Comme Appelé du Néant—As If Summoned from the Void: The Life of Alexandre Grothendieck

36 Pages·2011·2.11 MB·English

Comme Appelé du Néant—
As If Summoned from the Void:
The Life of Alexandre
Grothendieck
Allyn Jackson
This is the first part of a two-part article about
the life of Alexandre Grothendieck. The second
part of the article will appear in the next issue
of the Notices.
Et toute science, quand nous l’enten- the Institut des Hautes Études Scientifiques (IHÉS)
dons non comme un instrument de pou- and received the Fields Medal in 1966—suffice to
voir et de domination, mais comme secure his place in the pantheon of twentieth cen-
aventure de connaissance de notre es- tury mathematics. But such details cannot capture
pèce à travers les âges, n’est autre chose the essence of his work, which is rooted in some-
que cette harmonie, plus ou moins vaste thing far more organic and humble. As he wrote in
et plus ou moins riche d’une époque à his long memoir, Récoltes et Semailles(Reapings and
l’autre, qui se déploie au cours des Sowings, R&S), “What makes the quality of a re-
générations et des siècles, par le délicat searcher’s inventiveness and imagination is the
contrepoint de tous les thèmes apparus quality of his attention to hearing the voices of
tour à tour, comme appelés du néant. things” (emphasis in the original, page P27). Today
Grothendieck’s own voice, embodied in his written
And every science, when we understand works, reaches us as if through a void: now seventy-
it not as an instrument of power and six years old, he has for more than a decade lived
domination but as an adventure in in seclusion in a remote hamlet in the south of
knowledge pursued by our species France.
across the ages, is nothing but this har- Grothendieck changed the landscape of mathe-
mony, more or less vast, more or less matics with a viewpoint that is “cosmically general”,
rich from one epoch to another, which in the words of Hyman Bass of the University of
unfurls over the course of generations Michigan. This viewpoint has been so thoroughly
and centuries, by the delicate counter- absorbed into mathematics that nowadays it is dif-
point of all the themes appearing in ficult for newcomers to imagine that the field was
turn, as if summoned from the void. not always this way. Grothendieck left his deepest
mark on algebraic geometry, where he placed em-
—Récoltes et Semailles, page P20 phasis on discovering relationships among math-
ematical objects as a way of understanding the ob-
Alexandre Grothendieck is a mathematician of
jects themselves. He had an extremely powerful,
immense sensitivity to things mathematical, of
almost other-worldly ability of abstraction that al-
profound perception of the intricate and elegant
lowed him to see problems in a highly general con-
lines of their architecture. A couple of high points
text, and he used this ability with exquisite preci-
from his biography—he was a founding member of
sion. Indeed, the trend toward increasing generality
Allyn Jackson is senior writer and deputy editor of the No- and abstraction, which can be seen across the
tices. Her email address is [email protected]. whole field since the middle of the twentieth
1038 NOTICESOFTHEAMS VOLUME51, NUMBER4
century, is due in no small part to Grothendieck’s at least what is known of
influence. At the same time, generality for its own it—contains few clues
sake, which can lead to sterile and uninteresting that he was destined to
mathematics, is something he never engaged in. become a dominant fig-
Grothendieck’s early life during World War II had ure in that world. Many
a good deal of chaos and trauma, and his educa- of the details about
tional background was not the best. How he Grothendieck’s family
emerged from these deprived beginnings and background and early life
forged a life for himself as one of the leading math- are sketchy or unknown.
ematicians in the world is a story of high drama— Winfried Scharlau of the
as is his decision in 1970 to abruptly leave the Universität Münster is
mathematical milieu in which his greatest achieve- writing a biography of
ments blossomed and which was so deeply influ- Grothendieck and has
enced by his extraordinary personality. studied carefully this
part of his life. Much of
Early Life
the information in the
Grothendieck’s following biographical
Ce qui me satisfaisait le moins, dans nos
mother, Hanka, 1917. sketch comes from an in-
livres de maths [au lycée], c’était l’ab-
terview with Scharlau and
sence de toute définition sérieuse de la
from biographical materials he has assembled
notion de longueur (d’une courbe),
about Grothendieck [Scharlau].
d’aire (d’une surface), de volume (d’un
Grothendieck’s father, whose name may have
solide). Je me suis promis de combler
been Alexander Shapiro, was born into a Jewish
cette lacune, dès que j’en aurais le loisir.
family in Novozybkov in Ukraine on October 11,
1889. Shapiro was an anarchist and took part in var-
What was least satisfying to me in our
ious uprisings in czarist Russia in the early twen-
[high school] math books was the ab-
tieth century. Arrested at the age of seventeen, he
sence of any serious definition of the no-
managed to elude a
tion of length (of a curve), of area (of a
death sentence, but,
surface), of volume (of a solid). I
after escaping and
promised myself I would fill this gap
being recaptured a
when I had the chance.
few times, he spent a
total of about ten
—Récoltes et Semailles, page P3
years in prison.
Armand Borel of the Institute for Advanced Grothendieck’s father
Study in Princeton, who died in August 2003 at the has sometimes been
age of 80, remembered the first time he met confused with an-
Grothendieck, at a Bourbaki seminar in Paris in No- other more famous
vember 1949. During a break between lectures, activist also named
Borel, then in his mid-twenties, was chatting with Alexander Shapiro,
Charles Ehresmann, who at forty-five years of age who participated in
was a leading figure in French mathematics. As some of the same po-
Borel recalled, a young man strode up to Ehresmann litical movements. Grothendieck’s father,
and, without any preamble, demanded, “Are you This other Shapiro, Sascha, ca. 1922.
who was portrayed in
an expert on topological groups?” Ehresmann, not
John Reed’s book Ten
wanting to seem immodest, replied that yes, he
Days that Shook the World, emigrated to New York
knew something about topological groups. The
and died there in 1946, by which time Grothen-
young man insisted, “But I need a real expert!”
dieck’s father had already been dead for four years.
This was Alexandre Grothendieck, age twenty-
Another distinguishing detail is that Grothendieck’s
one—brash, intense, not exactly impolite but hav-
father had only one arm. According to Justine
ing little sense of social niceties. Borel remem-
Bumby, who lived with Grothendieck for a period
bered the question Grothendieck asked: Is every
in the 1970s and had a son by him, his father lost
local topological group the germ of a global topo-
his arm in a suicide attempt while trying to avoid
logical group? As it turned out, Borel knew a coun-
being captured by the police. Grothendieck himself
terexample. It was a question that showed Grothen-
may unwittingly have contributed to the confu-
dieck was already thinking in very general terms.
sion between the two Shapiros; for example, Pierre
Grothendieck’s time in Paris in the late 1940s
Cartier of the Institut des Hautes Études Scien-
was his first real contact with the world of math-
tifiques mentioned in [Cartier2] Grothendieck’s
ematical research. Up to that time, his life story—
OCTOBER2004 NOTICESOFTHEAMS 1039
maintaining that one of the figures in Reed’s book and he is mentioned briefly. Heydorn had been a
was his father. Lutheran priest and army officer, then left the
In 1921 Shapiro left Russia and was stateless for church and worked as an elementary school teacher
the rest of his life. To hide his political past, he ob- and a Heilpraktiker (which nowadays might be
tained identity papers with the name Alexander translated roughly as “practitioner of alternative
Tanaroff, and for the rest of his life he lived under medicine”). In 1930 he founded an idealistic polit-
this name. He spent time in France, Germany, and ical party called the “Menschheitspartei” (“Hu-
Belgium, where he associated with anarchist and manity Party”), which was outlawed by the Nazis.
other revolutionary groups. In the radical circles of Heydorn had four children of his own, and he and
Berlin in the mid-1920s, he met Grothendieck’s his wife Dagmar, following their sense of Christ-
mother, Johanna (Hanka) Grothendieck. She had ian duty, took in several foster children who were
been born on August 21, 1900, into a bourgeois separated from their families in the tumultuous pe-
family of Lutherans in Hamburg. Rebelling against riod leading up to World War II.
her traditional upbringing, she was drawn to Berlin, Grothendieck remained with the Heydorn fam-
which was then a hotbed of avant-garde culture and ily for five years, between the ages of five and
revolutionary social movements. eleven, and attended school. A memoir by Dagmar
Both she and Shapiro yearned to Heydorn recalled the young Alexandre as being
be writers. He never published any- very free, completely honest, and lacking in inhi-
thing, but she published some bitions. During his time with the Heydorns,
newspaper articles; in particular, Grothendieck received only a few letters from his
between 1920 and 1922, she wrote mother and no word at all from his father. Al-
for a leftist weekly newspaper though Hanka still had relatives in Hamburg, no one
called Der Pranger, which had ever came to visit her son. The sudden separation
taken up the cause of prostitutes from his parents was highly traumatic for Grothen-
living on the fringe of Hamburg dieck, as he indicated in Récoltes et Semailles(page
society. Much later, in the late 473). Scharlau speculated that the young Alexan-
1940s, she wrote an autobio- dre was probably not especially happy with the Hey-
graphical novel called Eine Frau, dorns. Having started life in a liberal home headed
which was never published. by a couple of anarchists, the stricter atmosphere
For most of his life, Tanaroff of the Heydorn household probably chafed. He
was a street photographer, an oc- was actually closer to some other families who
cupation that allowed him to earn lived near the Heydorns, and as an adult he con-
an independent living without tinued to write to them for many years. He also
A. Grothendieck as a child.
being in an employer-employee re- wrote to the Heydorns and visited Hamburg sev-
lationship that would have run eral times, the last time in the mid-1980s.
counter to his anarchist principles. He and Hanka By 1939, with war imminent, political pressure
had each been married before, and each had a child increased on the Heydorns, and they could no
from the previous marriage, she a daughter and he longer keep the foster children. Grothendieck was
a son. Alexandre Grothendieck was born in Berlin an especially difficult case, because he looked Jew-
on March 28, 1928, into a family consisting of ish. The exact whereabouts of his parents were
Hanka, Tanaroff, and Hanka’s daughter from her
unknown, but Dagmar Heydorn wrote to the French
first marriage, Maidi, who was four years older
consulate in Hamburg and managed to get a mes-
than Alexandre. He was known in the family, and
sage to Shapiro in Paris and to Hanka in Nîmes.
to his close friends later on, as Shurik; his father’s
Once contact with his parents was made, Grothen-
nickname was Sascha. Although he never met his
dieck, then 11 years old, was put on a train from
half-brother, Grothendieck dedicated to him the
Hamburg to Paris. He was reunited with his parents
manuscript A La Poursuite des Champs (Pursuing
in May 1939, and they spent a brief time together
Stacks), written in the 1980s.
before the war began.
In 1933, when the Nazis came to power, Shapiro It is not clear exactly what Grothendieck’s par-
fled Berlin for Paris. In December that year, Hanka ents were doing while he was in Hamburg, but they
decided to follow her husband, so she put her son remained politically active. They went to Spain to
in the care of a foster family in Blankenese, near fight in the Spanish Civil War and were among the
Hamburg; Maidi was left in an institution for hand- many who fled to France when Franco triumphed.
icapped children in Berlin, although she was not Because of their political activities, Hanka and her
handicapped (R&S, pages 472–473). The foster fam- husband were viewed in France as dangerous for-
ily was headed by Wilhelm Heydorn, whose re- eigners. Some time after Grothendieck joined them
markable life is outlined in his biography, Nur there, Shapiro was put into the internment camp
Mensch Sein![Heydorn]; the book contains a pho- Le Vernet, the worst of all the French camps. It is
tograph of Alexandre Grothendieck from 1934, probable that he never again saw his wife and son.
1040 NOTICESOFTHEAMS VOLUME51, NUMBER9
In August 1942 he was deported by the French au- maths avaient été résolus, il y avait
thorities to Auschwitz, where he was killed. What vingt ou trente ans, par un dénommé
happened to Maidi at this time is unclear, but even- Lebesgue. Il aurait développé justement
tually she married an American soldier and emi- (drôle de coïncidence, décidement!) une
grated to the United States; she passed away a cou- théorie de la mesure et de l’intégration,
ple of years ago. laquelle mettait un point final à la math-
In 1940 Hanka and her son were put into an in- ématique.
ternment camp in Rieucros, near Mende. As in-
ternment camps went, the one at Rieucros was one Mr. Soula [my calculus teacher] assured
of the better ones, and Grothendieck was permit- me that the final problems posed in
ted to go to the lycée(high school) in Mende. Nev- mathematics had been resolved, twenty
ertheless, it was a life of deprivation and uncer- or thirty years before, by a certain
tainty. He told Bumby that he and his mother were Lebesgue. He had exactly developed (an
sometimes shunned by French people who did not amusing coincidence, certainly!) a the-
know of Hanka’s opposition to the Nazis. Once he ory of measure and integration, which
ran away from the camp with the intention of as- was the endpoint of mathematics.
sassinating Hitler, but he was quickly caught and
returned. “This could easily have cost him his life”, —Récoltes et Semailles, page P4
Bumby noted. He had always been strong and a
By the time the war ended in Europe, in May
good boxer, attributes that were useful at this time,
1945, Alexandre Grothendieck was seventeen years
as he was sometimes the target of bullying.
old. He and his mother went to live in Maisargues,
After two years, mother and son were sepa-
a village in a wine-growing region outside of Mont-
rated; Hanka was sent to another internment camp,
pellier. He enrolled at the Université de Montpel-
and her son ended up in the town of Chambon-sur-
lier, and the two survived on his student scholar-
Lignon. André Trocmé, a Protestant pastor, had
ship and by doing seasonal work in the grape
transformed the mountain resort town of Cham-
harvest; his mother also worked at houseclean-
bon into a stronghold of resistance against the
ing. Over time he attended the university courses
Nazis and a haven for protecting Jews and others
less and less, as he found that the teachers were
endangered during the war [Hallie]. There Grothen-
mostly repeating what was in the textbooks. At the
dieck was taken into a children’s home supported
time, Montpellier “was among the most backward
by a Swiss organization. He attended the Collège
of French universities in the teaching of mathe-
Cévenol, set up in Chambon to provide an educa-
matics,” wrote Jean Dieudonné [D1].
tion for the young people, and earned a baccalau-
In this uninspiring environment, Grothendieck
réat. The heroic efforts of the Chambonnais kept
devoted most of his three years at Montpellier to
the refugees safe, but life was nevertheless pre-
filling the gap that he had felt in his high school
carious. In Récoltes et SemaillesGrothendieck men-
textbooks about how to provide a satisfactory de-
tioned the periodic roundups of Jews that would
finition of length, area, and volume. On his own,
send him and his fellow students scattering to
he essentially rediscovered measure theory and the
hide in the woods for a few days (page P2).
notion of the Lebesgue integral. This episode is one
He also related some of his memories of his
of several parallels between the life of Grothendieck
schooling in Mende and Chambon. It is clear that,
and that of Albert Einstein; as a young man Ein-
despite the difficulties and dislocation of his youth,
stein developed on his own ideas in statistical
he had a strong internal compass from an early age.
physics that he later found out had already been
In his mathematics classes, he did not depend on
discovered by Josiah Willard Gibbs.
his teachers to distinguish what was deep from
In 1948, having finished his Licenceès Sciences
what was inconsequential, what was right from
at Montpellier, Grothendieck went to Paris, the
what was wrong. He found the mathematics prob-
main center for mathematics in France. In an arti-
lems in the texts to be repetitive and presented in
cle about Grothendieck that appeared in a French
isolation from anything that would give them mean-
magazine in 1995 [Ikonicoff], a French education
ing. “These were the book’s problems, and not my
official, André Magnier, recalled Grothendieck’s
problems,” he wrote. When a problem did seize him,
application for a scholarship to go to Paris. Mag-
he lost himself in it completely, without regard to
nier asked him to describe the project he had been
how much time he spent on it (page P3).
working on at Montpellier. “I was astounded,” the
article quoted Magnier as saying. “Instead of a
From Montpellier to Paris to Nancy
meeting of twenty minutes, he went on for two
Monsieur Soula [mon professeur de cal- hours explaining to me how he had reconstructed,
cul] m’assurait…que les derniers prob- ‘with the tools available’, theories that had taken
lèmes qui s’étaient encore posés en decades to construct. He showed an extraordinary
OCTOBER2004 NOTICESOFTHEAMS 1041
sagacity.” Magnier also added: “Grothendieck gave time—such as Ehresmann, Leray, Chevalley, Del-
the impression of being an extraordinary young sarte, Dieudonné, and Weil—shared the common
man, but imbalanced by suffering and depriva- background of having been normaliens, meaning
tion.” Magnier immediately recommended Grothen- that they were graduates of the École Normale
dieck for the scholarship. Supérieure, the most prestigious institution of
Grothendieck’s calculus teacher at Montpellier, higher education in France.
Monsieur Soula, recommended he go to Paris and When Grothendieck joined Cartan’s seminar,
make contact with Cartan, who had been Soula’s he was an outsider: not only was he a German
teacher. Whether the name Cartan referred to the speaker living in postwar France, but his meager
father, Élie Cartan, who was then close to eighty educational background contrasted sharply with
years old, or his son, Henri Cartan, then in his mid- that of the group he found himself in. And yet in
forties, Grothendieck did not know (R&S, page 19). Récoltes et Semailles, Grothendieck said he did not
When he arrived in Paris, in the autumn of 1948, feel like a stranger in this milieu and related warm
he showed to mathematicians there the work he had memories of the “benevolent welcome” he received
done in Montpellier. Just as Soula had told him, the (pages 19–20). His outspokenness drew notice: in
results were already known. But Grothendieck was a tribute to Cartan for his 100th birthday, Jean Cerf
not disappointed. In fact, this early solitary effort recalled seeing in the Cartan seminar around this
was probably critical to his development as a math- time “a stranger (it was Grothendieck) who took
ematician. In Récoltes et Semailles,he said of this the liberty of speaking to Cartan, as if to his equal,
time: “Without knowing it, I learned in solitude from the back of the room” [Cerf]. Grothendieck
what is essential to the metier of a mathemati- felt free to ask questions, and yet, he wrote, he also
cian—something that no master can truly teach. found himself struggling to learn things that those
Without having been told, I nevertheless knew ‘in around him seemed to grasp instantly and play
my gut’ that I was a mathematician: someone who with “like they had known them from the cradle.”
‘does’ math, in the fullest sense of the word—like (R&S, page P6). This may have been one reason why,
one ‘makes’ love” (page P5). in October 1949, on the advice of Cartan and Weil,
He began attending the legendary seminar run he left the rarefied atmosphere of Paris for the
by Henri Cartan at the École Normale Supérieure. slower-paced Nancy. Also, as Dieudonné wrote
This seminar followed a pattern that Grothendieck [D1], Grothendieck was at this time showing more
was to take up with great vigor later in his career, interest in topological vector spaces than in alge-
in which a theme is investigated in lectures over braic geometry, so Nancy was the natural place for
the course of the year and the lectures are sys- him to go.
tematically written up and published. The theme
for the Cartan seminar for 1948–1949 was simpli- Apprenticeship in Nancy
cial algebraic topology and sheaf theory—then cut-
ting-edge topics that were not being taught any- …l’affection circulait…depuis ce pre-
where else in France [D1]. Indeed, this was not mier moment où j’ai été reçu avec af-
long after the notion of sheaves had been formu- fection à Nancy, en 1949, dans la mai-
lated by Jean Leray. In the Cartan seminar, Grothen- son de Laurent et Hélène Schwartz (où
dieck encountered for the first time many of the je faisais un peu partie de la famille),
outstanding mathematicians of the day, including celle de Dieudonné, celle de Godement
Claude Chevalley, Jean Delsarte, Jean Dieudonné, (qu’en un temps je hantais également
Roger Godement, Laurent Schwartz, and André régulièrement). Cette chaleur af-
Weil. Among Cartan’s students at this time was fectueuse qui a entouré mes premiers
Jean-Pierre Serre. In addition to attending the Car- pas dans le monde mathématique, et
tan seminar, Grothendieck went to a course on the que j’ai eu tendance un peu à oublier,
then-new notion of locally convex spaces, given by a été importante pour toute ma vie de
Leray at the Collège de France. mathématicien.
As the son of the geometer Élie Cartan, as an out-
standing mathematician in his own right, and as a …the affection circulated…from that
professor at the École Normale Supérieure, Henri first moment when I was received with
Cartan was in many ways the center of the Parisian affection in Nancy in 1949, in the house
mathematical elite. Also, he was one of the few of Laurent and Hélène Schwartz (where
French mathematicians who made efforts to reach I was somewhat a member of the fam-
out to German colleagues after the war. This was ily), in that of Dieudonné, in that of
despite his intimate knowledge of the war’s hor- Godement (which at that time also be-
rors: his brother, who had joined the Résistance, came one of my regular haunts). This
had been captured by the Germans and beheaded. affectionate warmth that surrounded
Cartan and many of the top mathematicians of the my first steps in the mathematical
1042 NOTICESOFTHEAMS VOLUME51, NUMBER9
world, and that I have had some ten- paper chosen for his the-
dency to forget, was important in my en- sis was “Produits ten-
tire life as a mathematician. soriels topologiques et
espaces nucléaires,”
—Récoltes et Semailles, page 42 which shows the first
signs of the generality of
In the late 1940s, Nancy was one of the strongest thinking that would
mathematical centers in France; indeed, the ficti- come to characterize
tious Nicolas Bourbaki was said to have come from Grothendieck’s entire
the “University of Nancago”, a name that makes ref- oeuvre. The notion of nu-
erence to Weil’s time at the University of Chicago clear spaces, which has
as well as to his fellow Bourbakists in Nancy. The had wide applications,
Nancy faculty included Delsarte, Godement, was first proposed in this
Dieudonné, and Schwartz. Among Grothendieck’s paper. Schwartz popu-
fellow students at Nancy were Jacques-Louis Lions larized Grothendieck’s
and Bernard Malgrange, who like Grothendieck results in a Paris semi-
were students of Schwartz, as well as Paulo Riben- nar, “Les produits ten-
boim, a Brazilian who at twenty-two years of age soriels d’après Grothen-
arrived in Nancy about the same time as Grothen- dieck,” published in 1954
dieck. [Schwartz]. In addition,
According to Ribenboim, who is today a pro- Grothendieck’s thesis ap-
fessor emeritus at Queen’s University in Ontario, peared as a monograph
the pace in Nancy was less hectic than in Paris, and in 1955 in the Memoirs of
professors had more time for the students. Riben- the AMS series; it was
boim said he had the impression that Grothen- reprinted for the seventh
dieck had come to Nancy because his lack of back- time in 1990 [Gthesis].
ground had made it hard for him to follow Cartan’s Grothendieck’s work
high-powered seminar. Not that Grothendieck came in functional analysis
out and said this: “He was not the guy who would “was quite remarkable,”
admit he didn’t understand!” Ribenboim remarked. commented Edward G. Ef-
Nevertheless, Grothendieck’s extraordinary talents fros of the University of
were apparent, and Ribenboim remembered look- California at Los Angeles.
ing up to him as an ideal. Grothendieck could be “He was arguably the first
extremely intense, sometimes expressing himself to realize that the alge- Top: Party at Hirzebruch home, 1961
in a brazen way, Ribenboim recalled: “He was not braic/categorical meth- Arbeitstagung (left to right)
mean, but very demanding of himself and every- ods that flourished after Dorothea von Viereck, Raoul Bott,
one else.” Grothendieck had very few books; rather the Second World War Grothendieck.
than learning things by reading, he would try to re- could be used in this Center, with Michael Atiyah.
construct them on his own. And he worked very highly analytic branch of Bottom: Bonn, 1961, excursion
hard. Ribenboim remembered Schwartz telling functional analysis.” In during Arbeitstagung, Ioan James,
him: You seem to be a nice, well-balanced young some ways, Grothendieck Michael Atiyah, Grothendieck.
man; you should make friends with Grothendieck was ahead of his time. Ef-
and do something so that he is not only working. fros noted that it took at least fifteen years before
Dieudonné and Schwartz were running a semi- Grothendieck’s work was fully incorporated into
nar in Nancy on topological vector spaces. As mainstream Banach space theory, partly because of
Dieudonné explained in [D1], by this time Banach a reluctance to adopt his more algebraic perspec-
spaces and their duality were well understood, but tive. The influence of his work has grown in recent
years, Effros said, with the “quantization” of Banach
locally convex spaces had only recently been in-
space theory, for which Grothendieck’s categorical
troduced, and a general theory for their duality had
approach is especially well suited.
not yet been worked out. In working in this area,
he and Schwartz had run into a series of problems, Although Grothendieck’s mathematical work
which they decided to turn over to Grothendieck. had gotten off to a promising start, his personal life
They were astonished when, some months later, he was unsettled. He lived in Nancy with his mother,
had solved every one of them and gone on to work who as Ribenboim recalled was occasionally bedrid-
on other questions in functional analysis. “When, den because of tuberculosis. She had contracted the
in 1953, it was time to grant him a doctor’s degree, disease in the internment camps. It was around this
it was necessary to choose from among six papers time that she was writing her autobiographical
he had written, any one of which was at the level novel Eine Frau. A liaison between Grothendieck and
of a good dissertation,” Dieudonné wrote. The an older woman who ran the boarding house where
OCTOBER2004 NOTICESOFTHEAMS 1043
he and his mother by methods different from those Grothendieck
rented rooms re- was attempting to use. This was “the only time in
sulted in the birth my life when doing mathematics became burden-
of his first child, a some for me!” he wrote. This frustration taught him
son named Serge; a lesson: always have several mathematical “irons
Serge was raised in the fire,” so that if one problem proves too stub-
mostly by his born, there is something else to work on.
mother. After he Chaim Honig, a professor at the Universidade de
finished his Ph.D., São Paulo, was an assistant in the mathematics de-
Grothendieck’s
partment when Grothendieck was there, and they
prospects for per-
became good friends. Honig said Grothendieck led
manent employment were bleak: he
a somewhat spartan and lonely existence, living off
was stateless, and at that time it was
of milk and bananas and completely immersing
difficult for noncitizens to get per-
himself in mathematics. Honig once asked Grothen-
manent jobs in France. Becoming a
dieck why he had gone into mathematics. Grothen-
French citizen would have entailed
dieck replied that he had two special passions,
military service, which Grothendieck
mathematics and piano, but he chose mathemat-
refused to do. Since 1950 he had
ics because he thought it would be easier to earn
had a position through the Centre
a living that way. His gift for mathematics was so
National de la Recherche Scientifique
abundantly clear, said Honig, “I was astonished
(CNRS), but this was more like a fel-
that at any moment he could hesitate between
lowship than a permanent job. At
mathematics and music.”
some point he considered learning
Grothendieck planned to write a book on topo-
carpentry as a way to earn money
(R&S, page 1246(*)). logical vector spaces with Leopoldo Nachbin, who
was in Rio de Janeiro, but the book never materi-
Laurent Schwartz visited Brazil
alized. However, Grothendieck taught a course in
in 1952 and told people there about
São Paulo on topological vector spaces and wrote
Top: Paris, with Karin his brilliant young student who was
up the notes, which were subsequently published
Tate, 1964. having trouble finding a job in
Bottom: with E. Luft, an France. As a result Grothendieck re- by the university. Barros-Neto was a student in the
excursion on the Rhine, ceived an offer of a visiting profes- course and wrote an introductory chapter for the
1961. sor position at the Universidade de notes, giving some basic prerequisites. Barros-Neto
São Paulo, which he held during recalled that at the time he was in Brazil Grothen-
1953 and 1954. According to José Barros-Neto, dieck was talking about changing fields. He was
who was then a student in São Paulo and is now a “very, very ambitious,” Barros-Neto said. “You could
professor emeritus at Rutgers University, Grothen- sense that drive—he had to do something funda-
dieck made a special arrangement so that he would mental, important, basic.”
be able to return to Paris to attend seminars that
took place in the fall. The second language for the A Rising Star
Brazilian mathematical community was French, so
La chose essentielle, c’était que Serre à
it was easy for Grothendieck to teach and converse
chaque fois sentait fortement la riche
with his colleagues. In going to São Paulo, Grothen-
substance derrière un énoncé qui, de but
dieck was carrying on a tradition of scientific ex-
en blanc, ne m’aurait sans doute fait ni
change between Brazil and France: in addition to
chaud ni froid—et qu’il arrivait à “faire
Schwartz, Weil, Dieudonné, and Delsarte had all vis-
ited Brazil in the 1940s and 1950s. Weil came to passer” cette perception d’une sub-
São Paulo in January 1945 and stayed until the fall stance riche, tangible, mystérieuse—
of 1947, when he went to the University of Chicago. cette perception qui est en même temps
The mathematical ties between France and Brazil désirde connaître cette substance, d’y
continue to this day. The Instituto de Matemática pénétrer.
Pura e Aplicada in Rio de Janeiro has a Brazil-
France cooperative agreement that brings many The essential thing was that Serre each
French mathematicians to IMPA. time strongly sensed the rich meaning
In Récoltes et Semailles, Grothendieck referred behind a statement that, on the page,
to 1954 as “the wearisome year” (“l’année pénible”) would doubtless have left me neither
(page 163). For the whole year he tried without hot nor cold—and that he could “trans-
success to make headway on the problem of ap- mit” this perception of a rich, tangible,
proximation in topological vector spaces, a prob- and mysterious substance—this per-
lem that was resolved only some twenty years later ception that is at the same time the
1044 NOTICESOFTHEAMS VOLUME51, NUMBER9
desireto understand this substance, to point, he asks Serre if
penetrate it. the Riemann zeta
function has infinitely
—Récoltes et Semailles, page 556 many zeros ([Corr],
page 204). “His knowl-
Bernard Malgrange of the Université de Greno- edge of classical alge-
ble recalled that after Grothendieck wrote his the- braic geometry was
sis he asserted that he was no longer interested in practically zero,” re-
topological vector spaces. “He told me, ‘There is called Serre. “My own
nothing more to do, the subject is dead’,” Mal- knowledge of classical
grange recalled. At that time, students were re- algebraic geometry
quired to prepare a “second thesis”, which did not was a little bit better,
contain original work but which was intended to but not very much, but
demonstrate depth of understanding of another I tried to help him with
area of mathematics far removed from the thesis that. But…there were
During an Arbeitstagung in 1961, an
topic. Grothendieck’s second thesis was on sheaf so many open ques-
evening at the Hirzebruch home in
theory, and this work may have planted the seeds tions that it didn’t
Bonn.
for his interest in algebraic geometry, where he was matter.” Grothendieck
to do his greatest work. After Grothendieck’s the- was not one for keep-
sis defense, which took place in Paris, Malgrange ing up on the latest literature, and to a large de-
recalled that he, Grothendieck, and Henri Cartan gree he depended on Serre to tell him what was
piled into a taxicab to go to lunch at the home of going on. In Récoltes et Semailles Grothendieck
Laurent Schwartz. They took a cab because Mal- wrote that most of what he learned in geometry,
grange had broken his leg skiing. “In the taxi Car- apart from what he taught himself, he learned
tan explained to Grothendieck some wrong things from Serre (pages 555–556). But Serre did not sim-
Grothendieck had said about sheaf theory,” Mal- ply teach Grothendieck things; he was able to di-
grange recalled. gest ideas and to discuss them in a way that
After leaving Brazil Grothendieck spent the year Grothendieck found especially compelling. Grothen-
of 1955 at the University of Kansas, probably at the dieck called Serre a “detonator,” one who provided
invitation of N. Aronszajn [Corr]. There Grothen- a spark that set the fuse burning for an explosion
dieck began to immerse himself in homological al- of ideas.
gebra. It was while he was at Kansas that he wrote Indeed, Grothendieck traced many of the cen-
“Sur quelques points d’algèbre homologique,” tral themes of his work back to Serre. For exam-
which came to be known informally among spe- ple, it was Serre who around 1955 described the
cialists as the “Tôhoku paper” after the name of Weil conjectures to Grothendieck in a cohomolog-
the journal in which it appeared, the Tôhoku Math- ical context—a context that was not made explicit
ematical Journal[To]. This paper, which became a in Weil’s original formulation of the conjectures and
classic in homological algebra, extended the work was the one that could hook Grothendieck (R&S,
of Cartan and Eilenberg on modules. Also while he page 840). Through his idea of a “Kählerian” ana-
logue of the Weil conjectures, Serre also inspired
was in Kansas, Grothendieck wrote “A general the-
Grothendieck’s conception of the so-called “stan-
ory of fiber spaces with structure sheaf,” which ap-
dard conjectures,” which are more general and
peared as a report of the National Science Foun-
would imply the Weil conjectures as a corollary
dation. This report developed his initial ideas on
(R&S, page 210).
nonabelian cohomology, a subject to which he later
returned in the context of algebraic geometry. When Grothendieck returned to France in 1956
Around this time, Grothendieck began corre- after his year in Kansas, he held a CNRS position
sponding with Jean-Pierre Serre of the Collège de and spent most of his time in Paris. He and Serre
France, whom he had met in Paris and later en- continued to correspond by letter and to talk reg-
countered in Nancy; a selection of their letters was ularly by telephone. This was when Grothendieck
published in the original French in 2001 and in a began to work more deeply in topology and alge-
dual French-English version in 2003 [Corr]. This was braic geometry. He “was bubbling with ideas,” re-
the beginning of a long and fruitful interaction. The called Armand Borel. “I was sure something first-
letters display a deep and vibrant mathematical rate would come out of him. But then what came
bond between two very different mathematicians. out was even much higher than I had expected. It
Grothendieck shows a high-flying imagination that was his version of Riemann-Roch, and that’s a fan-
is frequently brought back to earth by Serre’s in- tastic theorem. This is really a masterpiece of math-
cisive understanding and wider knowledge. Some- ematics.”
times in the letters Grothendieck displays a sur- The Riemann-Roch theorem was proved in its
prising level of ignorance: for example, at one classical form in the mid-nineteenth century. The
OCTOBER2004 NOTICESOFTHEAMS 1045
question it addresses is, What is the dimension of new kind of topological invariant. Grothendieck
the space of meromorphic functions on a compact himself called them K-groups, and they provided
Riemann surface having poles of at most given or- the starting point for the development of topolog-
ders at a specified finite set of points? The answer ical K-theory by Atiyah and Hirzebruch. Topologi-
is the Riemann-Roch formula, which expresses the cal K-theory then provided the inspiration for al-
dimension in terms of invariants of the surface— gebraic K-theory, and both have been active fields
thereby providing a profound link between the an- of research ever since.
alytic and topological properties of the surface. The Arbeitstagung, which means literally “work-
Friedrich Hirzebruch made a big advance in 1953, ing meeting,” was begun by Hirzebruch at the Uni-
when he generalized the Riemann-Roch theorem to versität Bonn and has been a forum for cutting-edge
apply not just to Riemann surfaces but to projec- mathematics research for more than forty years. It
tive nonsingular varieties over the complex num- was at the very first Arbeitstagungin July 1957 that
bers. The mathematical Grothendieck spoke
world cheered at this about his work on Rie-
tour de force, which mann-Roch. But in a cu-
might have seemed to be rious twist, the result
the final word on the was not published under
matter. his name; it appears in a
“Grothendieck came paper by Borel and Serre
along and said, ‘No, the [BS] (the proof also ap-
Riemann-Roch theorem peared later as an exposé
is not a theorem about in volume 6 of Séminaire
varieties, it’s a theorem de Géometrie Algébrique
about morphisms be- du Bois Mariefrom 1966-
tween varieties’,” said 67). While visiting the IAS
Nicholas Katz of Prince- in the fall of 1957, Serre
ton University. “This was received a letter from
a fundamentally new Grothendieck containing
point of view…the very an outline of the proof
statement of the theo- (November 1, 1957, letter
rem completely in [Corr]). He and Borel
changed.” The basic phi- organized a seminar to
losophy of category the- try to understand it. As
Bonn, around 1965.
ory, that one should pay Grothendieck was busy
more attention to the ar- with many other things,
rows between objects than to the objects them- he suggested to his colleagues that they write up
selves, was just then beginning to have an influ- and publish their seminar notes. But Borel specu-
ence. “What [Grothendieck] did is he applied this lated that there may have been another reason
philosophy on a very hard piece of mathematics,” Grothendieck was not interested in writing up the
Borel said. “This was really in the spirit of categories result himself. “The main philosophy of Grothen-
and functors, but no one had ever thought about dieck was that mathematics should be reduced to
doing this in such a hard topic…. If people had been a succession of small, natural steps,” Borel said. “As
given that statement and had understood it, there long as you have not been able to do this, you have
might have been others who would have been able not understood what is going on…. And his proof
to prove it. But the statement itself was ten years of Riemann-Roch used a trick, une astuce. So he
ahead of anybody else.” didn’t like it, so he didn’t want to publish it…. He
had many other things to do, and he was not in-
This theorem, which was also proved indepen-
terested in writing up this trick.”
dently by Gerard Washnitzer in 1959 [Washnitzer],
applies not just to a complex algebraic variety—the This was not the last time Grothendieck would rev-
case where the ground field has characteristic olutionize the viewpoint on a subject. “This just
zero—but to any proper smooth algebraic variety kept happening over and over again, where he would
regardless of the ground field. The Hirzebruch- come upon some problem that people had thought
Riemann-Roch theorem then follows as a special about for, in some cases, a hundred years…and just
case. A far-reaching generalization of the Riemann- completely transformed what people thought the
Roch theorem came in 1963, with the proof by subject was about,” Katz remarked. Grothendieck
Michael Atiyah and Isadore Singer of the Atiyah- was not only solving outstanding problems but re-
Singer Index Theorem. In the course of his proof, working the very questions they posed.
Grothendieck introduced what are now called
Grothendieck groups, which essentially provide a
1046 NOTICESOFTHEAMS VOLUME51, NUMBER9
A New World Opens marry a few years
later and with whom
[J’ai fini] par me rendre compte que he had three children,
cette idéologie du “nous, les grands et Johanna, Mathieu,
nobles esprits…”, sous une forme par- and Alexandre.
ticulièrement extrême et virulente, avait Mireille had been
sévi en ma mère depuis son enfance, et close to Grothen-
dominé sa relation aux autres, qu’elle se dieck’s mother and,
plaisait à regarder du haut de sa according to several
grandeur avec une commisération sou- people who knew
vent dédaigneuse, voire méprisante. them, was quite a bit
older than he was.
[I eventually] realized that this ideology John Tate of the
of “we, the grand and noble spirits…”, University of Texas at With Mireille and baby Mathieu, Paris,
in a particularly extreme and virulent Austin and his wife at May 1965.
form, raged in my mother since her the time, Karin Tate,
childhood and dominated her relations spent the academic year 1957–58 in Paris, where
to others, whom she liked to view from they met Grothendieck for the first time. Grothen-
the height of her grandeur with a pity dieck displayed none of the arrogance he attributed
that was frequently disdainful, even to his mother. “He was just friendly, and at the same
contemptuous. time rather naive and childlike,” John Tate recalled.
“Many mathematicians are rather childlike, un-
—Récoltes et Semailles, page 30 worldly in some sense, but Grothendieck more
than most. He just seemed like an innocent—not
According to Honig, Grothendieck’s mother was
very sophisticated, no pretense, no sham. He
with him at least part of the time that he was in
thought very clearly and explained things very pa-
Brazil, though Honig says he never met her.
tiently, without any sense of superiority. He wasn’t
Whether she was with him in Kansas is not clear.
contaminated by civilization or power or one-up-
When Grothendieck returned to France in 1956,
manship.” Karin Tate recalled that Grothendieck
they may not have continued living together. In a
had a great capacity for enjoyment, he was charm-
letter to Serre written in Paris in November 1957,
ing, and he loved to laugh. But he could also be ex-
Grothendieck asked whether he might be able to
tremely intense, seeing things in black-and-white
rent a Paris apartment that Serre was planning to
with no shades of gray. And he was honest: “You
vacate. “I am interested in it for my mother, who
always knew where you stood with him,” she said.
is not doing so well in Bois-Colombes, and is ter-
“He didn’t pretend anything. He was direct.” Both
ribly isolated,” Grothendieck explained [Corr]. In
she and her brother, Michael Artin of the Massa-
fact, his mother died before the year’s end.
chusetts Institute of Technology, saw similarities
Friends and colleagues say that Grothendieck
between Grothendieck’s personality and that of
spoke with great admiration, almost adulation, of
their father, Emil Artin.
both of his parents. And in Récoltes et Semailles
Grothendieck had “an incredible idealistic
Grothendieck expressed a deep and elemental love
streak,” Karin Tate remembered. For example, he
for them. For years he had in his office a striking
refused to have any rugs in his house because he
portrait of his father, painted by a fellow detainee
believed that rugs were merely a decorative luxury.
in the Le Vernet camp. As Pierre Cartier described
She also remembered him wearing sandals made
it, the portrait showed a man with his head shaved
out of tires. “He thought these were fantastic,” she
and a “fiery expression” in the eyes [Cartier1]; for
said. “They were a symbol of the kind of thing he
many years Grothendieck also shaved his head.
respected—you take what you have, and you make
According to Ribenboim, Hanka Grothendieck was
do.” In his idealism, he could also be wildly im-
very proud of her brilliant son, and he in turn had
practical. Before Grothendieck and Mireille visited
an extremely deep attachment to his mother.
Harvard for the first time in 1958, he gave her one
After her death, Grothendieck went through a
of his favorite novels so that she could improve her
period of soul-searching, during which he stopped
rather weak knowledge of English. The novel was
all mathematical activity and thought about be-
Moby Dick.
coming a writer. After several months, he decided
to return to mathematics, to finish work on some The Birth of the New Geometry
of the ideas he had begun developing. This was
1958, the year that, as Grothendieck put it, was Avec un recul de près de trente ans, je
“probably the most fecund of all my mathemati- peux dire maintenant que c’est l’année
cal life.” (R&S, page P24) By this time he was living [1958] vraiment où est née la vision de
with a woman named Mireille, whom he was to la géometrie nouvelle, dans le sillage
OCTOBER2004 NOTICESOFTHEAMS 1047