to have
few interests outside of mathematics; colleagues
have said that he never read a newspaper. Even
among mathematicians, who tend to be single-
minded and highly devoted to their work, Grothen-
dieck was an extreme case. “Grothendieck was
working on the foundations of algebraic geometry
seven days a week, twelve hours a day, for ten
years,” noted his IHÉS colleague David Ruelle. “He
had achieved level
−
1 and was working on level 0
of something that must be 10 levels high.…At a cer-
tain age it becomes clear you will never be able to
finish the building.”
The extremity of Grothendieck’s focus on math-
ematics is one reason for the “spiritual stagna-
tion” he referred to in
Récoltes et Semailles
, which
in turn is one of the reasons behind his departure,
in 1970, from the world of mathematics in which
he had been a leading figure. One step toward that
departure was a crisis within the IHÉS, which led
to his resignation. Starting in late 1969, Grothen-
dieck became embroiled in a conflict with the
founder and director of the IHÉS, Léon Motchane,
over military funding for the institute. As historian
of science David Aubin explained [Aubin], during
the 1960s, the IHÉS finances were rather precari-
ous, and in some years the institute received a
small portion of its budget, never more than about
5 percent, from sources within the French mili-
tary. All of the permanent IHÉS professors had
misgivings about military funding, and in late 1969
they insisted that Motchane quit accepting such
funding. Motchane agreed, but, as Aubin noted, he
went back on his word just a few months later,
when the IHÉS budget was stretched thin and he
accepted a grant from the minister of the army. Out-
raged, Grothendieck tried in vain to persuade the
other professors to resign along with him, but
none did. Less than a year earlier, Pierre Deligne
had joined the IHÉS faculty as a permanent pro-
fessor, largely on the recommendation of Grothen-
dieck, who now pressed his newly appointed
colleague to join him in resigning. Deligne too re-
fused. “Because I was very close to him mathe-
matically, Grothendieck was surprised and deeply
disappointed that this closeness of ideas did not
extend outside of mathematics,” Deligne recalled.
Grothendieck’s letter of resignation was dated
May 25, 1970.
His rupture with the IHÉS was the most visible
sign of a profound shift taking place in Grothen-
dieck’s life. Toward the end of the 1960s there
were other signs as well. Some were small. Mazur
recalled that when he was visiting the IHÉS in 1968,
Grothendieck told him he had gone to the movies—
for the first time in perhaps a decade. Other signs
were larger. In 1966, when he was to receive the Fields
Medal at the International Congress of Mathemati-
cians (ICM) in Moscow, Grothendieck refused to
attend as a protest against the Soviet government.
In 1967 Grothendieck made a three-week trip to Viet-
nam, which clearly left an impression on him. His
written account of the trip [Vietnam] described the
many air raid alerts and a bombing that left two math-
ematics teachers dead, as well as the valiant efforts
of the Vietnamese to cultivate a mathematical life in
their country. A friendship with a Romanian physician
named Mircea Dumitrescu led Grothendieck to make
N
OVEMBER
2004
N
OTICES OF THE
AMS
1199
in the late 1960s a fairly se-
rious foray into learning
some biology. He also dis-
cussed physics with Ruelle.
The events of the extra-
ordinary year of 1968 must
also have had an impact
on Grothendieck. That year
saw student protests and
social upheavals all over
the world, as well as the
Soviet Union’s brutal
crushing of the “Prague
Spring”. In France the boil-
ing point came in May
1968, when students ob-
jecting to university and
government policies car-
ried out massive protests
that soon turned into riots.
In Paris hundreds of thou-
sands of students, teach-
ers, and workers took to
the streets to protest police
brutality, and the French
government, fearing revo-
lution, stationed tanks
around the perimeter of
the city. Millions of workers
went on strike, paralyzing
the nation for about two
weeks. Karin Tate, who was
living in Paris with her hus-
band at the time, John Tate,
recalled the chaos that
reigned. “Paving stones, batons, and any other mis-
siles that were handy flew through the air,” she said.
“Soon the entire country was at a standstill. There
was no gasoline (truckers were on strike), there were
no trains (train workers were on strike), garbage was
piling up in Paris (sanitation workers were on
strike), there was very little food on the shelves.”
She and John fled to Bures-sur-Yvette, where her
brother, Michael Artin, was visiting the IHÉS. Many
Parisian mathematicians took the side of the stu-
dents in the conflict. Karin Tate said the protests
dominated conversations among the mathemati-
cians she knew, though she did not remember dis-
cussing the topic with Grothendieck.
Shortly after his resignation from the IHÉS,
Grothendieck plunged into a world completely new
to him, the world of protest politics. In a June 26,
1970, lecture at the Université de Paris in Orsay,
he spoke not about mathematics but about the
threat of nuclear proliferation to the survival
of humankind and called upon scientists and
mathematicians not to collaborate in any way with
the military. Nicholas Katz, who had recently arrived
for a visit at the IHÉS and was surprised to hear of
Grothendieck’s resignation, attended the lecture,
which he said drew an audience of hundreds in a
very crowded lecture hall. Katz remembered that
in the lecture Grothendieck went so far as to say
that doing mathematical research was actually
“harmful” (
“nuisible”
), given the impending threats
to the human race.
A written version of the lecture, “Responsabilité
du savant dans le monde d’aujourd’hui: Le savant
et l’appareil militaire” (“The responsibility of the
scholar in today’s world: The scholar and the mili-
tary apparatus”), circulated as an unpublished man-
uscript. An appendix described the hostile reactions
of the students who attended the lecture and who
handed out flyers mocking Grothendieck. One of
the flyers is reproduced in the appendix; a typical
slogan: “Réussissez, ossifiez-vous, détruisez-vous
vous-mêmes: devenez un petit schéma télécom-
mandé par Grothendieck” (“Succeed, ossify, self-
destruct: become a little scheme remote-controlled
by Grothendieck”). He was clearly seen as a detested
member of the establishment.
In another appendix in this manuscript, Grothen-
dieck called for the founding of a group to fight
Grothendieck wrote this abstract into the colloquium book at the Universität Bielefeld
when he spoke there in 1971. The abstract says: “Witch's Kitchen 1971. Riemann-Roch
Theorem: The ‘dernier cri’: The diagram [displayed] is commutative! To give an
approximate sense to the statement about
f
:
X
→
Y
, I had to abuse the listeners’
patience for almost two hours. Black on white (in Springer Lecture Notes) it probably
takes about 400, 500 pages. A gripping example of how our thirst for knowledge and
discovery indulges itself more and more in a logical delirium far removed from life,
while life itself is going to Hell in a thousand ways—and is under the threat of final
extermination. High time to change our course!”
1200
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
for the survival of the human race against envi-
ronmental degradation and the dangers of military
conflict. This group, called “Survival” (“Survivre
et Vivre” in French) came into being in July 1970
when Grothendieck delivered his Orsay lecture a
second time, at a summer school on algebraic
geometry at the University of Montreal. The main
activity of Survival was the publication of a newslet-
ter by the same name, the first issue of which was
written in English by Grothendieck and is dated Au-
gust 1970. The newsletter describes an ambitious
agenda of publication of books on science, orga-
nization of public courses on science aimed at non-
experts, and boycotts of scientific institutions that
accept military funds.
That first issue carried a list of the names, pro-
fessions, and addresses of the group’s members,
who numbered twenty-five at the time. On the list
were several mathematicians, Grothendieck’s
mother-in-law, and his son Serge. The directors of
the group were Grothendieck and three other math-
ematicians: Claude Chevalley, Denis Guedj, and
Pierre Samuel (
R&S
, page 758). Survival was one of
many leftist groups that emerged in the wake of
the tumultuous 1960s; a similar organization in the
United States was the Mathematics Action Group.
Too small and diffuse to accumulate much influ-
ence, Survival was more active in Paris than in the
United States and Canada, due mostly to Grothen-
dieck’s presence. When he moved out of Paris in
1973, the group petered out.
At the ICM in Nice in the summer of 1970,
Grothendieck tried to recruit members for Sur-
vival. He wrote, “I expected massive enrollments—
there were (if I remember correctly) two or three”
(
R&S
, page 758). Nevertheless, his proselytizing
drew a good deal of attention. “First of all, he was
one of the world stars in mathematics at that time,”
said Pierre Cartier of the IHÉS, who attended the
congress. “Also, you have to remember the politi-
cal climate at the time.” Many mathematicians
opposed the Vietnam War and sympathized with
Survival’s antimilitary stance. During the congress,
Cartier said, Grothendieck sneaked a table in be-
tween two publishers’ booths in the exhibit area
and, assisted by his son Serge, began to hand out
the Survival newsletter. This caused a heated row
between him and his old colleague and friend, Jean
Dieudonné, who had become the first dean of the
science faculty at the Université de Nice when it was
founded in 1964 and who was responsible for the
ICM being held there. Cartier said that he and oth-
ers tried unsuccessfully to persuade Dieudonné
to permit this “unofficial booth”. Eventually
Grothendieck took the table out to the street in
front of the hall in which the congress was being
held. But another problem loomed: in delicate
negotiations with the mayor of Nice, the congress
organizers had promised there would be no street
demonstrations. Police officers began to question
Grothendieck, and finally the chief of police showed
up. Grothendieck was asked to move his table just
a few yards back so that it was off the sidewalk.
“But he refused,” Cartier recalled. “He wanted to
be put in jail. He really wanted to be put in jail!”
Finally, Cartier said, he and some others moved the
table back sufficiently to satisfy the police.
Although Grothendieck’s plunge into politics
was sudden, he was by no means alone. His good
friend Cartier has a long history of political ac-
tivism. For example, he was among the mathe-
maticians who used the holding of the ICM in
Warsaw in 1983 to negotiate the release of one
hundred fifty political prisoners in Poland. Cartier
traces his activism to the example set by his teacher
and mentor, Laurent Schwartz, who was one of the
most politically vocal and active academics in
France. Schwartz was the thesis adviser of Grothen-
dieck. Another mathematician Grothendieck knew
well, Pierre Samuel, is one of the founders of the
French Green Party. Outside of France, many math-
ematicians were politically active. Among the best-
known examples in North America are Chandler
Davis and Stephen Smale, who were deeply in-
volved in protests against the Vietnam War.
But despite his strong convictions, Grothen-
dieck was never effective in the real world of pol-
itics. “He was always an anarchist at heart,” Cartier
observed. “On many issues, my basic positions are
not very far from his positions. But he was so naive
that it was totally impossible to do anything with
him politically.” He was also rather ignorant. Cartier
recalled that, after an inconclusive presidential
election in France in 1965, the newspapers carried
Tata Institute International Colloquium in 1968.
Grothendieck (standing, left) and Armand Borel
(seated, facing camera).
N
OVEMBER
2004
N
OTICES OF THE
AMS
1201
headlines saying that de Gaulle had not been
elected. Grothendieck asked if this meant that
France would no longer have a president. Cartier
had to explain to him what a runoff election is.
“Grothendieck was politically illiterate,” Cartier
said. But he did want to help people: it was not un-
usual for Grothendieck to give shelter for a few
weeks to homeless people or others in need. “He
was very generous, he has always been very gen-
erous,” Cartier said. “He remembered his youth, his
difficult youth, when his mother had nothing, and
he was always ready to help—but in a nonpolitical
way.”
The Wild ’70s
[In 1970 J]’ai alors quitté un milieu pour
entrer dans un autre—le milieu des gens
“des premiers rangs” pour le “marais”;
soudain, la plupart de mes nouveaux
amis étaient de ceux justement qu’un an
avant encore j’aurais tacitement situés
dans cette contrée sans nom et sans
contours. Le soi-disant marais soudain
s’animait et prenait vie par les visages
d’amis liés à moi par une aventure com-
mune—une autre aventure!
[In 1970] I left one milieu to enter an-
other—the milieu of people “of the first
rank” for the “swamp”; suddenly, the
majority of my new friends were those
who just a year before I had tacitly sit-
uated in this region without name and
without shape. The so-called swamp
suddenly moved around and took on
life through the faces of friends tied to
me by a common adventure—another
adventure!
—
Récoltes et Semailles
, page 38
“Légion d’Honneur! Légion d’Honneur!” Grothen-
dieck was shouting from the back of the auditorium,
waving a paper facsimile of the Légion d’Honneur
cross, a distinction conferred by the French gov-
ernment. The scene was the opening day of a sum-
mer school on modular functions, held in Antwerp
in the summer of 1972 and supported by the North
Atlantic Treaty Organization (NATO). Grothen-
dieck’s longtime friend Jean-Pierre Serre of the
Collège de France, who had recently received the
Légion d’Honneur, was presenting the opening
speech. Grothendieck approached Serre and asked,
“Do you mind if I go to the blackboard and say
something?” Serre replied, “Yes, I mind” and left
the room. Grothendieck then mounted the podium
and began speaking against NATO support for the
conference. Other mathematicians sympathized
with this view: One example was Roger Godement,
who in April 1971 wrote an open letter explaining
his reasons for refusing to attend the conference.
Unbeknownst to Grothendieck, Cartier and some
other mathematicians who were uncomfortable
about the NATO support had conducted extensive
negotiations to have a NATO representative come
to the conference for a public debate. Cartier and
others eased Grothendieck off the podium, but the
damage had been done: Cartier soon received an
angry phone call from the NATO representative,
who had heard about the outburst and refused to
come, believing that conditions for an orderly de-
bate had been ruined. “To me, it was sad, because
from what I remember, I think that the audience
was mostly on Grothendieck’s political side,” Cartier
noted. “Even people who were close to his politi-
cal views or his social views were antagonized by
his behavior.…He behaved like a wild teenager.”
By the time of the Antwerp meeting, Grothen-
dieck had cut many of the ties that had bound him
to an orderly life focused on mathematics. For one
thing, he no longer had a permanent position. After
he left the IHÉS in 1970, Serre arranged for him to
have a visiting position at the Collège de France for
two years. This elite institution operates differ-
ently from other universities in France (or any-
where else for that matter). Each professor at the
Collège must submit for approval by the assembly
of all the professors a program of the lectures he
or she plans to deliver during the year. Serre re-
called that Grothendieck offered two possible
programs: one on mathematics and one on the
political themes that occupied the Survival group.
The committee approved the mathematical pro-
gram and rejected the other one. So Grothendieck
presented mathematical lectures prefaced by long
discourses about politics. After two years he applied
for a permanent position at the Collège de France,
a position that had become vacant with the retire-
ment of Szolem Mandelbrojt. The curriculum vitae
Grothendieck, center, University of Montreal,
around 1970.
1202
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
Grothendieck submitted plainly showed that he
intended to give up mathematics to focus on tasks
he believed to be far more urgent: “the imperatives
of survival and the promotion of a stable and hu-
mane order on our planet.” How could the Collège
appoint to a position in mathematics someone who
had declared that he would no longer do any math-
ematics? “He was rightly refused,” Serre said.
It was also during the period just after he left
the IHÉS that Grothendieck’s family life crumbled
and he and his wife separated. In the two years after
he left the IHÉS, Grothendieck spent a fair amount
of time lecturing in mathematics departments in
North America. He spread the gospel of Survival by
insisting he would give a mathematics lecture only
if arrangements were made for him also to give a
political lecture. On one such trip in May 1972, he
visited Rutgers University and met Justine Bumby
(Skalba), then a graduate student of Daniel Goren-
stein. Captivated by Grothendieck’s charismatic
personality, Bumby left behind her life as a grad-
uate student to follow him, first on the remainder
of his trip in the United States, and then on to
France, where she lived with him for two years.
“He’s the most intelligent person I’ve ever met,”
she said. “I was very much in awe of him.”
Their life together was in some ways emblem-
atic of the counterculture years of the 1970s. Once,
at a peaceful demonstration in Avignon, the police
intervened, harassing and pushing away the demon-
strators. Grothendieck got angry when they started
pestering him, Bumby recalled. “He was a good
boxer, so he was very fast,” she said. “We see the
policemen approaching us, and we are all scared,
and then the next thing we know, the two police-
men are on the ground.” Grothendieck had single-
handedly decked two police officers. After some
other officers had subdued him, Bumby and
Grothendieck were bundled into a wagon and
taken to the police station. When his identification
papers revealed that he was a professor at the
Collège de France, the two were taken in to see the
chief of police, who spoke to them in English, as
Bumby spoke no French. After a short conversation,
in which the police chief expressed his desire to
avoid trouble between police and professors, the
two were released and no charges were brought.
Shortly after Bumby came to France with
Grothendieck, he started a commune in a large
house he had rented just south of Paris in Chate-
nay-Malabry, and they lived there together. She
said he sold organically grown vegetables and sea
salt out of the basement of the house. The com-
mune was a bustling place: Bumby said that
Grothendieck held meetings, which might attract
up to a hundred people, about the issues raised in
the Survival group, and these attracted consider-
able media attention. However, the commune dis-
solved fairly rapidly as a result of complicated
personal relationships among the members. It
was around this time that Grothendieck’s position
ended at the Collège de France, and in the fall of
1972 he took a temporary position teaching for
one year at the Université de Paris in Orsay. After
that, Grothendieck obtained a position called
pro-
fesseur à titre personnel
, which is attached to a
single individual and can be taken to any univer-
sity in France. Grothendieck took his to the Uni-
versité de Montpellier, where he was to remain
until his retirement in 1988.
In early 1973 he and Bumby moved to Olmet-
le-sec, 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 to the land.
Here Grothendieck again attempted 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. After the latter commune dis-
solved, he moved with Bumby and his children to
Villecun, a short distance away. Bumby noted that
Grothendieck had a hard time adjusting to the
ways of the people attracted to the counterculture
movement. “His students in mathematics had been
very serious, and they were very disciplined, very
hardworking people,” she said. “In the counter-
culture he was meeting people who would loaf
around all day listening to music.” Having been an
undisputed leader in mathematics, Grothendieck
now found himself in a very different milieu, in
which his views were not always taken seriously.
“He was used to people agreeing with his opinions
when he was doing algebraic geometry,” Bumby
remarked. “When he switched to politics all the
people who would have agreed with him before
suddenly disagreed with him.... It was something
he wasn’t used to.”
Although most of the time Grothendieck was
very warm and affectionate, Bumby said, he
A. Grothendieck with children Serge (left) and
Johanna in 1960.
N
OVEMBER
2004
N
OTICES OF THE
AMS
1203
sometimes had violent outbursts followed by pe-
riods of silent withdrawal. There were also dis-
turbing episodes in which he would launch into a
monologue in German, even though she under-
stood no German. “He would just go on as if I
wasn’t there,” she said. “It was kind of scary.” He
was frugal, sometimes compulsively so: one time,
to avoid throwing away three quarts of leftover cof-
fee, he drank it—with the predictable result that
he got quite sick afterward. Bumby said she believes
that his speaking German and his extreme frugal-
ity may have been connected psychologically to the
hardships he endured as a child, especially the
time when he lived with his mother in the intern-
ment camps.
Grothendieck may have been experiencing some
kind of psychological breakdown, and Bumby today
wonders whether she should have sought treatment
for him. Whether he would have submitted to such
treatment is unclear. They parted ways not long
after their son, John, was born in the fall of 1973. Af-
ter spending some time in Paris, Bumby moved back
to the United States. She married a mathematician
who was a widower, Richard Bumby of Rutgers Uni-
versity, and they raised John and Richard’s two
daughters. John exhibited a good deal of mathemat-
ical talent and was a mathematics major at Harvard
University. He recently finished his Ph.D. in statistics
at Rutgers. Grothendieck has had no contact with
this son.
During the early 1970s, Grothendieck’s interests
were very far from those of the mathematical world
he had left behind. But that world intruded in a ma-
jor way in the summer of 1973, when, at a conference
in honor of W. V. D. Hodge in Cambridge, England,
Pierre Deligne presented a series of lectures about
his proof of the last and most stubborn of the Weil
conjectures. Grothendieck’s former student Luc
Illusie was at the conference and wrote to him with
the news. Wanting to know more, Grothendieck, ac-
companied by Bumby, visited the IHÉS in July 1973.
In 1959 Bernard Dwork proved by
p
-adic meth-
ods the first Weil conjecture (which says that the
zeta function of a variety over a finite field is a ra-
tional function). Grothendieck’s 1964
l
-adic proof
of this conjecture was more general and intro-
duced his “formalism of the six operations.” In
the 1960s Grothendieck also proved the second
Weil conjecture (which says that the zeta function
of a variety satisfies a functional equation). Find-
ing a way to prove the last Weil conjecture (some-
times called the “congruence Riemann Hypothesis”)
was a major inspiration for much of his work. He
formulated what he called the “standard conjec-
tures,” which, if they could be proved, would imply
all of the Weil conjectures. The standard conjectures
were also formulated independently around the
same time by Enrico Bombieri. To this day, the
standard conjectures remain inaccessible. Deligne
found a clever way to circumvent them when he
proved the last Weil conjecture. One of the key ideas
he used came from a paper by R. A. Rankin [Rankin],
which is about the classical theory of modular
forms and of which Grothendieck was unaware. As
John Tate put it, “For the proof of the last Weil con-
jecture, you needed another ingredient that was
more classical. That was Grothendieck’s blind spot.”
When Bumby and Grothendieck turned up at
the IHÉS that summer, among the visitors was
William Messing of the University of Minnesota.
Messing first met Grothendieck in 1966, when as
a graduate student at Princeton he attended a se-
ries of lectures Grothendieck gave at Haverford
College. These lectures made a deep impression on
Messing, and Grothendieck became his informal
thesis adviser. In 1970 Messing joined the Survival
group at the Montreal meeting at which it was
founded. The following year, while Grothendieck
was visiting Kingston University in Ontario, he and
Messing made a car trip to visit Alex Jameson, an
Indian activist living on a reservation near Buffalo,
New York. Grothendieck was pursuing a quixotic
hope of helping the Indians resolve a dispute over
a land treaty.
In the summer of 1973 Messing was living in a
small studio in the Ormaille, the housing complex
for IHÉS visitors. Excitement was bubbling among
the mathematicians over Deligne’s breakthrough.
“Grothendieck was with Justine,” Messing recalled.
“They came for dinner, and Katz and I spent the
evening explaining to Grothendieck the main new
and different things in Deligne’s proof of the last
of the Weil conjectures. He was pretty excited.” At
the same time, Grothendieck expressed disap-
pointment that the proof bypassed the question of
whether or not the standard conjectures were true.
“I think he certainly would have been very happy
to have proven [all the Weil conjectures] himself,”
Katz remarked. “But in his mind, the Weil conjec-
tures were important because they were the tip of
the iceberg reflecting some fundamental struc-
tures in mathematics that he wanted to discover
and develop.” A proof of the standard conjectures
would reveal that structure in a much deeper way.
Later during that visit Grothendieck also met
with Deligne to discuss the proof. Deligne recalled
that Grothendieck was not as interested in the
proof as he would have been had it used the the-
ory of motives. “If I had done it using motives, he
would have been very interested, because it would
have meant the theory of motives had been devel-
oped,” Deligne remarked. “Since the proof used a
trick, he did not care.” In trying to develop the the-
ory of motives, Grothendieck had run into a major
technical difficulty. “The most serious problem
was that, for his idea of motives to work, one had
to be able to construct enough algebraic cycles,”
Deligne explained. “I think he tried very hard and
1204
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
he failed. And since then nobody has been able to
succeed.” According to Deligne, this technical ob-
stacle to developing the theory of motives was
probably far more frustrating to Grothendieck than
his inability to prove the last Weil conjecture.
A Distant Voice
[J]’ai quitté “le grand monde” mathé-
matique en 1970.…Après quelques an-
nées de militantisme anti-militariste et
écologique, style “révolution culturelle”,
dont tu as sans doute eu quelque écho
ici et là, je disparais pratiquement de la
circulation, perdu dans une université
de province Dieu sait où. La rumeur dit
que je passe mon temps à garder des
moutons et à forer des puits. La vérité
est qu’à part beaucoup d’autres occu-
pations, j’allais bravement, comme tout
le monde, faire mes cours à la Fac (c’était
là mon peu original gagne-pain, et ça
l’est encore aujourd’hui).
I left “the great world” of mathematics
in 1970.…After several years of anti-
military and ecological militancy, “cul-
tural revolution”-style, of which you
have no doubt heard an echo here and
there, I just about disappeared from
circulation, lost in a university in some
province, God knows where. Rumor has
it that I pass my time tending sheep
and drilling wells. The truth is that apart
from many other occupations, I bravely
went, like everyone else, to teach my
courses in the Department (this was the
way I originally earned my bread, and
it’s the same today).
—
Récoltes et Semailles
, page L3
When Grothendieck came to the Université de
Montpellier in 1973, Yves Ladegaillerie, then twenty-
five years old, was a
maître des conferences
there,
having finished his doctorate at the Institut Henri
Poincaré in Paris three years earlier. Grothendieck
proposed that Ladegaillerie do a
thèse d’état
with
him in topology and spent a great deal of time ini-
tiating the younger mathematician into his vision
and methods. In a brief memoir about Grothen-
dieck, Ladegaillerie wrote: “I had had as professors
in Paris some of the great mathematicians of the
day, from Schwartz to Cartan, but Grothendieck was
completely different, an extra-terrestrial. Rather
than translating things into another language, he
thought and spoke directly in the language of mod-
ern structural mathematics, to whose creation he
had contributed greatly” [Ladegaillerie]. Once, in
order to verify a certain algebraic computation
involving braids, Ladegaillerie made a little model
using some string and a small plank with holes. This
made Grothendieck laugh out of sheer delight: “At
that moment, he was like a child before a wizard
who performed a trick, and he told me: ‘I would
never have thought of doing that’.”
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 Bumby
and her baby there, though she soon was gone.
Many friends, acquaintances, and students went
to visit Grothendieck, including people from
the ecology movement. In 1974 the leader of a
group of Buddhist missionaries from Japan came
to visit Grothendieck, and after that many other
adherents of Buddhism passed through his
home (
R&S
, page 759). Once, after being host to a
Buddhist monk whose travel documents were not
in order, Grothendieck became the first person in
France ever to be charged under an obscure 1949
law against “gratuitously lodging and feeding a
stranger in an irregular situation” (
R&S
, page 53).
As someone who had been stateless all his life,
Grothendieck was outraged at the charge and tried
to launch a campaign against it. He even traveled
to Paris to speak about it at a Bourbaki seminar. His
campaign made headlines in French national news-
papers. Ultimately he paid a fine and received a sus-
pended sentence.
It was around this time that Grothendieck
learned to drive. He had an ancient Citroën of a
model called 2CV and known informally as a
deux
chevaux
. One of his students, Jean Malgoire, now
a
maître des conferences
at Montpellier, recalled a
terrifying journey through a torrential rainstorm
with Grothendieck at the wheel. In addition to
being a poor driver, Grothendieck was far more oc-
cupied with the discourse he was presenting to
his passengers than with the condition of the road.
“I was sure we would never get there alive!” Mal-
goire said. “I understood then that Alexandre had
a very special relationship with reality.… Rather
than adapting to what was real, he believed that re-
ality would adapt itself to him.” One time, while dri-
ving a moped, Grothendieck collided head-on with
an automobile. According to Ladegaillerie, he had
turned his eyes from the road to get an apricot out
of a bag that was behind him. Although he had a
leg fracture serious enough to require surgery, he
requested acupuncture as the only anesthetic. He
agreed to take antibiotics only when the surgeon
told him that the alternative was to amputate the
broken leg.
At the Université de Montpellier, Grothendieck
had a regular faculty position and taught at all lev-
els. Although the students were not as strong as the
ones he had had in Paris, he nevertheless poured a
great deal of energy, enthusiasm, and patience into
N
OVEMBER
2004
N
OTICES OF THE
AMS
1205
his teaching. He had an unconventional teaching
style. For an examination involving polyhedra, he
had students submit paper-and-glue models, much
to the dismay of those who had to shepherd the
exam papers through the grading process. One
person who took undergraduate courses from him
at Montpellier is Susan Holmes, now a statistician
at Stanford University. “I found him very inspiring,
as he was both unconventional and kind to the
students, who really didn’t understand at all that
he was a great mathematician,” she recalled. He
showed up in the worn-out attire of a hippie and
distributed his homegrown organic apples in class.
“He definitely did not explain in a linear fashion
suited to undergraduates, but his teaching was
very inspiring, and one got the impression of some
wonderful mysterious ‘big picture’,” Holmes said.
Grothendieck was never one for reading as a way
to learn about and understand mathematics. Talk-
ing to others had always been his primary way of
finding out what was going on in the field. His de-
parture from the intense, stimulating atmosphere
of the IHÉS, where oral exchanges were his pri-
mary mode of communication about mathematics,
was an enormous change for him. Compared with
the pace he kept during the 1960s, Grothendieck’s
later mathematical work was sporadic. Although he
had several Ph.D. students at Montpellier, he did
not establish anything like the thriving school he
had headed at the IHÉS. Some of Grothendieck’s for-
mer students and colleagues from his Paris days
traveled to Montpellier to visit him. The most fre-
quent of these visitors was Deligne, who during the
1970s was the main person keeping Grothendieck
aware of new developments.
At Montpellier, Grothendieck did not have a
seminar that met consistently. He formed a small
working group with Ladegaillerie, Malgoire, and
some of his other students, but according to Lade-
gaillerie it never really got off the ground. During
1980–81, he ran a seminar, whose sole attendee
was Malgoire, on relations between Galois groups
and fundamental groups. This is the subject of his
1,300-page manuscript
La Longue Marche à Tra-
vers la Théorie de Galois
(
The Long March through
Galois Theory
), completed in 1981. Grothendieck
did not publish
La Longue Marche
, but through
Malgoire’s efforts part of it was published in 1995
by the Université de Montpellier [Marche]. There was
also a small working seminar in which Ladegaillerie
gave some lectures on William Thurston’s work on
Teichmüller spaces, which stimulated Grothen-
dieck’s interest in this subject.
By the 1980s Grothendieck felt he had done all
he could in trying to motivate the less-than-
enthusiastic students at Montpellier and decided
to apply for a position as a researcher in the Cen-
tre National de la Recherche Scientifique (CNRS).
The CNRS, an agency of the French government,
employs mathematicians and scientists to do re-
search. Based at universities or research institu-
tions, CNRS positions usually entail no teaching. In
the 1950s, before he went to the IHÉS, Grothendieck
had held a CNRS position. In the 1970s he applied
to reenter the CNRS but was turned down. At that
time, Michel Raynaud of the Université de Paris-
Orsay was on the committee of mathematicians that
reviewed CNRS applications. Raynaud said the
CNRS administration had been hesitant to take
Grothendieck on, arguing that it was unclear
whether he would continue doing mathematics.
The committee could not contradict this argument,
and the application was turned down.
When Grothendieck reapplied to the CNRS in
1984, his application was once again controversial.
Jean-Pierre Bourguignon, now director of the IHÉS,
chaired the committee in charge of reviewing ap-
plications in mathematics, among which was
Grothendieck’s. According to Bourguignon, in the
handwritten letter required for the application,
Grothendieck listed several tasks he would not
perform, such as supervising research students. Be-
cause CNRS contracts obligate researchers to per-
form some of these tasks, this letter was viewed
by the CNRS administration as proof of Grothen-
dieck’s ineligibility. Bourguignon said he tried to
get Grothendieck to amend his application so that
it did not state explicitly all the tasks he refused
to carry out, but Grothendieck would not budge.
After considerable effort on the part of several
people, Grothendieck was eventually put on a spe-
cial kind of position, called a
position asterisquée
,
that was acceptable to him and to the CNRS. The
CNRS did not actually hire him but was in charge
only of paying his salary, and he retained his uni-
versity affiliation. So for his last few years at Mont-
pellier before his retirement in 1988, Grothendieck
did not teach and spent less and less time at the
university.
The mathematical part of Grothendieck’s 1984
application to the CNRS was the now-famous man-
uscript
Esquisse d’un Programme
. In it he outlines,
in a somewhat mysterious but nevertheless pene-
trating and visionary fashion, a new area that he
called “anabelian algebraic geometry”. He also
muses on the inadequacy of general topology and
presents ideas for a renewal in the form of what
he called “tame topology”. The
Esquisse
also con-
tains his ideas about
dessins d’enfants
, which he
originally developed in order to have a simple way
of explaining to students some notions in alge-
braic geometry and which have since spawned a
good deal of research. Grothendieck sent the
Es-
quisse
to mathematicians who he thought might
take an interest in it, and the manuscript circulated
unpublished for several years.
Leila Schneps of the Université de Paris VI read
the
Esquisse
in 1991. Before that she had identified
1206
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
Grothendieck with the foundational works of
EGA
and
SGA
, and she found that the
Esquisse
was
completely different. “It was a wild expression of
mathematical imagination,” she recalled. “I loved
it. I was bowled over, and I wanted to start work-
ing on it right away.” She became an enthusiastic
evangelist for the research program described in
the
Esquisse
, and she and others have made a good
deal of progress on it. She said, “Some of it doesn’t
even seem to make sense at first, but then you
work for two years, and you go back and look at
it, and you say, ‘He
knew
this’.” She edited a book
on
dessins d’enfants
, which appeared in 1994
[Schneps1], and in 1995 she and Pierre Lochak,
also of the Université de Paris VI, organized a con-
ference around the
Esquisse
. The
Esquisse
appeared
for the first time in print in the proceedings of that
conference [Schneps2].
Aside from the
Esquisse
and
La Longue Marche
,
Grothendieck wrote at least one other mathematical
work during the 1980s.
À la Poursuite des Champs
(
Pursuing Stacks
), which runs 1,500 pages, began as
a letter to Daniel Quillen of the University of Oxford.
Completed in 1983, it sketches Grothendieck’s
vision of a synthesis of homotopical algebra, ho-
mological algebra, and topos theory.
À la Poursuite
des Champs
circulated widely among mathemati-
cians but was never published. Although its topic is
mathematics, the style of
À la Poursuite des Champs
is completely different from the style of his earlier
mathematical writings. It was written as a sort of
“log book” on a mathematical voyage of discovery,
which includes all the false starts, wrong turns, and
sudden inspirations that characterize mathemati-
cal discovery but that are typically omitted from
written mathematical works. When nonmathemati-
cal matters drew his attention, they become part of
the log book too: for example,
À la Poursuite des
Champs
contains a digression about the birth of one
of his grandchildren. During the 1990s he wrote a
2,000-page mathematical work on the foundations
of homotopy theory called
Les Dérivateurs
, which
he gave to Malgoire in 1995 and which is now being
made available on the Web [Deriv].
While he was at Montpellier, Grothendieck’s un-
compromising, “anti-establishment” bent seems
to have become more pronounced. After Ladegail-
lerie’s thesis was finished, Grothendieck wrote to
Springer-Verlag to suggest that it be published in
the Lecture Notes series. He was outraged when he
received the reply that the series no longer pub-
lished theses. The thesis was submitted for publi-
cation anyway, with the predictable result that it
was rejected. According to Ladegaillerie, Grothen-
dieck wrote letters about this to colleagues, in an
effort to build a campaign against Springer. Lade-
gaillerie decided to publish his thesis in the form
of several papers rather than as a whole, and the
main part appeared in
Topology
. Grothendieck
reproached him for having cut the work into
publishable pieces. As Ladegaillerie put it, Grothen-
dieck tried to enlist him in his “fight against the
establishment,” but Ladegaillerie resisted, believ-
ing that such a fight was unreasonable and unjus-
tified.
“Despite such disagreements, we have stayed
friends, with highs and lows,” Ladegaillerie said. Of
his work with Grothendieck, Ladegaillerie said, “It
was fascinating to work with a genius. I don’t like
this word, but for Grothendieck there is no other
word possible.…It was fascinating, but it was also
frightening, because the man was not ordinary.”
Memories of working on mathematics with Grothen-
dieck long into the night, by the light of a kerosene
lamp, are “the greatest memories of my life as a
mathematician.”
Reaping and Sowing
Il y a beaucoup de choses dans
Récoltes
et Semailles
, et les uns et les autres y ver-
ront sans doute beaucoup de choses
différentes: un
voyage
à la découverte
d’un passé; une
méditation
sur l’exis-
tence; un
tableau de moeurs
d’un milieu
et d’une époque (ou le tableau du glisse-
ment insidieux et implacable d’une
époque à une autre…); une
enquête
(qua-
siment policière par moments, et en
d’autres frisant le roman de cape et
d’épée dans les bas-fonds de la mé-
gapolis mathématique…); une vaste
di-
vagation mathématique
(qui sèmera
plus d’un…); un traité pratique de psy-
chanalyse appliquée (ou, au choix, un
livre de
“psychanalyse-fiction”
); une
panégyrique de la
connaissance de soi
;
“Mes
confessions
”; un
journal
intime;
une psychologie de la
découverte et de
la création
; un
réquisitoire
(impitoyable,
comme il se doit…), voire un
règlement
de comptes
dans “le beau monde math-
ématique” (et sans faire de cadeaux).
There are many things in
Récoltes et
Semailles
, and different people will no
doubt see in it many different things:
a
voyage
to the discovery of a past; a
med-
itation
on existence; a
portrait of the morals
of a milieu and of an era (or the portrait
of an insidious and relentless sliding of
one era into another…); an
inquest
(almost
detective-style at times, and at others
bordering on cloak-and-dagger fiction set
in the underbelly of the mathematical
megapolis); a vast
mathematical ramble
(which will leave more than one reader in
the dust…); a practical treatise on applied
psychology (or, if you like, a book of
N
OVEMBER
2004
N
OTICES OF THE
AMS
1207
“psychoanalytic-fiction”
); a panegyric on
self-knowledge
; “My
confessions
”; a pri-
vate
diary
; a psychology of
discovery and
creation
; an indictment (pitiless, as is fit-
ting), even a
settling of scores
in “the world
of elite mathematics” (and without any
gifts).
—
Récoltes et Semailles
, page L2
Between June 1983 and February 1986, Grothen-
dieck wrote
Récoltes et Semailles: Réflexions et té-
moignage sur un passé de mathématicien
(
Reapings
and Sowings: Reflections and testimony about the
past of a mathematician
). It is a work that defies
categorization. The title suggests a memoir, but
Ré-
coltes et Semailles
is something more and less than
a memoir. It is more, in that it contains not only
memories of events in his life but also analyses,
often quite deep and minute, of the moral and psy-
chological significance of those events and his at-
tempts to reconcile their meaning with his view of
himself and the world. These analyses lead him into
philosophical musings about the role of discovery
and creativity in mathematics and in life more gen-
erally. At the same time,
Récoltes et Semailles
is
something less than a memoir, in that it does not
attempt a systematic and comprehensive account
of events in Grothendieck’s life. He is not writing
for future biographers or historians, but primarily
for himself.
Récoltes et Semailles
is a probing ex-
amination of matters closest to his heart. He brings
to this work the searching curiosity, the same drive
to get to the very bottom of things, that he brought
to his mathematics. The result is a dense, multi-
layered work that reveals a great and sometimes
terrifying mind carrying out the difficult work of
trying to understand itself and the world.
Needless to say,
Récoltes et Semailles
is not an
easy read, and Grothendieck makes a lot of de-
mands on his readers. Much of it has a quotidian
feel, and in some parts he is obviously setting down
his thoughts as they evolve from one day to the next.
As a result, within the space of a page there can be
sudden and sometimes disconcerting changes in
mood and topic. The organization is complex. The
main text is divided into numbered sections, each
with its own carefully chosen and evocative title.
Within each section there are cross-references to
other sections, as well as numerous footnotes,
some quite long and substantial, and sometimes
even footnotes to the footnotes. The wide-ranging
vocabulary presents special challenges for those
whose native language is not French, as does his pen-
chant for using colloquialisms, some of them rather
vulgar. Through it all Grothendieck writes with
great care, insight, and clarity, in a pungent and ar-
resting style. He often succeeds at describing things
that at first glance would seem quite ineffable.
One of the reasons for the complexity of the
structure of
Récoltes et Semailles
, and for its spon-
taneity, is that Grothendieck wrote it without a
definite plan in mind. He started writing it as an
introduction to
À la Poursuite des Champs
, which
was to mark his return to making a serious in-
vestment of time and energy in doing and pub-
lishing mathematics. The introduction was intended
to explain the new spirit of his research, which
would not focus on the precise and exhaustive
foundation-building of his earlier work, but would
take readers on a “voyage of discovery” of new
mathematical worlds. Grothendieck envisioned
Ré-
coltes et Semailles
as the first volume of a series
called
Réflexions
, which would contain his thoughts
and reflections on things mathematical and oth-
erwise. The second volume was to have been
À la
Poursuite des Champs
, and
La Longue Marche à
Travers la Théorie de Galois
and
Esquisse d’un
Programme
were also to have been included.
In the first part of
Récoltes et Semailles
, which he
called “Fatuité et Renouvellement” (“Complacency
and Renewal”), Grothendieck does a lot of soul-
searching about the mathematical community in
which he worked. The welcoming atmosphere he
encountered upon joining that community as a
newcomer in 1948 began to disappear, he says, as
mathematicians came to use their reputations to set
themselves in a superior position. Mathematics be-
came a way to gain power, and the elite mathemati-
cians of the day became smug, feared figures who
used that power to discourage and disdain when
it served their interests. He ruefully recounts some
instances in which he himself displayed attitudes of
conceit and haughtiness and realizes that these
attitudes had coalesced into a “sportive” or com-
petitive approach to mathematics that had begun
to hamper his ability to open himself to the beauty
of mathematical things.
It was after writing “Fatuité et Renouvellement”
that he was suddenly struck by “the insidious re-
ality of a
Burial
of my oeuvre and at the same time
of my person, which suddenly imposed itself on me,
with an irresistible force and with this very name,
‘The Burial’, on [April 19, 1984].” (
R&S
, page L8).
On that date he began writing what eventually
became a three-part series called “L’Enterrement”
(“The Burial”), comprising more than one thou-
sand pages. In it he strongly attacks some of his
former students and colleagues, whom he believes
tried to “bury” his work and his style of mathe-
matics by pilfering his ideas and not according
proper credit to him. He also champions the work
of Zoghman Mebkhout, who during the 1970s de-
veloped some of Grothendieck’s ideas and whose
work Grothendieck believes was unfairly margin-
alized and ignored. “L’Enterrement” presents
six mathematical areas, or “construction sites”
(
“chantiers”
), that he says were abandoned when
1208
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
he left the IHÉS in 1970 and that he believes his
students should have developed. Throughout “L’En-
terrement” he closely analyzes his relationship
with Pierre Deligne, the most brilliant of all of his
students and the one with whom he had the clos-
est mathematical affinity.
“L’Enterrement (II) ou La Clef du Yin et du Yang”
(“The Burial (II) or the Key to Yin and
Yang”) is rather different from the
other two parts of “L’Enterrement”
in being less directly concerned with
the investigation of the “burial”. This
second part, which Grothendieck
notes is the most personal and deep-
est part of
Récoltes et Semailles
, con-
stitutes a wide-ranging meditation
on diverse themes such as creativity,
intuition, violence, conflict, and the
self. He uses the “yin-yang” dialectic
to analyze different styles of doing
mathematics, concluding that his
own style is fundamentally “yin”, or
feminine. This style is captured in
one especially evocative section called
“La mer qui monte…” (“The rising
sea…”). He likens his approach to
mathematics to a sea: “The sea ad-
vances imperceptibly and without
sound, nothing seems to happen and
nothing is disturbed, the water is so
far off one hardly hears it. But it ends up surround-
ing the stubborn substance, which little by little be-
comes a peninsula, then an island, then an islet, which
itself is submerged, as if dissolved by the ocean stretch-
ing away as far as the eye can see” (
R&S
, page 553).
In “L’Enterrement” he pursues some of the
themes established in “Fatuité et Renouvellement”
concerning the competitive, snobbish attitudes of
the upper crust of the mathematical world. For
example, he notes that much of his work in math-
ematics was marked by an “attitude of service”:
service to the mathematical community of writing
clear and complete expositions that make funda-
mental and foundational ideas widely accessible.
Although he candidly admits that his own conceit
sometimes led him into elitist attitudes, he says that
he never lost this spontaneous sense of service, “ser-
vice to all those who leaped with me into a common
adventure” (
R&S
, page 630, (*)). He believes that the
mathematical community lost this sense of ser-
vice as personal aggrandizement and the develop-
ment of an exclusionary elite became the order of
the day. He also decries the devaluation of vision
and intuition in favor of technical mastery.
Apart from “Fatuité et Renouvellement” and the
three parts that make up “L’Enterrement”,
Récoltes
et Semailles
has two introductory volumes, as well
as an appendix to “La Clef du Yin et du Yang”. About
two hundred copies were sent out to his
mathematical colleagues. Despite Grothendieck’s
intention to publish it, the original French-language
version of
Récoltes et Semailles
has never appeared
in print, as the strong attacks it contains could be
deemed libelous. Nevertheless, it has circulated
widely. Copies can be found on bookshelves in math-
ematicians’ offices all over the world, especially in
France, and in some libraries in uni-
versities and mathematics institutes.
Historian of science Alain Herreman
of the Université de Rennes has un-
dertaken an effort to post on the
Web html files containing the entire
French original, and partial transla-
tions into English, Russian, and Span-
ish have appeared there too [
R&S
].
A Japanese translation of a large por-
tion of
Récoltes et Semailles
was pre-
pared by Yu
¯ichi Tsuji, who knew
Grothendieck through the Survival
group, and was published in the
1990s by Gendaisu
¯gakusha, a math-
ematics publisher. According to
Michel Waldschmidt of the Univer-
sité de Paris VI, who was president
of the Société Mathématique de
France (SMF) during 2001–04, the
society considered, during his pres-
idency, the question of whether to
publish
Récoltes et Semailles
. The
question raised strong opinions both for and against,
Waldschmidt said, and ultimately the SMF decided
against publication.
Many mathematicians, especially some of
Grothendieck’s former students, were shocked and
hurt by the accusations in
Récoltes et Semailles
. One
of them, Luc Illusie of the Université de Paris-Orsay,
recalled that he talked to another former student,
Jean-Louis Verdier, about whether they should try
to discuss the accusations with Grothendieck. Ac-
cording to Illusie, Verdier, who died in 1989, felt
that Grothendieck’s state of mind was such that
there was no sound basis for discussion. But, Illusie
said, “I thought, ‘It is not possible that Grothendieck
has become like that. I will try to reason and to dis-
cuss with him. Maybe I will agree with him on some
points that he is right and on others he is not
right.’ Eventually, we settled the material points, but
nothing really emerged, and he remained convinced
that everyone was against him.”
In
Récoltes et Semailles
Grothendieck says that,
after he left the mathematical world in 1970, his
style of doing mathematics was held in contempt
and that many of the paths he had broken went
undeveloped. It is true that after that time, re-
search in algebraic geometry began to shift, mix-
ing the highly general approach that characterized
his work with investigation of specific problems.
Deligne’s proof of the Weil conjecture, which was
Grothendieck in a
photograph from the 1950s.
N
OVEMBER
2004
N
OTICES OF THE
AMS
1209
very much in the spirit of Grothendieck but which
also incorporated many new ideas, was one of the
great advances of the 1970s. Along with develop-
ments in the theory of D-modules and Deligne’s
mixed Hodge theory, greater attention began to
be paid to more specific problems, such as the
classification theory of varieties and questions
about low-dimensional varieties. Also, after the
Antwerp meeting of 1972, collaborations grew
between algebraic geometry and representation
theory, leading to advances in the theory of auto-
morphic forms and the Langlands program. As
Illusie put it, all these developments show that
there has been “quite a natural balance between
general theory and the study of specific examples
at great length, to enrich the theory itself.”
Récoltes et Semailles
also contains the accusation
that Grothendieck’s work was not always properly
credited. Indeed, his work was so well known and
fundamental that credit was not always specifi-
cally accorded to him. “It is true that everybody
knew he had invented motives, for instance, or
l
-adic cohomology, and so there was no need to
quote his name every time one used them,” re-
marked Jean-Pierre Serre. “His name was rarely
mentioned because of that. On the other hand, it
was well known that it was due to him. Nobody was
saying that it was due to someone else.” Serre
noted that Grothendieck’s complaining about lack
of credit is in sharp contrast to his behavior dur-
ing the 1960s, when he shared his ideas with great
generosity and in some cases attached other peo-
ple’s names to ideas he himself had come up with.
“It was sad to read
Récoltes et Semailles
because of
that,” Serre said.
Even granting that there was a shift away from
Grothendieck’s style of mathematics and that credit
was not always specifically accorded to him, it is
a long leap from there to the deliberate “burial”
that he asserts took place. “In retrospect, very few
mathematical ideas have been as widely used as
Grothendieck’s,” said Illusie. “Everybody who is
doing algebraic or arithmetic geometry now uses
Grothendieck’s language, ideas, theorems, and so
on. So when you think one second, it is completely
ridiculous that he suggested that he could have
been buried.” There is no question that mathe-
matics suffered a great loss when Grothendieck
halted his research career in 1970. But mathemat-
ics did not stop; others continued to work, fol-
lowing their own ideas and interests. In February
1986, after receiving a copy of
Récoltes et Semailles
,
Serre wrote to Grothendieck: “You are surprised
and indignant that your former students did not
continue the work that you had undertaken and
largely completed. But you do not ask the most ob-
vious question, the one every reader expects you
to answer: and you, why did you abandon the work
in question?” [Corr].
Although the accusations of a “burial” have gen-
erated a good deal of notoriety, there is much more
to
Récoltes et Semailles
. Those who have read
beyond those parts have been deeply touched by
the work’s beauty and insights. Grothendieck’s
critique of how the highly competitive atmosphere
of the mathematical world stifles creativity and
renewal of the field resonated with many. In
Récoltes
et Semailles
Grothendieck puts the highest value
on the innocent, childlike curiosity that gives birth
to the creative impulse, and he mourns the way it
is trampled on by competitiveness and the desire
for power and prestige.
“I am one of quite probably a minority who
think that
Récoltes et Semailles
is a miraculous
document,” said William Messing. “That is not to
say that there are not parts that are excessive and
have aspects of what might be referred to as para-
noia. But it’s very striking that the person who
created
EGA
and
SGA
would write in such a style.
The systematic and soul-searching aspect is of a
piece with his approach to mathematics. Those
who have really read it—as opposed to looking at
five pages of negative comments—tend to think of
it as an extraordinary document.”
Lightness Descending
[A]ujourd’hui je ne suis plus, comme
naguère, le prisonnier de tâches
Contents of
Récoltes et Semailles
Présentation des Themes—ou Prélude en Quatre
Mouvements
•
En guise d’avant-propos (January 1986: pages A1–A6)
•
Promenade à travers une oeuvre—ou l’enfant et la mère (Jan-
uary 1986: pages P1–P65 )
•
Epilogue en post-scriptum—ou contexte et préalables d’un
débat (February 1986: pages L44–156)
Lettre—Introduction
•
Une lettre, May–June 1985: pages L1–L43
•
Table des matières (pages T1–T10)
•
Introduction (March 1984: sections 1–5, pages i–xi)
•
Introduction (May–June 1985: sections 6–10, pages xi–xxii)
Première Partie: Fatuité et Renouvellement
(June 1983, February 1984: pages 1–171)
Deuxième Partie: L’Enterrement I, ou la Robe de l’Empereur
de Chine
(April–June 1984: pages 173–420)
Troisième Partie: L’Enterrement II, ou la Clef du Yin et du
Yang
(September 1984–January 1985: pages 421–774)
Quatrième Partie: L’Enterrement III, ou les Quatre Opérations
(February 1985–June 1985: pages 775–1252)
Les Portes sur l’Univers (Appendice à la Clef du Yin et du
Yang)
(March–April 1986: pages PU1–PU127)
1210
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
interminables, qui si souvent m’avaient
interdit de m’élancer dans l’inconnu,
mathématique ou non. Le temps des
tâches
pour moi est révolu. Si l’âge m’a
apporté quelque chose, c’est d’être plus
léger.
Today I am no longer, as I once was, the
prisoner of interminable tasks, which so
often prevented me from leaping into
the unknown, mathematical or other-
wise. The time of
tasks
for me is over.
If age has brought me anything, it is
lightness.
—
Esquisse d’un Programme
“[T]he ethics of the scientific profession (espe-
cially among mathematicians) have degraded to
such a degree that pure and simple theft between
colleagues (especially at the expense of those who
have no position of power to defend themselves)
has almost become the general rule and is in any
case tolerated by all, even in the most flagrant and
iniquitous cases.” So wrote Grothendieck in an
April 19, 1988, letter to the Royal Swedish Acad-
emy of Sciences in which he declined the 1988
Crafoord Prize. He also sent to the academy the
introductory volumes of
Récoltes et Semailles
.
The academy had awarded the prize of around
US$200,000 to him and Pierre Deligne. Grothen-
dieck’s letter became widely known when it was
published in
Le Monde
on May 4, 1988 [LeMonde].
To play into the game of accepting prizes and hon-
ors, Grothendieck wrote, would be to validate “a
spirit and an evolution in the scientific world that
I see as profoundly unhealthy, and condemned to
disappear soon, so suicidal is it, spiritually as well
as intellectually and materially.” Evidently these sen-
timents resonated with many readers of
Le Monde
.
One of the newspaper’s editors told Jean-Pierre
Bourguignon that the paper had received more re-
actions to Grothendieck’s letter than to any other
preceding it and that most of the letters registered
approval that finally a scientist had recognized
how corrupt the scientific milieu had become. News
of the letter appeared in other magazines and
newspapers, and it was avidly discussed within
the mathematical community. An English transla-
tion was published in the
Mathematical Intelli-
gencer
[Intell], and a short item appeared in the
Notices
[Notices].
The same year in which he turned down the
Crafoord Prize, Grothendieck retired from the Uni-
versité de Montpellier at the age of sixty. Also that
year, six mathematicians decided to assemble
a collection of articles as a “Festschrift” on the
occasion of Grothendieck’s sixtieth birthday
[Festschrift] (there was also a special issue of the
journal
K-Theory
dedicated to Grothendieck). The
Festschrift seems to have been an attempt to make
amends with Grothendieck and to show that he had
not been “buried”, as he asserted in
Récoltes et
Semailles
. Some of the people contributing papers
were among those he had most heavily criticized.
When the Festschrift appeared in 1990, Illusie, who
was one of the editors, sent a copy to Grothendieck,
whose reaction was extremely bitter. In a letter to
Illusie, he objected strongly to the brief foreword
of the volume and also to the fact that he had
not been told that the volume would appear. He
said his work had been used like “confetti,” like
bright, worthless bits one throws into the air to
give the pretense of happiness and celebration
while ignoring the malaise underneath. Grothen-
dieck submitted this letter for publication in the
Bulletin de la Société Mathématique de France
. When
the SMF told him that the
Bulletin
carries only
mathematics articles but that the letter could
appear instead in the
SMF Gazette
, Grothendieck
refused. The letter was never published.
After he retired, Grothendieck spent little time
at the Université de Montpellier, though he con-
tinued to live in the area, in a village called Les
Aumettes. At this time, Ladegaillerie said, Grothen-
dieck seemed to be going through a deep spiritual
crisis and wrote strange letters “that made us fear
the worst about his condition.” During 1987–88,
Grothendieck wrote
La Clef des Songes ou Dialogue
avec le Bon Dieu
(
The Key to Dreams or Dialogue
with the Good Lord
), which expresses his convic-
tion that God exists and that He speaks to people
through their dreams. It also contains a good deal
of material about Grothendieck’s early life.
La Clef
des Songes
runs about three hundred pages and is
accompanied by another five hundred pages of
notes. According to a lecture given in the summer
of 2004 by Winfried Scharlau of the Universität
Münster, Grothendieck subsumed
La Clef des
Songes
under a collection of works that he called
Méditations
and that included the material making
up
Réflexions
, as well as a poetical work called
“Eloge de l’Inceste” (“The Eulogy to Incest”). Nei-
ther that work nor
La Clef des Songes
was ever
widely distributed.
Many of Grothendieck’s friends and colleagues
became aware of his increasing preoccupation with
spiritual matters when they received “La Lettre de
la Bonne Nouvelle” (“The Letter of Good News”),
which is dated January 26, 1990, and which he
sent to about two hundred fifty people. The letter
states: “You are part of a group of two to three thou-
sand people, personally known to me, whom God
destines for a great mission: That of announcing
and preparing the
‘New Age’
(or
Age of Libera-
tion
…), which will commence on the
‘Day of Truth’
,
14 October 1996.” He says that God manifested
Himself to him for the first time in 1986 and
N
OVEMBER
2004
N
OTICES OF THE
AMS
1211
communicated to him through dreams. He also
describes encounters with a deity named Flora,
who imparts revelations but also cruelly tests his
faith. Although the content of the letter is baffling,
the way it is written is perfectly lucid. Three months
later Grothendieck sent a “correction”, stating
that he was no longer certain of the truth of the
revelations described in “La Lettre de la Bonne
Nouvelle”. He writes: “That I was the victim of a
mystification by one of more ‘spirits’ (among which
my limited capacity could not distinguish), invested
with prodigious powers over my body and in my
psyche, I no longer have the least doubt.” Together,
the two letters impart an impression of deep dis-
turbance and suffering.
In July 1990 Grothendieck asked Malgoire to
take possession of all of his mathematical papers,
including books, preprints, correspondence, and
manuscripts in various states of preparation.
Grothendieck wanted to “lighten” himself of many
things, as Malgoire put it. He burned a huge amount
of material, most of it nonmathematical, including
letters that his parents had exchanged in the 1930s.
He showed Malgoire a 200-liter oil drum filled with
cinders and estimated he had destroyed a total of
25,000 pages. Grothendieck also left many papers
and other items, including his mother’s death mask,
with a friend named Yolande Levine, to whom he
had been very close for the preceding decade. He
then disappeared into the Pyrenées to live in com-
plete isolation. A small number of people knew
where he was, and he instructed them not to for-
ward any mail that arrived for him at the univer-
sity. Malgoire said that even today, close to fifteen
years after Grothendieck went into seclusion, the
university still gets a great deal of correspondence
addressed to him. In 1995 Grothendieck formally
conferred the legal rights to his mathematical
papers to Malgoire.
Grothendieck has had very little contact with
mathematicians in the past fifteen years. Among
the few who have seen him are Leila Schneps and
Pierre Lochak, who met him in the mid-1990s. They
told him about the progress made on the program
he had outlined in the
Esquisse d’un Programme
,
and he was surprised to learn that people were still
interested in his work. He had developed a strong
interest in physics but expressed frustration with
what he felt was a lack of rigor in that field. Lochak
and Schneps exchanged some letters with him and
also sent him some books on physics that he had
asked for. In one letter he asked a disarmingly
simple question: What is a meter? His letters began
to swing between warm friendliness and cold sus-
picion, and eventually he severed all contact with
them. Although the friendship with Grothendieck
could not be sustained, Lochak and Schneps retain
a fervent admiration and a deep attachment to the
man and his work. Together they painstakingly
typed into TEX a large chunk of the handwritten
La
Longue Marche à Travers la Théorie de Galois
. They
have also started a website, the Grothendieck Cir-
cle, which contains a wealth of material about
Grothendieck, his life, and his work [Circle].
The Dancing Star
Ich sage euch: man muß noch Chaos in
sich haben, um einen tanzenden Stern
gebären zu können. Ich sage euch: ihr
habt noch Chaos in euch.
I tell you: one must have chaos inside,
to give birth to a dancing star. I tell you:
you have yet chaos in you.
—Friedrich Nietzsche,
Also sprach
Zarathustra
The work of Alexandre Grothendieck has had a
profound impact on modern mathematics and,
more broadly, ranks among the most important ad-
vances in human knowledge during the twentieth
century. The stature of Grothendieck can be com-
pared to that of, for example, Albert Einstein. Each
of them opened revolutionary new perspectives
that transformed the terrain of exploration, and
each sought fundamental, unifying connections
among phenomena. Grothendieck’s propensity for
investigating how mathematical objects behave
relative to one another echoes the relativistic view-
point proposed by Einstein. Grothendieck’s work
also has parallels with another great twentieth-
century advance, that of quantum mechanics, which
turned conventional notions upside down by re-
placing point particles by “probability clouds”.
“[T]hese ‘probability clouds’, replacing the reas-
suring material particles of before, remind me
strangely of the elusive ‘open neighborhoods’ that
populate the toposes, like evanescent phantoms,
to surround the imaginary ‘points’,” he wrote (
R&S
,
page P60).
Yet, as extraordinary as Grothendieck’s achieve-
ments are, he traced his creative capacity to some-
thing rather humble: the naive, avid curiosity of a
child. “Discovery is the privilege of the child,” he
wrote in
Récoltes et Semailles
(page 1), “the child
who has no fear of being once again wrong, of
looking like an idiot, of not being serious, of
not doing things like everyone else.” For the work
of discovery and creation, Grothendieck saw
intellectual endowment and technical power as
secondary to the child’s simple thirst to know and
understand. This child is inside each of us, though
it may be marginalized, neglected, or drowned out.
“Each of us can rediscover what discovery and
creation are, and no one can invent them” (
R&S
,
page 2).
1212
N
OTICES OF THE
AMS
V
OLUME
51, N
UMBER
10
One aspect of this childlike curiosity is a scrupu-
lous fidelity to truth. Grothendieck taught his stu-
dents an important discipline when writing about
mathematics: never say anything false. Statements
that were almost or essentially true were not per-
mitted. It was acceptable to be vague, but when one
gives precise details, one must say only things that
are true. Indeed, Grothendieck’s life has been a
constant search for truth. From his mathematical
work through
Récoltes et Semailles
and even “La Let-
tre de la Bonne Nouvelle”, Grothendieck wrote with
the unblinking honesty of a child. He spoke the
truth—
his
truth, as he perceived it. Even when he
made factual mistakes or was misled by incorrect
assumptions, he presented candidly what was in
his mind. He has never tried to hide who he is and
what he thinks.
Grothendieck’s search for truth took him to the
very roots of mathematical ideas and to the far
reaches of human psychological perception. He
has had a long journey. “In his solitary retirement
in the Pyrenées, Alexandre Grothendieck has the
right to rest after all he has been through,” wrote
Yves Ladegaillerie [Ladegaillerie]. “He deserves our
admiration and our respect but, above all, in think-
ing of what we owe him, we must leave him in
peace.”
References
[Circle] The Grothendieck Circle,
http://www.
grothendieck-circle.org
.
[Corr]
Correspondence Grothendieck-Serre.
Société Math-
ématique de France, 2001. (Bilingual French-English
edition, AMS, 2003.)
[Deriv]
Les Dérivateurs
, by Alexandre Grothendieck, edited
by M. Künzer, J. Malgoire, and G. Maltsiniotis. Avail-
able at
http://www.math.jussieu.fr/~maltsin/
groth/Derivateurs.html
.
[Aubin] D. A
UBIN
,
A Cultural History of Catastrophes and
Chaos: Around the “Institut des Hautes Études Scien-
tifiques,” France
, doctoral thesis, Princeton Univer-
sity, 1998.
[Festschrift]
The Grothendieck Festschrift: A Collection of
Articles Written in Honor of the 60th Birthday of Alexan-
der Grothendieck
, Volumes I–III (P. Cartier, L. Illusie,
N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, eds.),
Progress in Mathematics, vol. 87, Birkhäuser Boston,
Inc., Boston, MA, 1990.
[Herreman] A. H
ERREMAN
, Découvrir et transmettre: La
dimension collective des mathématiques dans
Récoltes et Semailles
d’Alexandre Grothendieck,
Prépublications de l’IHÉS, 2000. Available at
http://
name.math.univ-rennes1.fr/alain.herreman/
.
[Infeld] L. I
NFELD
,
Whom the Gods Love. The Story of Évariste
Galois,
Whittlesey House, New York, 1948.
[Intell] English translation of Grothendieck’s letter de-
clining the 1988 Crafoord Prize,
Math. Intelligencer
11
(1989).
[Ladegaillerie] Y. L
ADEGAILLERIE
, Alexandre Grothendieck
après 1970. Personal communication.
[LeMonde] Lettre à l’Académie Royale des Sciences de
Suède: Le mathématicien français Alexandre Grothen-
dieck refuse le prix Crafoord,
Le Monde
, May 4, 1988.
[Marche] A. G
ROTHENDIECK
,
La Longue Marche à Travers
la Théorie de Galois,
volume 1, edited and with a
foreword by Jean Malgoire, Université Montpellier II,
Département des Sciences Mathématiques, 1995.
[Notices] Crafoord Prize recipients named,
Notices Amer.
Math. Soc.
(July/August 1988), 811–812.
[R&S] A. G
ROTHENDIECK
,
Récoltes et Semailles: Réflexions et
témoignages sur un passé de mathématicien
, Univer-
sité des Sciences et Techniques du Languedoc, Mont-
pellier, et Centre National de la Recherche Scientifique,
1986. (Parts available in the original French at
http:// mapage.noos.fr/recoltesetsemailles/
.
Partial translations are available in English at
http://www.fermentmagazine.org/home5.html
,
in Russian at
http://elenakosilova.narod.ru/
studia/groth.htm
, and in Spanish at
http://
kolmogorov.unex.es/navarro/res
.)
[Rankin] R. A. R
ANKIN
, Contributions to the theory of Ra-
manujan’s function
τ
(
n
)
and similar arithmetical func-
tions. I. The zeros of the function
∞
n
=1
τ
(
n
)
/n
s
on the
line
R
s
= 13
/
2
. II. The order of the Fourier coefficients
of integral modular forms,
Proc. Cambridge Philos. Soc.
35
(1939), 351–372.
[Schneps1]
The Grothendieck Theory of Dessins d’Enfants
(L. Schneps, ed.), London Math. Soc. Lecture Note Ser.,
vol. 200, Cambridge University Press, 1994.
[Schneps2]
Geometric Galois Actions
(L. Schneps and
P. Lochak, eds.), London Math. Soc. Lecture Note Ser.,
vols. 242 and 243, Cambridge University Press, 1997.
[Vietnam] A. G
ROTHENDIECK
, La vie mathématique en
République Democratique du Vietnam, text of a
lecture presented in Paris on December 20, 1967.
Unpublished.
Acknowledgments
The assistance of the following individuals is
gratefully acknowledged: Norbert A’Campo, James
Arthur, Michael Artin, Hyman Bass, Armand Borel,
Jean-Pierre Bourguignon, Felix Browder, Justine
Bumby, Richard Bumby, Pierre Cartier, Pierre
Deligne, Edward Effros, Gerd Faltings, Momota
Ganguli, Robin Hartshorne, Alain Herreman,
Friedrich Hirzebruch, Susan Holmes, Chaim Honig,
Luc Illusie, Nicholas Katz, Dieter Kotschick, Klaus
Künnemann, Yves Ladegaillerie, Pierre Lochak,
Andy Magid, Jean Malgoire, Bernard Malgrange,
Barry Mazur, Colin McLarty, Vikram Mehta, William
Messing, Shigeyuki Morita, David Mumford, Jose
Barros Neto, Arthur Ogus, Michel Raynaud, Paulo
Ribenboim, David Ruelle, Winfried Scharlau, Leila
Schneps, Jean-Pierre Serre, John Tate, Karin Tate,
Jacques Tits, and Michel Waldschmidt.
Photographs used in this article are courtesy of
Michael Artin, the Tata Institute, Karin Tate, and
the website of the Grothendieck Circle (
http://
www.grothendieck-circle.org
).