The Life of Alexandre Grothendieck, Volume 51, Number 9

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- Alexandre Grothendieck is a mathematician of ematical objects as a way of understanding the ob- immense sensitivity to things mathematical, of jects themselves. He had an extremely powerful, profound perception of the intricate and elegant almost other-worldly ability of abstraction that al- lines of their architecture. A couple of high points lowed him to see problems in a highly general con- from his biography—he was a founding member of text, and he used this ability with exquisite preci- 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 NOTICES OF THE AMS VOLUME 51, NUMBER 4 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- litical movements. Borel recalled, a young man strode up to Ehresmann Grothendieck’s father, This other Shapiro, and, without any preamble, demanded, “Are you Sascha, ca. 1922. an expert on topological groups?” Ehresmann, not who was portrayed in wanting to seem immodest, replied that yes, he John Reed’s book Ten knew something about topological groups. The Days that Shook the World, emigrated to New York young man insisted, “But I need a real expert!” and died there in 1946, by which time Grothen- This was Alexandre Grothendieck, age twenty- dieck’s father had already been dead for four years. one—brash, intense, not exactly impolite but hav- Another distinguishing detail is that Grothendieck’s ing little sense of social niceties. Borel remem- father had only one arm. According to Justine bered the question Grothendieck asked: Is every Bumby, who lived with Grothendieck for a period local topological group the germ of a global topo- in the 1970s and had a son by him, his father lost logical group? As it turned out, Borel knew a coun- his arm in a suicide attempt while trying to avoid terexample. It was a question that showed Grothen- being captured by the police. Grothendieck himself dieck was already thinking in very general terms. may unwittingly have contributed to the confu- Grothendieck’s time in Paris in the late 1940s sion between the two Shapiros; for example, Pierre was his first real contact with the world of math- Cartier of the Institut des Hautes Études Scien- ematical research. Up to that time, his life story— tifiques mentioned in [Cartier2] Grothendieck’s

OCTOBER 2004 NOTICES OF THE AMS 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 NOTICES OF THE AMS VOLUME 51, NUMBER 9 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 good boxer, attributes that were useful at this time, By the time the war ended in Europe, in May as he was sometimes the target of bullying. 1945, Alexandre Grothendieck was seventeen years After two years, mother and son were sepa- old. He and his mother went to live in Maisargues, rated; Hanka was sent to another internment camp, a village in a wine-growing region outside of Mont- 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 Semailles Grothendieck 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 From Montpellier to Paris to Nancy article quoted Magnier as saying. “Instead of a 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

OCTOBER 2004 NOTICES OF THE AMS 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 NOTICES OF THE AMS VOLUME 51, NUMBER 9 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 locally convex spaces had only recently been in- years, Effros said, with the “quantization” of Banach troduced, and a general theory for their duality had space theory, for which Grothendieck’s categorical not yet been worked out. In working in this area, approach is especially well suited. 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

OCTOBER 2004 NOTICES OF THE AMS 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 (CNRS), but this was more like a fel- abundantly clear, said Honig, “I was astonished lowship than a permanent job. At that at any moment he could hesitate between some point he considered learning mathematics and music.” carpentry as a way to earn money Grothendieck planned to write a book on topo- (R&S, page 1246(*)). logical vector spaces with Leopoldo Nachbin, who Laurent Schwartz visited Brazil was in Rio de Janeiro, but the book never materi- in 1952 and told people there about alized. However, Grothendieck taught a course in Top: Paris, with Karin his brilliant young student who was São Paulo on topological vector spaces and wrote Tate, 1964. having trouble finding a job in up the notes, which were subsequently published 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 with his colleagues. In going to São Paulo, Grothen- chaque fois sentait fortement la riche dieck was carrying on a tradition of scientific ex- substance derrière un énoncé qui, de but change between Brazil and France: in addition to en blanc, ne m’aurait sans doute fait ni Schwartz, Weil, Dieudonné, and Delsarte had all vis- chaud ni froid—et qu’il arrivait à “faire 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 désir de 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 NOTICES OF THE AMS VOLUME 51, NUMBER 9 desire to 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- was in Kansas, Grothendieck wrote “A general the- logue of the Weil conjectures, Serre also inspired ory of fiber spaces with structure sheaf,” which ap- Grothendieck’s conception of the so-called “stan- peared as a report of the National Science Foun- dard conjectures,” which are more general and dation. This report developed his initial ideas on would imply the Weil conjectures as a corollary nonabelian cohomology, a subject to which he later (R&S, page 210). 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

OCTOBER 2004 NOTICES OF THE AMS 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 Arbeitstagung in 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 Marie from 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 This theorem, which was also proved indepen- had many other things to do, and he was not in- dently by Gerard Washnitzer in 1959 [Washnitzer], terested in writing up this trick.” 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 NOTICES OF THE AMS VOLUME 51, NUMBER 9 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

OCTOBER 2004 NOTICES OF THE AMS 1047 des deux maître-outils de cette géome- corresponding numbers for each finite extension trie: les schémas (qui représentent une field. These numbers are then incorporated into a métamorphose de l’ancienne notion de generating function, which is the zeta function of “variété algébrique”), et les topos (qui V. Weil proved for both curves and abelian varieties représentent une métamorphose, plus three facts about this zeta function: it is rational, profonde encore, de la notion d’espace). it satisfies a functional equation, and its zeros and poles have a certain specific form. This form, once With hindsight of thirty years, I can a change of variables is made, corresponds exactly now say that [1958] is the year where to the Riemann hypothesis. Moreover, Weil ob- the vision of the new geometry was re- served that, if V arose from reduction modulo p ally born, in the wake of two master- of a variety W in characteristic zero, then the Betti tools of this geometry: schemes (which numbers of W can be read off the zeta function of represent a metamorphosis of the old V, when the zeta function is expressed as a ratio- notion of “algebraic variety”), and nal function. The Weil conjectures ask whether toposes (which represent a metamor- these same facts hold true if one defines such a phosis, yet more profound, of the no- zeta function for a projective nonsingular alge- tion of space). braic variety. In particular, would topological data such as the Betti numbers emerge in the zeta func- —Récoltes et Semailles, page P23 tion? This conjectured link between algebraic geom- etry and topology hinted that some of the new In August 1958, Grothendieck gave a plenary lec- tools, such as cohomology theory, that were then ture at the International Congress of Mathemati- being developed for topological spaces, could be cians in Edinburgh [Edin]. The talk outlined, with adapted for use with algebraic varieties. Because a remarkable prescience, many of the main themes of its similarity to the classical Riemann hypothe- that he was to work on for the next dozen years. sis, the third of the Weil conjectures is sometimes It was clear by this time that he was aiming to prove called the “congruence Riemann hypothesis”; this the famous conjectures of André Weil, which hinted one turned out to be the most difficult of the three at a profound unity between the discrete world of to prove. algebraic varieties and the continuous world of “As soon as [the Weil] conjectures were made, topology. it was clear that they were somehow going to play At this time, algebraic geometry was evolving a central role,” Katz said, “both because they were rapidly, with many open questions that did not re- fabulous just as ‘black-box’ statements, but also be- quire a great deal of background. Originally the cause it seemed obvious that solving them required main objects of study were varieties over the com- developing incredible new tools that would some- plex numbers. During the early part of the twen- how have to be incredibly valuable in their own tieth century, this area was a specialty of Italian way—which turned out to be completely correct.” mathematicians, such as Guido Castelnuovo, Fed- Pierre Deligne of the Institute for Advanced Study erigo Enriques, and Francesco Severi. Although said that it was the conjectured link between al- they developed many ingenious ideas, not all of gebraic geometry and topology that attracted their results were proved rigorously. In the 1930s Grothendieck. He liked the idea of “turning this and 1940s, other mathematicians, among them dream of Weil into powerful machinery,” Deligne B. L. van der Waerden, André Weil, and Oscar remarked. Zariski, wanted to work with varieties over arbitrary Grothendieck was not interested in the Weil fields, particularly varieties over fields of charac- conjectures because they were famous or because teristic p, which are important in number theory. other people considered them to be difficult. In- But, because of the lack of rigor of the Italian deed, he was not motivated by the challenge of hard school of algebraic geometry, it was necessary to problems. What interested him were problems that build new foundations for the field. This is what seemed to point to larger, hidden structures. “He Weil did in his 1946 book Foundations of Alge- would aim at finding and creating the home which braic Geometry [Weil1]. was the problem’s natural habitat,” Deligne noted. Weil’s conjectures appeared in his 1949 paper “That was the part that interested him, more than [Weil2]. Motivated by problems in number theory, solving the problem.” This approach contrasts with Weil studied a certain zeta function that had been that of another great mathematician of the time, introduced in special cases by Emil Artin; it is John Nash. In his mathematical prime, Nash called a zeta function because it was defined in searched out specific problems considered by his analogy to the Riemann zeta function. Given an al- colleagues to be the most important and chal- gebraic variety V defined over a finite field of char- lenging [Nasar]. “Nash was like an Olympian ath- acteristic p, one can count the number of points lete,” remarked Hyman Bass of the University of of V that are rational over this field, as well as the Michigan. “He was interested in enormous

1048 NOTICES OF THE AMS VOLUME 51, NUMBER 9 personal challenges.” If Nash is an ideal example some way of formulating a problem, stripping ap- of a problem-solver, then Grothendieck is an ideal parently everything away from it, so you don’t example of a theory-builder. Grothendieck, said think anything is left. And yet something is left, and Bass, “had a sweeping vision of what mathematics he could find real structure in this seeming vac- could be.” uum.” In the fall of 1958, Grothendieck made the first of his several visits to the mathematics depart- The Heroic Years ment at Harvard University. Tate was a professor there, and the chairman was Oscar Zariski. By this Pendant les années héroiques de l’IHÉS, time Grothendieck had reproved, by recently de- Dieudonné et moi en avons été les seuls veloped cohomological methods, the connectedness membres, et les seuls aussi à lui don- theorem that was one of Zariski’s biggest results, ner crédibilité et audience dans le proved in the 1940s. According to David Mumford monde scientifique. …Je me sentais un of Brown University, who was Zariski’s student at peu comme un cofondateur “scien- the time, Zariski never took up the new methods tifique”, avec Dieudonné, de mon insti- himself, but he understood their power and wanted tution d’attache, et je comptais bien y his students to be exposed to them, and this was finir mes jours! J’avais fini par m’iden- why he invited Grothendieck to Harvard. tifier fortement à l’IHÉS…. Zariski and Grothendieck got along well, Mum- ford noted, though as mathematicians they were During the heroic years of the IHÉS, very different. It was said that Zariski, when he got Dieudonné and I were the only mem- stuck, would go to the blackboard and draw a pic- bers, and the only ones also giving it ture of a self-intersecting curve, which would allow credibility and an audience in the sci- him to refresh his understanding of various ideas. entific world. …I felt myself a bit like a “The rumor was that he would draw this in the cor- “scientific” co-founder, with Dieudonné, ner of the blackboard, and then he would erase it of the institution where I was on the fac- and then he would do his algebra,” explained Mum- ulty, and I counted on ending my days ford. “He had to clear his mind by creating a geo- there! I ended up strongly identifying metric picture and reconstructing the link from the with the IHÉS…. geometry to the algebra.” According to Mumford, this is something Grothendieck would never do; he —Récoltes et Semailles, page 169 seemed never to work from examples, except for ones that were extremely simple, almost trivial. In June 1958, the Institut des Hautes Études He also rarely drew pictures, apart from homo- Scientifiques (IHÉS) was formally established in a logical diagrams. meeting of its sponsors at the Sorbonne in Paris. When Grothendieck was first invited to Har- The founder, Léon Motchane, a businessman with vard, Mumford recalled, he had some correspon- a doctoral degree in physics, had a vision of es- dence with Zariski before the visit. This was not long tablishing in France an independent research in- after the era of the House Un-American Activities stitution akin to the Institute for Advanced Study Committee, and one requirement for getting a visa in Princeton. The original plan for the IHÉS was to was swearing that one would not work to overthrow focus on fundamental research in three areas: the government of the United States. Grothendieck mathematics, theoretical physics, and the method- told Zariski he would refuse to take such a pledge. ology of human sciences. While the third area never When told he might end up in jail, Grothendieck gained a foothold, within a decade the IHÉS had es- said jail would be acceptable as long as students tablished itself as one of the world’s top centers could visit and he could have as many books as he for mathematics and theoretical physics, with a wanted. small but stellar faculty and an active visitor pro- In Grothendieck’s lectures at Harvard, Mumford gram. found the leaps into abstraction to be breathtak- According to the doctoral thesis of historian of ing. Once he asked Grothendieck how to prove a science David Aubin [Aubin], it was at the Edinburgh certain lemma and got in reply a highly abstract ar- Congress in 1958, or possibly before, that Motchane gument. Mumford did not at first believe that such persuaded Dieudonné and Grothendieck to accept an abstract argument could prove so concrete a professorships at the newly established IHÉS. lemma. “Then I went away and thought about it for Cartier wrote in [Cartier2] that Motchane originally a couple of days, and I realized it was exactly right,” wanted to hire Dieudonné, who made it a condi- Mumford recalled. “He had more than anybody tion of his taking the position that an offer also be else I’ve ever met this ability to make an absolutely made to Grothendieck. Because the IHÉS has been startling leap into something an order of magni- from the start independent of the state, there was tude more abstract…. He would always look for no problem in hiring Grothendieck despite his

OCTOBER 2004 NOTICES OF THE AMS 1049 being stateless. The two professors formally took dents to read EGA. And for many mathematicians up their positions in March 1959, and Grothendieck today, EGA remains a useful and comprehensive ref- started his seminar on algebraic geometry in May erence. The current IHÉS director, Jean-Pierre Bour- of that year. René Thom, who had received a Fields guignon, says that the institute still sells over 100 Medal at the 1958 Congress, joined the faculty in copies of EGA every year. October 1963, and the IHÉS section on theoretical Grothendieck’s plans for what EGA would cover physics was launched with the appointments of were enormous. In a letter to Serre from August Louis Michel in 1962 and of David Ruelle in 1964. 1959, he gave a brief outline, which included the Thus by the mid-1960s, Motchane had assembled fundamental group, category theory, residues, du- an outstanding group of researchers for his new ality, intersections, Weil cohomology, and “God institute. willing, a little homotopy.” Up to 1962, the IHÉS had “Unless there are unexpected no permanent quarters. Of- difficulties or I get bogged fice space was rented from down, the multiplodocus the Fondation Thiers, and should be ready in 3 years’ seminars were given there or time, or 4 at the outside,” at universities in Paris. Aubin Grothendieck optimistically reported that an early visitor wrote, using his and Serre’s to the IHÉS, Arthur Wight- joking term “multiplodocus,” man, was expected to work meaning a very long paper. from his hotel room. It is said “We will be able to start that, when a visitor noted the doing algebraic geometry!” inadequate library, Grothen- he crowed. As it turned out, dieck replied, “We don’t read EGA ran out of steam after books, we write them!” In- almost exponential growth: deed, in the early years, chapters one and two are much of the institute’s ac- each one volume, chapter tivity centered on the “Pub- three is two volumes, and lications mathématiques de the last, chapter four, runs l’IHÉS,” which began with the four volumes. Altogether, initial volumes of the foun- they comprise 1,800 pages. Around 1965. dational work Éléments de Despite its falling short of Géométrie Algébrique, uni- Grothendieck’s plans, EGA is versally known by its acronym EGA. In fact, the writ- a monumental work. ing of EGA had begun half a year before Dieudonné It is no coincidence that the title of EGA echoes and Grothendieck formally took their positions at the title of the series by Nicolas Bourbaki, Éléments the IHÉS; a reference in [Corr] dates the beginning de Mathématique, which in turn echoes Euclid’s El- of the writing to the autumn of 1958. ements: Grothendieck was a member of Bourbaki The authorship of EGA is attributed to Grothen- for several years, starting in the late 1950s and was dieck, “with the collaboration of Jean Dieudonné.” close to many of the other members. Bourbaki was Grothendieck wrote notes and drafts, which were the pseudonym for a group of mathematicians, fleshed out and polished by Dieudonné. As Ar- most of them French, who collaborated on writing mand Borel explained it, Grothendieck was the a series of foundational treatises on mathematics. one who had the global vision for EGA, whereas Dieudonné was a founder of the Bourbaki group, Dieudonné had only a line-by-line understanding. together with Henri Cartan, Claude Chevalley, Jean “Dieudonné put this in a rather heavy style,” Borel Delsarte, and André Weil. Usually there were about remarked. At the same time, “Dieudonné was of ten members, and the group’s composition evolved course fantastically efficient. No one else could over the years. The first Bourbaki book appeared have done it without ruining his own work.” For in 1939, and the group’s influence was at its height some wanting to enter the field at that time, learn- during the 1950s and 1960s. The purpose of the ing from EGA could be a daunting challenge. Nowa- books was to provide axiomatic treatments of cen- days it is seldom used as an introduction to the tral areas of mathematics at a level of generality field, as there are many other, more approachable that would make the books useful to the largest texts to choose from. But those texts do not do what number of mathematicians. The books were born EGA aims to do, which is to explain fully and sys- in a crucible of animated and sometimes heated dis- tematically some of the tools needed to investigate cussions among the group’s members, many of schemes. When he was at Princeton University, whom had strong personalities and highly indi- Gerd Faltings, now at the Max-Planck-Institut für vidual points of view. Borel, who was a member of Mathematik in Bonn, encouraged his doctoral stu- Bourbaki for 25 years, wrote that this collaboration

1050 NOTICES OF THE AMS VOLUME 51, NUMBER 9 may have been “a unique occurrence in the history remaining volumes were published by Springer- of mathematics” [Borel]. Bourbaki pooled the efforts Verlag. SGA 1 dates from the seminars of of some of the top mathematicians of the day, who 1960–1961, and the last in the series, SGA 7, dates unselfishly and anonymously devoted a good deal from 1967–1969. In contrast to EGA, which is in- of time and energy to writing texts that would tended to set foundations, SGA describes ongoing make a wide swath of the field accessible. The research as it unfolded in Grothendieck’s seminar. texts had a large impact, and by the 1970s and He presented many of his results in the Bourbaki 1980s, there were grumblings that Bourbaki had too Seminar in Paris, and they were collected in FGA, much influence. Also, some criticized the style of Fondements de la Géométrie Algébrique, which ap- the books as being excessively abstract and gen- peared in 1962. Together, EGA, SGA, and FGA total eral. around 7,500 pages. The work of Bourbaki and that of Grothendieck bear some similarities in the level of generality The Magic Fan and abstraction and also in the aim of being foun- [S]’il y a une chose en mathématique dational, thorough, and systematic. The main dif- qui (depuis toujours sans doute) me ference is that Bourbaki covered a range of math- fascine plus que toute autre, ce n’est ni ematical areas, while Grothendieck focused on “le nombre”, ni “la grandeur”, mais tou- developing new ideas in algebraic geometry, with jours la forme. Et parmi les mille-et-un the Weil conjectures as a primary goal. In addition, visages que choisit la forme pour se Grothendieck’s work was very much centered on révéler à nous, celui qui m’a fasciné his own internal vision, whereas Bourbaki was a col- plus que tout autre et continue à me laborative effort that forged a synthesis of the fasciner, c’est la structure cachée dans viewpoints of its members. les choses mathématiques. Borel described in [Borel] the March 1957 meet- ing of Bourbaki, dubbed the “Congress of the in- [I]f there is one thing in mathematics flexible functor” because of Grothendieck’s pro- that fascinates me more than anything posal that a Bourbaki draft on sheaf theory be else (and doubtless always has), it is redone from a more categorical viewpoint. Bour- neither “number” nor “size”, but always baki abandoned this idea, believing it could lead form. And among the thousand-and- to an endless cycle of foundation-building. Grothen- one faces whereby form chooses to re- dieck “could not really collaborate with Bourbaki veal itself to us, the one that fascinates because he had his big machine, and Bourbaki was me more than any other and continues not general enough for him,” Serre recalled. In ad- to fascinate me, is the structure hidden dition, Serre remarked, “I don’t think he liked very in mathematical things. much the system of Bourbaki, where we would re- ally discuss drafts in detail and criticize them. —Récoltes et Semailles, page P27 …That was not his way of doing mathematics. He wanted to do it himself.” Grothendieck left Bour- In the first volume of Récoltes et Semailles, baki in 1960, though he remained close to many Grothendieck presents an expository overview of of its members. his work intended to be accessible to nonmathe- Stories have circulated that Grothendieck left maticians (pages P25–48). There he writes that, at Bourbaki because of clashes with Weil, but in fact its most fundamental level, this work seeks a uni- the two had only a brief overlap: following the fication of two worlds: “the arithmetic world, in edict that members must retire at age 50, Weil left which live the (so-called) ‘spaces’ having no notion the group in 1956. Nevertheless, it is true that of continuity, and the world of continuous size, in Grothendieck and Weil were very different as math- which live the ‘spaces’ in the proper sense of the ematicians. As Deligne put it, “Weil felt somewhat term, accessible to the methods of the analyst”. The that Grothendieck was too ignorant of what the Ital- reason the Weil conjectures were so tantalizing is ian geometers had done and what all the classical exactly that they provided clues about this unity. literature was, and Weil did not like the style of Rather than trying to solve the Weil conjectures di- building a big machine. …Their styles were quite rectly, Grothendieck greatly generalized their en- different.” tire context. Doing so allowed him to perceive the Apart from EGA, the other major part of larger structures in which the conjectures lived Grothendieck’s oeuvre in algebraic geometry is and of which they provided only a fleeting glimpse. Séminaire de Géométrie Algébrique du Bois Marie, In this section of Récoltes et Semailles, Grothendieck known as SGA, which contains written versions of explained some of the key ideas in his work, in- lectures presented in his IHÉS seminar. They were cluding scheme, sheaf, and topos. originally distributed by the IHÉS. SGA 2 was co- Basically, a scheme is a generalization of the no- published by North-Holland and Masson, while the tion of an algebraic variety. Given the array of

OCTOBER 2004 NOTICES OF THE AMS 1051 finite fields of prime characteristic, a scheme pro- categorical setting, where the category of sheaves duces in turn an array of varieties, each with its dis- lives. A topos, then, can be described as a category tinct geometry. “The array of these different vari- that, without necessarily arising from an ordinary eties of different characteristics can be visualized space, nevertheless has all of the “nice” properties as a sort of ‘infinite fan of varieties’ (one for each of a category of sheaves. The notion of topos, characteristic),” Grothendieck wrote. “The ‘scheme’ Grothendieck wrote, highlights the fact that “what is this magic fan, which links, like so many differ- really counts in a topological space is not at all its ent ‘branches’, the ‘avatars’ or ‘incarnations’ of all ‘points’ or its subsets of points and their proxim- the possible characteristics.” The generalization ity relations and so forth, but rather the sheaves to a scheme allows one to study in a unified way on the space and the category that they form.” all the different “incarnations” of a variety. Before To come up with the idea of topos, Grothendieck Grothendieck, “I don’t think people really believed “thought very deeply about the notion of space,” you could do that,” commented Michael Artin. “It Deligne commented. “The theory he created to un- was too radical. No one had had the courage to even derstand those conjectures of Weil was first to think this may be the way to work, to work in com- create the concept of topos, a generalization of the plete generality. That was very remarkable.” notion of space, then to define a topos adapted to Starting with the insight of the nineteenth- the problem,” he explained. Grothendieck also century Italian mathematician Enrico Betti, ho- showed that “one can really work with it, that the mology and its dual cohomology were developed intuition we have about ordinary space works [on as tools for studying topological spaces. Basically, a topos] also. …This was a very deep idea.” cohomology theories provide invariants, which can In Récoltes et Semailles Grothendieck commented be thought of as “yardsticks” for measuring this that from a technical point of view much of his or that aspect of a space. The great hope, sparked work in mathematics consisted in developing the by the insight implicit in the Weil conjectures, was cohomology theories that were lacking. Étale co- that cohomological methods for topological spaces homology was one such theory, developed by could be adapted for use with varieties and Grothendieck, Michael Artin, and others, specifi- schemes. This hope was realized to a great extent cally to apply to the Weil conjectures, and indeed in the work of Grothendieck and his collaborators. it was one of the key ingredients in their proof. But “It was like night and day to [bring] these coho- Grothendieck went yet further, developing the con- mological techniques” into algebraic geometry, cept of a motive, which he described as the “ulti- Mumford noted. “It completely turned the field mate cohomological invariant” of which all others upside down. It’s like analysis before and after are different realizations or incarnations. A full the- Fourier analysis. Once you get Fourier techniques, ory of motives has remained out of grasp, but the suddenly you have this whole deep insight into a concept has generated a good deal of mathemat- way of looking at a function. It was similar with co- ics. For example, in the 1970s Deligne and Robert homology.” Langlands of the IAS conjectured precise rela- The notion of a sheaf was conceived by Jean tionships between motives and automorphic rep- Leray and further developed by Henri Cartan and resentations. This conjecture, now part of the so- Jean-Pierre Serre. In his groundbreaking paper called Langlands Program, first appeared in print known as FAC (“Faisceaux algébriques cohérents”, in [Langlands]. James Arthur of the University of [FAC]), Serre showed how sheaves could be used Toronto said that proving this conjecture in full in algebraic geometry. Without saying exactly what generality is decades away. But, he pointed out, a sheaf is, Grothendieck described in Récoltes et Se- what Andrew Wiles did in the proof of Fermat’s Last mailles how this notion changed the landscape: Theorem was essentially to prove this conjecture When the idea of a sheaf came along, it was as if in the case of two-dimensional motives that come the good old standard cohomology “yardstick” from elliptic curves. Another example is the work suddenly multiplied into an infinite array of new of Vladimir Voevodsky of the IAS on motivic co- “yardsticks”, in all sizes and forms, each perfectly homology, for which he received the Fields Medal suited to its own unique measuring task. What is in 2002. This work builds on some of Grothen- more, the category of all sheaves over a space car- dieck’s original ideas about motives. ries so much information that one can essentially In looking back on this brief retrospective of his “forget” what the space is. All the information is mathematical work, Grothendieck wrote that what in the sheaf—what Grothendieck called the “silent makes up its essence and power is not results or and sure guide” that led him on the path to his dis- big theorems, but “ideas, even dreams” (page P51). coveries. The notion of topos, Grothendieck wrote, is “a The Grothendieck School metamorphosis of the notion of a space.” The con- cept of a sheaf provides a way of translating from Jusqu’au moment du premier “réveil,” the topological setting, where the space lives, to the en 1970, les relations à mes élèves, tout

1052 NOTICES OF THE AMS VOLUME 51, NUMBER 9 comme ma relation à mon propre tra- seminar. The atmosphere was “fantastic”, Artin vail, était une source de satisfaction et recalled. The seminar was well populated by the de joie, un des fondements tangibles, ir- leading lights of Parisian mathematics, as well as récusables, d’un sentiment d’harmonie mathematicians visiting from other places. A group dans ma vie, qui continuait à lui don- of brilliant and eager students began to collect ner un sens…. around Grothendieck and to write their theses under his direction (the IHÉS does not give de- Until the moment of the first “awaken- grees, so formally they were students at universi- ing”, in 1970, the relations with my stu- ties in and around Paris). By 1962 the IHÉS had dents, just like my relation to my own moved to its permanent home in the middle of a work, was a source of satisfaction and serene, tree-filled park called the Bois-Marie, in joy, one of the tangible, unimpeachable the Paris suburb of Bures-sur-Yvette. The gazebo- bases of a sense of harmony in my life, like building where the seminar was held, with its which continued to give it meaning…. large picture windows and open, airy feel, pro- vided an unusual and dramatic setting. Grothen- —Récoltes et Semailles, page 63 dieck was the dynamic center of the activities. “The seminars were highly interactive,” recalled During a visit to Harvard in the fall of 1961, Hyman Bass, who visited the IHÉS in the 1960s, “but Grothendieck wrote to Serre: “The mathematical at- Grothendieck dominated whether he was the mosphere at Harvard is ravishing, a real breath of speaker or not.” He was extremely rigorous and fresh air compared with Paris, which is getting could be rather tough on people. “He was not un- more gloomy every year. Here, there are a fair num- kind, but also not coddling,” Bass said. ber of intelligent students who are beginning to be Grothendieck developed a certain pattern of familiar with the language of schemes and ask for working with students. A typical example is that nothing more than to work on interesting problems, of Luc Illusie of the Université de Paris-Sud, who which obviously are not in short supply” [Corr]. Michael Artin was at Harvard at that time as a Ben- became a student of Grothendieck’s in 1964. Illusie jamin Peirce instructor, after having finished his had been participating in the Paris seminar of thesis with Zariski in 1960. Immediately after his Henri Cartan and Laurent Schwartz, and it was thesis, Artin set about learning the new language Cartan who suggested that Illusie might do a the- of schemes, and he also became interested in the sis with Grothendieck. Illusie, who had until that idea of étale cohomology. When Grothendieck came time worked only in topology, was apprehensive to Harvard in 1961, “I asked him to tell me the de- about meeting this “god” of algebraic geometry. As finition of étale cohomology,” Artin recalled with it turned out, Grothendieck was quite kind and a laugh. The definition had not yet been formulated friendly and asked Illusie to explain what he had precisely. Said Artin, “Actually we argued about the been working on. After Illusie had spoken for a definition for the whole fall.” short time, Grothendieck went to the board and After moving to the Massachusetts Institute of launched into a discussion of sheaves, finiteness Technology in 1962, Artin gave a seminar on étale conditions, pseudo-coherence, and the like. “It was cohomology. He spent much of the following two like a sea, like a continuous stream of mathemat- years at the IHÉS working with Grothendieck. Once ics on the board,” Illusie recalled. At the end of it the definition of étale cohomology was in hand, Grothendieck said that the next year he would de- there was still a lot of work to be done to tame the vote his seminar to L-functions and l-adic coho- theory and make it into a tool that could really be mology and that Illusie should help to write up the used. “The definition looked marvelous, but it notes. When Illusie protested that he knew noth- came with no guarantees that anything was finite, ing about algebraic geometry, Grothendieck said or could ever be computed, or anything,” Mumford it didn’t matter: “You will learn quickly.” commented. This was the work that Artin and And Illusie did. “His lectures were so clear, and Grothendieck plunged into; one product was the he made so many efforts to recall what was nec- Artin representability theorem. Together with Jean- essary, all the prerequisites,” Illusie remarked. Louis Verdier, they directed the 1963–64 seminar, Grothendieck was an excellent teacher, very patient which focused on étale cohomology. That seminar and adept at explaining things clearly. “He took was written up in the three volumes of SGA 4, time to explain very simple examples showing how which total nearly 1,600 pages. the machinery works,” Illusie said. Grothendieck There might be disagreement with Grothen- discussed formal properties that are often glossed dieck’s “gloomy” assessment of the Parisian math- over as being “trivial” and therefore too obvious ematical scene of the early 1960s, but there is no to require explanation. Usually “you don’t specify question that it got an enormous boost when he them, you don’t spend time,” Illusie said, but such returned to the IHÉS in 1961 and restarted his things are pedagogically very useful. “Sometimes

OCTOBER 2004 NOTICES OF THE AMS 1053 it was a bit lengthy, but it was very good for un- showing that the two manifolds could be different, derstanding.” and Mazur went on to do some work in homotopy Grothendieck gave Illusie the assignment of theory with Artin that was inspired by this ques- writing up notes for some exposés of the semi- tion. But at the time Grothendieck posed it, Mazur nars—namely, exposés I, II, and III of SGA 5. The was a dedicated differential topologist, and such notes done, “I was shivering when I handed them a question would not have occurred to him. “For to him,” Illusie recalled. A few weeks later, Grothen- [Grothendieck], it was a natural question,” Mazur dieck asked Illusie to come to his home to discuss said. “But for me, it was precisely the right kind of the notes; he often worked at home with colleagues motivation to get me to begin to think about alge- and students. When Grothendieck took the notes bra.” Grothendieck had a real talent for “matching out and set them on the table, Illusie saw that they people with open problems. He would size you up were blackened with penciled comments. The two and pose a problem that would be just the thing sat there for several hours as Grothendieck went to illuminate the world for you. It’s a mode of per- over each comment. “He could criticize for a ceptiveness that’s quite wonderful, and rare.” comma, for a period, he could criticize for an ac- In addition to work with students and colleagues cent, he could criticize also on the substance of the at the IHÉS, Grothendieck maintained correspon- thing very deeply and propose another organiza- dence with a large number of mathematicians out- tion—it was all kinds of comments,” Illusie said. side Paris, some of whom were working on parts “But all his comments were very up to the point.” of his program in other places. For example, Robin This kind of line-by-line critique of written notes Hartshorne of the University of California at Berke- was typical of Grothendieck’s way of working with ley was at Harvard in 1961 and got the idea for his students. Illusie recalled that a couple of students thesis topic, concerning Hilbert schemes, from could not bear this kind of close criticism and Grothendieck’s lectures there. Once the thesis was ended up writing their theses with someone else. done, Hartshorne sent a copy to Grothendieck, One was nearly reduced to tears after an encounter who was by then back in Paris. In a reply dated Sep- with Grothendieck. Illusie said, “Some people I re- tember 17, 1962, Grothendieck made some brief member didn’t like it so much. You had to comply positive remarks about the thesis. “The next three with that. …[But] they were not petty criticisms.” or four pages [of the letter] are all of his ideas Nicholas Katz was also given an assignment about further theorems that I might be able to de- when he visited the IHÉS as a postdoc in 1968. velop and other things that one might like to know Grothendieck suggested that Katz could give a lec- about the subject,” Hartshorne said. Some of the ture in the seminar about Lefschetz pencils. “I had things the letter suggested are “impossibly diffi- heard of Lefschetz pencils but really knew as lit- cult,” he noted; others show a remarkable pre- tle as is possible to know about them except for science. After this outpouring of ideas, Grothen- having heard of them,” Katz recalled. “But by the end of the year I had given a few talks in the sem- dieck returned to the thesis and offered three inar, which now exist as part of SGA 7. I learned a pages of detailed comments. tremendous amount from it, and it had a big effect In his 1958 talk at the Edinburgh Congress, on my future.” Katz said that Grothendieck would Grothendieck had outlined his ideas for a theory come to the IHÉS perhaps one day a week to talk of duality, but because he was busy with other top- to the visitors. “What was completely amazing is ics in his IHÉS seminar, it did not get treated there. he would then somehow get them interested in So Hartshorne offered to give a seminar on dual- something, give them something to do,” Katz ex- ity at Harvard and write up the notes. Over the sum- plained. “But with, it seemed to me, a kind of amaz- mer of 1963, Grothendieck fed Hartshorne about ing insight into what was a good problem to give 250 pages of “pre-notes” that formed the basis for to that particular person to think about. And he was the seminar, which Hartshorne began in the fall of somehow mathematically incredibly charismatic, so 1963. Questions from the audience helped that it seemed like people felt it was almost a priv- Hartshorne to develop and refine the theory, which ilege to be asked to do something that was part of he began to write up in a systematic fashion. He Grothendieck’s long range vision of the future.” would send each chapter to Grothendieck to cri- Barry Mazur of Harvard University still remem- tique. “It would come back covered with red ink all bers today the question that Grothendieck posed over,” Hartshorne recalled. “So I’d fix everything he to him in one of their first conversations at the IHÉS said, and then I would send him the new version. in the early 1960s, a question that Gerard Wash- And it would come back again covered with more nitzer had originally asked Grothendieck. The ques- red ink.” After realizing that this was a potentially tion: Can an algebraic variety defined over a field endless process, Hartshorne decided one day to give topologically different manifolds by two dif- send the manuscript off to be published; it ap- ferent embeddings of the field into the complex peared in the Springer Lecture Notes series in 1966 numbers? Serre had given some early examples [Hartshorne].

1054 NOTICES OF THE AMS VOLUME 51, NUMBER 9 Grothendieck “had so many ideas that he kept times, though Grothendieck himself never pursued basically all the serious people working in algebraic abstraction for abstraction’s sake. Reid also noted geometry in the world busy during that time,” that, apart from the small number of students of Hartshorne observed. How did he keep such an Grothendieck who could “take the pace and sur- enterprise going? “I don’t think there is a simple vive,” the people who benefited most from his answer,” Artin replied. But certainly Grothendieck’s ideas were those influenced at a distance, partic- energy and breadth were factors. “He was very dy- ularly American, Japanese, and Russian mathe- namic, and he did cover a lot of territory,” Artin maticians. Pierre Cartier sees Grothendieck’s her- said. “One thing that was remarkable was that he itage in the work of such Russian mathematicians had complete control of the field, which was not as Vladimir Drinfeld, Maxim Kontsevich, Yuri Manin, inhabited by slouches, for a period of 12 years or and Vladimir Voevodsky. Said Cartier, “They cap- so.” ture the true spirit of Grothendieck, but they are During his IHÉS years, Grothendieck’s devotion able to combine it with other things.” to mathematics was total. His tremendous energy and capacity for work, combined with a tenacious Photographs used in this article are courtesy of fidelity to his internal vision, produced a flood of Friedrich Hirzebruch, Karin Tate, and the website ideas that swept many into its currents. He did not of the Grothendieck Circle (http://www. shrink from the daunting program he had set for grothendieck-circle.org). himself, but plunged straight in, taking on tasks The second part of this article will appear in the great and small. “His mathematical agenda was next issue of the Notices. much more than any single human being could do,” Bass commented. He parceled out much of the work to his students and collaborators, while also References taking on a great deal himself. What motivated [Aubin] D. AUBIN, A Cultural History of Catastrophes and him, as he explained in Récoltes et Semailles, was Chaos: Around the “Institut des Hautes Études Sci- simply the desire to understand, and indeed those entifiques,”France, doctoral thesis, Princeton Uni- who knew him then confirm that he was not pro- versity, 1998. pelled by any sense of competition. “At the time, [Borel] A. BOREL, Twenty-five years with Nicolas Bourbaki, there was never a thought of proving something be- 1949–1973, Notices, Amer. Math. Soc. 45 (1998), 373–380. fore somebody else,” Serre explained. And in any [BS] A. BOREL and J.-P. SERRE, Le théorème de Riemann-Roch, case, “he could not be in competition with anybody, Bull. Soc. Math. France 86 (1958) 97–136. in a sense, because he wanted to do things his own [Cartier1] P. CARTIER, A mad day’s work: From Grothen- way, and essentially nobody else wanted to do the dieck to Connes and Kontsevich. The evolution of same. It was too much work.” concepts of space and symmetry, Bull. Amer. Math. The dominance of the Grothendieck school had Soc. 38 (4) 389-408; published electronically July some detrimental effects. Even Grothendieck’s dis- 2001. tinguished IHÉS colleague, René Thom, felt the [Cartier2] ——— , Un pays dont on ne connaîtrait que le pressure. In [Fields], Thom wrote that his relations nom: Les ‘motifs’ de Grothendieck, Le Réel en Math- with Grothendieck were “less agreeable” than with ématiques (P. Cartier and N. Charraud eds.), Agalma, his other IHÉS colleagues. “His technical superior- 2004. ity was crushing,” Thom wrote. “His seminar at- [Cerf] J. CERF, Trois quarts de siècle avec Henri Cartan, Gazette des Mathématiciens, April 2004, Société tracted the whole of Parisian mathematics, whereas Mathématique de France. I had nothing new to offer. That made me leave the [Corr] Corréspondance Grothendieck-Serre. Société Math- strictly mathematical world and tackle more gen- ématique de France, 2001. (Published in a bilingual eral notions, like morphogenesis, a subject which French-English version by the Amer. Math. Soc., interested me more and led me towards a very 2003, under the title Grothendieck-Serre Corre- general form of ‘philosophical’ biology.” spondence.) In the historical remarks that appear at the end [D1] J. DIEUDENNÉ, A. Grothendieck’s early work (1950- of his 1988 textbook Undergraduate Algebraic 1960), K-theory, 3 (1989) 299–306. (This issue of Geometry, Miles Reid wrote: “[T]he Grothendieck K-Theory was devoted to Grothendieck on the oc- personality cult had serious side effects: many peo- casion of his 60th birthday.) ple who had devoted a large part of their lives to [D2] ——— , Les travaux de Alexander Grothendieck, Proc. mastering Weil foundations suffered rejection and Internat. Congr. Math. (Moscow, 1966), pp. 21–24. Izdat. “Mir”, Moscow, 1968. humiliation. …[A] whole generation of students [Edin] A. GROTHENDIECK, The cohomology theory of ab- (mainly French) got themselves brainwashed into stract algebraic varieties, 1960 Proc. Internat. Con- the foolish belief that a problem that can’t be gress Math. (Edinburgh, 1958), pp. 103–118, Cam- dressed up in high-powered abstract formalism is bridge Univ. Press, New York. unworthy of study.” Such “brainwashing” was per- [FAC] J.-P. SERRE, Faisceaux algébriques cohérents, Ann. haps an inevitable by-product of the fashions of the of Math. 61 (1955), 197–278.

OCTOBER 2004 NOTICES OF THE AMS 1055 [Fields] Fields Medalists’ Lectures, (M. Atiyah and D. Iagol- nitzer, eds.), World Scientific, second edition, 2003. [Gthesis] A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucléaires, Memoirs of the AMS (1955), no. 16. [Hallie] P. HALLIE, Lest Innocent Blood Be Shed, Harper- Collins, 1994. [Hartshorne] R. HARTSHORNE, Residues and Duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, 1966. [Heydorn] W. HEYDORN, Nur Mensch Sein!, Memoirs from 1873 to 1958, (I. Groschek and R. Hering, eds.), Dölling and Galitz Verlag, 1999. [Ikonicoff] R. Ikonicoff, Grothendieck, Science et Vie, Au- gust 1995, number 935, pages 53–57. [Langlands] R. P. LANGLANDS, Automorphic representa- tions, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and L -func- tions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Part 2, pp. 205–246. Amer. Math. Soc., 1979. [Nasar] S. NASAR, A Beautiful Mind, Simon and Schuster, 1998. [R&S] Récoltes et semailles: Réflexions et témoignages sur un passé de mathématicien, by Alexandre Grothen- dieck. Université des Sciences et Techniques du Languedoc, Montpellier, et Centre National de la Recherche Scientifique, 1986. [Scharlau] Materialen zu einer Biographie von Alexander Grothendieck, compiled by Winfried Scharlau. Avail- able at http://www.math.uni-muenster. de/math/u/charlau/scharlau. [Schwartz] L. SCHWARTZ, Les produits tensoriels d’après Grothendieck, Séminaire Secrétariat mathématique, Paris, 1954. [To] A. GROTHENDIECK, Sur quelques points d’algèbre ho- mologique,” Tôhoku Math. J. (2) 9 (1957), 119–221. [Washnitzer] G. WASHNITZER, Geometric syzygies, Ameri- can Journal of Mathematics, 81 (1959) 171-248. [Weil1] A. Weil, Foundations of Algebraic Geometry, AMS Colloquium Publications, No. 29, 1946. [Weil2] ——— , Numbers of solutions of equations in fi- nite fields, Bulletin of the Amer. Math. Soc., 55 (1949) 497–508.

1056 NOTICES OF THE AMS VOLUME 51, NUMBER 9 Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre Grothendieck

Allyn Jackson

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.

A Different Way of Thinking he first encountered Grothendieck’s way of think- ing, it seemed completely different and new. But Dans le travail de découverte, cette at- it is hard to articulate what the difference was. As tention intense, cette sollicitude ardente Katz put it, the change in point of view was so fun- sont une force essentielle, tout comme damental and profound and, once adopted, so la chaleur du soleil pour l’obscure ges- completely natural “that it’s sort of hard to imag- tation des semences enfouies dans la ine the time before you thought that way.” terre nourricière, et pour leur humble Although Grothendieck approached problems et miraculeuse éclosion à la lumière du from a very general point of view, he did so not for jour. generality’s sake but because he was able to use gen- erality in a very fruitful way. “It’s a kind of approach In the work of discovery, this intense at- that in less gifted hands just leads to what most tention, this ardent solicitude, are an es- sential force, just like the warmth of people would say are sterile generalities,” Katz the sun for the obscure gestation of commented. “He somehow knew what general seeds covered in nourishing soil, and for things to think about.” Grothendieck always sought their humble and miraculous blossom- the precise level of generality that would provide ing in the light of day. precisely the right leverage to gain insight into a problem. “He seemed to have the knack, time after —Récoltes et Semailles, page P49 time, of stripping away just enough so that it wasn’t a special case, but it wasn’t a vacuum either,” com- Grothendieck had a mathematical style all his mented John Tate of the University of Texas at own. As Michael Artin of the Massachusetts Insti- Austin. “It’s streamlined; there is no baggage. It’s tute of Technology commented, in the late 1950s just right.” and 1960s “the world needed to get used to him, One striking characteristic of Grothendieck’s to his power of abstraction.” Nowadays Grothen- mode of thinking is that it seemed to rely so little dieck’s point of view has been so thoroughly ab- on examples. This can be seen in the legend of the sorbed into algebraic geometry that it is standard so-called “Grothendieck prime”. In a mathematical fare for graduate students starting in the field, many of whom do not realize that things were conversation, someone suggested to Grothendieck once quite different. Nicholas Katz of Princeton Uni- that they should consider a particular prime num- versity said that when as a young mathematician ber. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual Allyn Jackson is senior writer and deputy editor of the prime number. Grothendieck suggested, “All right, Notices. Her email address is [email protected]. take 57.”

1196 NOTICES OF THE AMS VOLUME 51, NUMBER 10 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. Grothendieck lecturing at the IHÉS. One thing Grothendieck said was that one should never try to prove anything that is not almost ob- term motif (“motive” in English) is intended to vious. This does not mean that one should not be evoke both meanings of the word: a recurrent ambitious in choosing things to work on. Rather, theme and something that causes action. “if you don’t see that what you are working on is Grothendieck’s attention to choosing names almost obvious, then you are not ready to work on meant that he loathed terminology that seemed un- that yet,” explained Arthur Ogus of the University suitable: In Récoltes et Semailles, he said he felt an of California at Berkeley. “Prepare the way. And that “internal recoiling” upon hearing for the first time was his approach to mathematics, that everything the term perverse sheaf. “What an idea to give such should be so natural that it just seems completely a name to a mathematical thing!” he wrote. “Or to straightforward.” Many mathematicians will choose any other thing or living being, except in sternness a well-formulated problem and knock away at it, towards a person—for it is evident that of all the an approach that Grothendieck disliked. In a ‘things’ in the universe, we humans are the only well-known passage of Récoltes et Semailles, he ones to whom this term could ever apply” (page describes this approach as being comparable to 293). cracking a nut with a hammer and chisel. What he Although Grothendieck possessed great tech- prefers to do is to soften the shell slowly in water, nical power, it was always secondary; it was a or to leave it in the sun and the rain, and wait for means for carrying out his larger vision. He is the right moment when the nut opens naturally known for certain results and for developing cer- (pages 552–553). “So a lot of what Grothendieck did tain tools, but it is his creation of a new viewpoint looks like the natural landscape of things, because on mathematics that is his greatest legacy. In this it looks like it grew, as if on its own,” Ogus noted. regard, Grothendieck resembles Evariste Galois; Grothendieck had a flair for choosing striking, indeed, in various places in Récoltes et Semailles evocative names for new concepts; indeed, he saw Grothendieck wrote that he strongly identified with the act of naming mathematical objects as an in- Galois. He also mentioned that as a young man he tegral part of their discovery, as a way to grasp them read a biography of Galois by Leopold Infeld [In- even before they have been entirely understood feld] (page P63). (R&S, page P24). One such term is étale, which in Ultimately, the wellspring of Grothendieck’s French is used to describe the sea at slack tide, that achievement in mathematics is something quite is, when the tide is neither going in nor out. At slack humble: his love for the mathematical objects he tide, the surface of the sea looks like a sheet, which studied. evokes the notion of a covering space. As Grothen- dieck explained in Récoltes et Semailles, he chose A Spirit in Stagnation the word topos, which means “place” in Greek, to suggest the idea of “the ‘object par excellence’ to [P]endant vingt-cinq ans, entre 1945 which topological intuition applies” (pages 40–41). (quand j’avais dix-sept ans) et 1969 Matching the concept, the word topos suggests the (quand j’allais sur les quarante-deux), most fundamental, primordial notion of space. The j’ai investi pratiquement la totalité de

NOVEMBER 2004 NOTICES OF THE AMS 1197 mon énergie dans la recherche mathé- had achieved level −1 and was working on level 0 matique. Investissement démesuré, of something that must be 10 levels high.…At a cer- certes. Je l’ai payé par une longue stag- tain age it becomes clear you will never be able to nation spirituelle, par un “épaississe- finish the building.” ment” progressif, que j’aurai plus d’une The extremity of Grothendieck’s focus on math- fois l’occasion d’évoquer dans les pages ematics is one reason for the “spiritual stagna- de Récoltes et Semailles. tion” he referred to in Récoltes et Semailles, which in turn is one of the reasons behind his departure, [F]or twenty-five years, between 1945 in 1970, from the world of mathematics in which (when I was seventeen years old) and he had been a leading figure. One step toward that 1969 (when I reached forty-two), I in- departure was a crisis within the IHÉS, which led vested practically my entire energy into to his resignation. Starting in late 1969, Grothen- mathematical research. An excessive in- dieck became embroiled in a conflict with the vestment, certainly. I paid for it with a founder and director of the IHÉS, Léon Motchane, long spiritual stagnation, with a pro- over military funding for the institute. As historian gressive “dulling”, that I have more than of science David Aubin explained [Aubin], during once found occasion to evoke in the the 1960s, the IHÉS finances were rather precari- pages of Récoltes et Semailles. ous, and in some years the institute received a small portion of its budget, never more than about —Récoltes et Semailles, page P17 5 percent, from sources within the French mili- tary. All of the permanent IHÉS professors had During the 1960s, Barry Mazur of Harvard Uni- misgivings about military funding, and in late 1969 versity visited the Institut des Hautes Études Sci- they insisted that Motchane quit accepting such entifiques (IHÉS) with his wife. Although by that funding. Motchane agreed, but, as Aubin noted, he time Grothendieck had a family and a house of his went back on his word just a few months later, own, he also kept an apartment in the same build- when the IHÉS budget was stretched thin and he ing where the Mazurs were living and frequently accepted a grant from the minister of the army. Out- worked there late into the night. Because the apart- raged, Grothendieck tried in vain to persuade the ment keys did not open the outside doors, which other professors to resign along with him, but were locked at 11:00 p.m., one might have trouble none did. Less than a year earlier, Pierre Deligne getting into the building after an evening in Paris. had joined the IHÉS faculty as a permanent pro- But “I remember we never had any problems,” fessor, largely on the recommendation of Grothen- Mazur recalled. “We would take the last train back, dieck, who now pressed his newly appointed absolutely certain that there would be Grothen- colleague to join him in resigning. Deligne too re- dieck working, his desk by the window. We would fused. “Because I was very close to him mathe- throw some gravel at his window and he would matically, Grothendieck was surprised and deeply open the outside door for us.” Grothendieck’s disappointed that this closeness of ideas did not apartment was sparely furnished; Mazur remem- extend outside of mathematics,” Deligne recalled. bered a wire sculpture in the outline of a goat and Grothendieck’s letter of resignation was dated an urn filled with Spanish olives. May 25, 1970. This somewhat lonely image of Grothendieck His rupture with the IHÉS was the most visible working away into the night in a spartan apartment sign of a profound shift taking place in Grothen- captures one aspect of his life during the 1960s. dieck’s life. Toward the end of the 1960s there At this time he did mathematics nonstop. He was were other signs as well. Some were small. Mazur talking to colleagues, advising students, lecturing, recalled that when he was visiting the IHÉS in 1968, carrying on extensive correspondence with math- Grothendieck told him he had gone to the movies— ematicians outside of France, and writing the seem- for the first time in perhaps a decade. Other signs ingly endless volumes of EGA and SGA. It is no were larger. In 1966, when he was to receive the Fields exaggeration to say that he was single-handedly Medal at the International Congress of Mathemati- leading a large and thriving segment of worldwide cians (ICM) in Moscow, Grothendieck refused to research in algebraic geometry. He seemed to have attend as a protest against the Soviet government. few interests outside of mathematics; colleagues In 1967 Grothendieck made a three-week trip to Viet- have said that he never read a newspaper. Even nam, which clearly left an impression on him. His among mathematicians, who tend to be single- written account of the trip [Vietnam] described the minded and highly devoted to their work, Grothen- many air raid alerts and a bombing that left two math- dieck was an extreme case. “Grothendieck was ematics teachers dead, as well as the valiant efforts working on the foundations of algebraic geometry of the Vietnamese to cultivate a mathematical life in seven days a week, twelve hours a day, for ten their country. A friendship with a Romanian physician years,” noted his IHÉS colleague David Ruelle. “He named Mircea Dumitrescu led Grothendieck to make

1198 NOTICES OF THE AMS VOLUME 51, NUMBER 10 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- Grothendieck wrote this abstract into the colloquium book at the Universität Bielefeld lution, stationed tanks when he spoke there in 1971. The abstract says: “Witch's Kitchen 1971. Riemann-Roch around the perimeter of Theorem: The ‘dernier cri’: The diagram [displayed] is commutative! To give an the city. Millions of workers approximate sense to the statement about f : X → Y, I had to abuse the listeners’ went on strike, paralyzing patience for almost two hours. Black on white (in Springer Lecture Notes) it probably the nation for about two takes about 400, 500 pages. A gripping example of how our thirst for knowledge and weeks. Karin Tate, who was discovery indulges itself more and more in a logical delirium far removed from life, living in Paris with her hus- while life itself is going to Hell in a thousand ways—and is under the threat of final band at the time, John Tate, extermination. High time to change our course!” recalled the chaos that reigned. “Paving stones, batons, and any other mis- Grothendieck’s resignation, attended the lecture, siles that were handy flew through the air,” she said. which he said drew an audience of hundreds in a “Soon the entire country was at a standstill. There very crowded lecture hall. Katz remembered that was no gasoline (truckers were on strike), there were in the lecture Grothendieck went so far as to say no trains (train workers were on strike), garbage was that doing mathematical research was actually piling up in Paris (sanitation workers were on “harmful” (“nuisible” ), given the impending threats strike), there was very little food on the shelves.” to the human race. She and John fled to Bures-sur-Yvette, where her A written version of the lecture, “Responsabilité brother, Michael Artin, was visiting the IHÉS. Many du savant dans le monde d’aujourd’hui: Le savant Parisian mathematicians took the side of the stu- et l’appareil militaire” (“The responsibility of the dents in the conflict. Karin Tate said the protests scholar in today’s world: The scholar and the mili- dominated conversations among the mathemati- tary apparatus”), circulated as an unpublished man- cians she knew, though she did not remember dis- uscript. An appendix described the hostile reactions cussing the topic with Grothendieck. of the students who attended the lecture and who Shortly after his resignation from the IHÉS, handed out flyers mocking Grothendieck. One of Grothendieck plunged into a world completely new the flyers is reproduced in the appendix; a typical to him, the world of protest politics. In a June 26, slogan: “Réussissez, ossifiez-vous, détruisez-vous 1970, lecture at the Université de Paris in Orsay, vous-mêmes: devenez un petit schéma télécom- he spoke not about mathematics but about the mandé par Grothendieck” (“Succeed, ossify, self- threat of nuclear proliferation to the survival destruct: become a little scheme remote-controlled of humankind and called upon scientists and by Grothendieck”). He was clearly seen as a detested mathematicians not to collaborate in any way with member of the establishment. the military. Nicholas Katz, who had recently arrived In another appendix in this manuscript, Grothen- for a visit at the IHÉS and was surprised to hear of dieck called for the founding of a group to fight

NOVEMBER 2004 NOTICES OF THE AMS 1199 (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 Tata Institute International Colloquium in 1968. front of the hall in which the congress was being Grothendieck (standing, left) and Armand Borel held. But another problem loomed: in delicate (seated, facing camera). negotiations with the mayor of Nice, the congress organizers had promised there would be no street for the survival of the human race against envi- demonstrations. Police officers began to question ronmental degradation and the dangers of military Grothendieck, and finally the chief of police showed conflict. This group, called “Survival” (“Survivre up. Grothendieck was asked to move his table just et Vivre” in French) came into being in July 1970 a few yards back so that it was off the sidewalk. when Grothendieck delivered his Orsay lecture a “But he refused,” Cartier recalled. “He wanted to second time, at a summer school on algebraic be put in jail. He really wanted to be put in jail!” geometry at the University of Montreal. The main Finally, Cartier said, he and some others moved the activity of Survival was the publication of a newslet- table back sufficiently to satisfy the police. ter by the same name, the first issue of which was Although Grothendieck’s plunge into politics written in English by Grothendieck and is dated Au- was sudden, he was by no means alone. His good gust 1970. The newsletter describes an ambitious friend Cartier has a long history of political ac- agenda of publication of books on science, orga- tivism. For example, he was among the mathe- nization of public courses on science aimed at non- maticians who used the holding of the ICM in experts, and boycotts of scientific institutions that Warsaw in 1983 to negotiate the release of one accept military funds. hundred fifty political prisoners in Poland. Cartier That first issue carried a list of the names, pro- traces his activism to the example set by his teacher fessions, and addresses of the group’s members, and mentor, Laurent Schwartz, who was one of the who numbered twenty-five at the time. On the list most politically vocal and active academics in were several mathematicians, Grothendieck’s France. Schwartz was the thesis adviser of Grothen- mother-in-law, and his son Serge. The directors of dieck. Another mathematician Grothendieck knew the group were Grothendieck and three other math- well, Pierre Samuel, is one of the founders of the ematicians: Claude Chevalley, Denis Guedj, and French Green Party. Outside of France, many math- Pierre Samuel (R&S, page 758). Survival was one of ematicians were politically active. Among the best- many leftist groups that emerged in the wake of known examples in North America are Chandler the tumultuous 1960s; a similar organization in the Davis and Stephen Smale, who were deeply in- United States was the Mathematics Action Group. volved in protests against the Vietnam War. Too small and diffuse to accumulate much influ- But despite his strong convictions, Grothen- ence, Survival was more active in Paris than in the dieck was never effective in the real world of pol- United States and Canada, due mostly to Grothen- itics. “He was always an anarchist at heart,” Cartier dieck’s presence. When he moved out of Paris in observed. “On many issues, my basic positions are 1973, the group petered out. not very far from his positions. But he was so naive At the ICM in Nice in the summer of 1970, that it was totally impossible to do anything with Grothendieck tried to recruit members for Sur- him politically.” He was also rather ignorant. Cartier vival. He wrote, “I expected massive enrollments— recalled that, after an inconclusive presidential there were (if I remember correctly) two or three” election in France in 1965, the newspapers carried

1200 NOTICES OF THE AMS VOLUME 51, NUMBER 10 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 Grothendieck, center, University of Montreal, entrer dans un autre—le milieu des gens around 1970. “des premiers rangs” pour le “marais”; soudain, la plupart de mes nouveaux who in April 1971 wrote an open letter explaining amis étaient de ceux justement qu’un an his reasons for refusing to attend the conference. avant encore j’aurais tacitement situés Unbeknownst to Grothendieck, Cartier and some dans cette contrée sans nom et sans other mathematicians who were uncomfortable contours. Le soi-disant marais soudain about the NATO support had conducted extensive s’animait et prenait vie par les visages negotiations to have a NATO representative come d’amis liés à moi par une aventure com- to the conference for a public debate. Cartier and mune—une autre aventure! others eased Grothendieck off the podium, but the damage had been done: Cartier soon received an [In 1970] I left one milieu to enter an- angry phone call from the NATO representative, other—the milieu of people “of the first who had heard about the outburst and refused to rank” for the “swamp”; suddenly, the come, believing that conditions for an orderly de- majority of my new friends were those bate had been ruined. “To me, it was sad, because who just a year before I had tacitly sit- from what I remember, I think that the audience uated in this region without name and was mostly on Grothendieck’s political side,” Cartier without shape. The so-called swamp noted. “Even people who were close to his politi- suddenly moved around and took on cal views or his social views were antagonized by life through the faces of friends tied to his behavior.…He behaved like a wild teenager.” me by a common adventure—another By the time of the Antwerp meeting, Grothen- adventure! dieck had cut many of the ties that had bound him to an orderly life focused on mathematics. For one —Récoltes et Semailles, page 38 thing, he no longer had a permanent position. After “Légion d’Honneur! Légion d’Honneur!” Grothen- he left the IHÉS in 1970, Serre arranged for him to dieck was shouting from the back of the auditorium, have a visiting position at the Collège de France for waving a paper facsimile of the Légion d’Honneur two years. This elite institution operates differ- cross, a distinction conferred by the French gov- ently from other universities in France (or any- ernment. The scene was the opening day of a sum- where else for that matter). Each professor at the mer school on modular functions, held in Antwerp Collège must submit for approval by the assembly in the summer of 1972 and supported by the North of all the professors a program of the lectures he Atlantic Treaty Organization (NATO). Grothen- or she plans to deliver during the year. Serre re- dieck’s longtime friend Jean-Pierre Serre of the called that Grothendieck offered two possible Collège de France, who had recently received the programs: one on mathematics and one on the Légion d’Honneur, was presenting the opening political themes that occupied the Survival group. speech. Grothendieck approached Serre and asked, The committee approved the mathematical pro- “Do you mind if I go to the blackboard and say gram and rejected the other one. So Grothendieck something?” Serre replied, “Yes, I mind” and left presented mathematical lectures prefaced by long the room. Grothendieck then mounted the podium discourses about politics. After two years he applied and began speaking against NATO support for the for a permanent position at the Collège de France, conference. Other mathematicians sympathized a position that had become vacant with the retire- with this view: One example was Roger Godement, ment of Szolem Mandelbrojt. The curriculum vitae

NOVEMBER 2004 NOTICES OF THE AMS 1201 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 A. Grothendieck with children Serge (left) and the Survival group, and these attracted consider- Johanna in 1960. able media attention. However, the commune dis- solved fairly rapidly as a result of complicated Grothendieck submitted plainly showed that he personal relationships among the members. It intended to give up mathematics to focus on tasks was around this time that Grothendieck’s position he believed to be far more urgent: “the imperatives ended at the Collège de France, and in the fall of of survival and the promotion of a stable and hu- 1972 he took a temporary position teaching for mane order on our planet.” How could the Collège one year at the Université de Paris in Orsay. After appoint to a position in mathematics someone who that, Grothendieck obtained a position called pro- had declared that he would no longer do any math- fesseur à titre personnel, which is attached to a ematics? “He was rightly refused,” Serre said. single individual and can be taken to any univer- It was also during the period just after he left sity in France. Grothendieck took his to the Uni- the IHÉS that Grothendieck’s family life crumbled versité de Montpellier, where he was to remain and he and his wife separated. In the two years after until his retirement in 1988. he left the IHÉS, Grothendieck spent a fair amount In early 1973 he and Bumby moved to Olmet- of time lecturing in mathematics departments in le-sec, a rural village in the south of France. This North America. He spread the gospel of Survival by area was at the time a magnet for hippies and insisting he would give a mathematics lecture only others in the counterculture movement who wanted if arrangements were made for him also to give a to return to a simpler lifestyle close to the land. political lecture. On one such trip in May 1972, he Here Grothendieck again attempted to start up a visited Rutgers University and met Justine Bumby commune, but personality conflicts led to its (Skalba), then a graduate student of Daniel Goren- collapse. At various times three of Grothendieck’s stein. Captivated by Grothendieck’s charismatic children came to live in the Paris commune and in personality, Bumby left behind her life as a grad- the one in Olmet. After the latter commune dis- uate student to follow him, first on the remainder solved, he moved with Bumby and his children to of his trip in the United States, and then on to Villecun, a short distance away. Bumby noted that France, where she lived with him for two years. Grothendieck had a hard time adjusting to the “He’s the most intelligent person I’ve ever met,” ways of the people attracted to the counterculture she said. “I was very much in awe of him.” movement. “His students in mathematics had been Their life together was in some ways emblem- very serious, and they were very disciplined, very atic of the counterculture years of the 1970s. Once, hardworking people,” she said. “In the counter- at a peaceful demonstration in Avignon, the police culture he was meeting people who would loaf intervened, harassing and pushing away the demon- around all day listening to music.” Having been an strators. Grothendieck got angry when they started undisputed leader in mathematics, Grothendieck pestering him, Bumby recalled. “He was a good now found himself in a very different milieu, in boxer, so he was very fast,” she said. “We see the which his views were not always taken seriously. policemen approaching us, and we are all scared, “He was used to people agreeing with his opinions and then the next thing we know, the two police- when he was doing algebraic geometry,” Bumby men are on the ground.” Grothendieck had single- remarked. “When he switched to politics all the handedly decked two police officers. After some people who would have agreed with him before other officers had subdued him, Bumby and suddenly disagreed with him.... It was something Grothendieck were bundled into a wagon and he wasn’t used to.” taken to the police station. When his identification Although most of the time Grothendieck was papers revealed that he was a professor at the very warm and affectionate, Bumby said, he

1202 NOTICES OF THE AMS VOLUME 51, NUMBER 10 sometimes had violent outbursts followed by pe- found a clever way to circumvent them when he riods of silent withdrawal. There were also dis- proved the last Weil conjecture. One of the key ideas turbing episodes in which he would launch into a he used came from a paper by R. A. Rankin [Rankin], monologue in German, even though she under- which is about the classical theory of modular stood no German. “He would just go on as if I forms and of which Grothendieck was unaware. As wasn’t there,” she said. “It was kind of scary.” He John Tate put it, “For the proof of the last Weil con- was frugal, sometimes compulsively so: one time, jecture, you needed another ingredient that was to avoid throwing away three quarts of leftover cof- more classical. That was Grothendieck’s blind spot.” fee, he drank it—with the predictable result that When Bumby and Grothendieck turned up at he got quite sick afterward. Bumby said she believes the IHÉS that summer, among the visitors was that his speaking German and his extreme frugal- William Messing of the University of Minnesota. ity may have been connected psychologically to the Messing first met Grothendieck in 1966, when as hardships he endured as a child, especially the a graduate student at Princeton he attended a se- time when he lived with his mother in the intern- ries of lectures Grothendieck gave at Haverford ment camps. College. These lectures made a deep impression on Grothendieck may have been experiencing some Messing, and Grothendieck became his informal kind of psychological breakdown, and Bumby today thesis adviser. In 1970 Messing joined the Survival wonders whether she should have sought treatment group at the Montreal meeting at which it was for him. Whether he would have submitted to such founded. The following year, while Grothendieck treatment is unclear. They parted ways not long was visiting Kingston University in Ontario, he and after their son, John, was born in the fall of 1973. Af- Messing made a car trip to visit Alex Jameson, an ter spending some time in Paris, Bumby moved back Indian activist living on a reservation near Buffalo, to the United States. She married a mathematician New York. Grothendieck was pursuing a quixotic who was a widower, Richard Bumby of Rutgers Uni- hope of helping the Indians resolve a dispute over versity, and they raised John and Richard’s two a land treaty. daughters. John exhibited a good deal of mathemat- In the summer of 1973 Messing was living in a ical talent and was a mathematics major at Harvard small studio in the Ormaille, the housing complex University. He recently finished his Ph.D. in statistics for IHÉS visitors. Excitement was bubbling among at Rutgers. Grothendieck has had no contact with the mathematicians over Deligne’s breakthrough. this son. “Grothendieck was with Justine,” Messing recalled. During the early 1970s, Grothendieck’s interests “They came for dinner, and Katz and I spent the were very far from those of the mathematical world evening explaining to Grothendieck the main new he had left behind. But that world intruded in a ma- and different things in Deligne’s proof of the last jor way in the summer of 1973, when, at a conference of the Weil conjectures. He was pretty excited.” At in honor of W. V. D. Hodge in Cambridge, England, the same time, Grothendieck expressed disap- Pierre Deligne presented a series of lectures about pointment that the proof bypassed the question of his proof of the last and most stubborn of the Weil whether or not the standard conjectures were true. conjectures. Grothendieck’s former student Luc “I think he certainly would have been very happy Illusie was at the conference and wrote to him with to have proven [all the Weil conjectures] himself,” the news. Wanting to know more, Grothendieck, ac- Katz remarked. “But in his mind, the Weil conjec- companied by Bumby, visited the IHÉS in July 1973. tures were important because they were the tip of In 1959 Bernard Dwork proved by p-adic meth- the iceberg reflecting some fundamental struc- ods the first Weil conjecture (which says that the tures in mathematics that he wanted to discover zeta function of a variety over a finite field is a ra- and develop.” A proof of the standard conjectures tional function). Grothendieck’s 1964 l-adic proof would reveal that structure in a much deeper way. of this conjecture was more general and intro- Later during that visit Grothendieck also met duced his “formalism of the six operations.” In with Deligne to discuss the proof. Deligne recalled the 1960s Grothendieck also proved the second that Grothendieck was not as interested in the Weil conjecture (which says that the zeta function proof as he would have been had it used the the- of a variety satisfies a functional equation). Find- ory of motives. “If I had done it using motives, he ing a way to prove the last Weil conjecture (some- would have been very interested, because it would times called the “congruence Riemann Hypothesis”) have meant the theory of motives had been devel- was a major inspiration for much of his work. He oped,” Deligne remarked. “Since the proof used a formulated what he called the “standard conjec- trick, he did not care.” In trying to develop the the- tures,” which, if they could be proved, would imply ory of motives, Grothendieck had run into a major all of the Weil conjectures. The standard conjectures technical difficulty. “The most serious problem were also formulated independently around the was that, for his idea of motives to work, one had same time by Enrico Bombieri. To this day, the to be able to construct enough algebraic cycles,” standard conjectures remain inaccessible. Deligne Deligne explained. “I think he tried very hard and

NOVEMBER 2004 NOTICES OF THE AMS 1203 he failed. And since then nobody has been able to involving braids, Ladegaillerie made a little model succeed.” According to Deligne, this technical ob- using some string and a small plank with holes. This stacle to developing the theory of motives was made Grothendieck laugh out of sheer delight: “At probably far more frustrating to Grothendieck than that moment, he was like a child before a wizard his inability to prove the last Weil conjecture. who performed a trick, and he told me: ‘I would never have thought of doing that’.” A Distant Voice Grothendieck lived an ascetic, unconventional life in an old house without electricity in Villecun, [J]’ai quitté “le grand monde” mathé- about thirty-five miles outside of Montpellier. matique en 1970.…Après quelques an- Ladegaillerie remembered seeing Justine Bumby nées de militantisme anti-militariste et and her baby there, though she soon was gone. écologique, style “révolution culturelle”, Many friends, acquaintances, and students went dont tu as sans doute eu quelque écho to visit Grothendieck, including people from ici et là, je disparais pratiquement de la the ecology movement. In 1974 the leader of a circulation, perdu dans une université group of Buddhist missionaries from Japan came de province Dieu sait où. La rumeur dit to visit Grothendieck, and after that many other que je passe mon temps à garder des adherents of Buddhism passed through his moutons et à forer des puits. La vérité home (R&S, page 759). Once, after being host to a est qu’à part beaucoup d’autres occu- Buddhist monk whose travel documents were not pations, j’allais bravement, comme tout in order, Grothendieck became the first person in le monde, faire mes cours à la Fac (c’était France ever to be charged under an obscure 1949 là mon peu original gagne-pain, et ça law against “gratuitously lodging and feeding a l’est encore aujourd’hui). stranger in an irregular situation” (R&S, page 53). As someone who had been stateless all his life, I left “the great world” of mathematics Grothendieck was outraged at the charge and tried in 1970.…After several years of anti- to launch a campaign against it. He even traveled military and ecological militancy, “cul- to Paris to speak about it at a Bourbaki seminar. His tural revolution”-style, of which you campaign made headlines in French national news- have no doubt heard an echo here and papers. Ultimately he paid a fine and received a sus- there, I just about disappeared from pended sentence. circulation, lost in a university in some It was around this time that Grothendieck province, God knows where. Rumor has learned to drive. He had an ancient Citroën of a it that I pass my time tending sheep model called 2CV and known informally as a deux and drilling wells. The truth is that apart chevaux. One of his students, Jean Malgoire, now from many other occupations, I bravely a maître des conferences at Montpellier, recalled a went, like everyone else, to teach my terrifying journey through a torrential rainstorm courses in the Department (this was the with Grothendieck at the wheel. In addition to way I originally earned my bread, and being a poor driver, Grothendieck was far more oc- it’s the same today). cupied with the discourse he was presenting to his passengers than with the condition of the road. —Récoltes et Semailles, page L3 “I was sure we would never get there alive!” Mal- When Grothendieck came to the Université de goire said. “I understood then that Alexandre had Montpellier in 1973, Yves Ladegaillerie, then twenty- a very special relationship with reality.… Rather five years old, was a maître des conferences there, than adapting to what was real, he believed that re- having finished his doctorate at the Institut Henri ality would adapt itself to him.” One time, while dri- Poincaré in Paris three years earlier. Grothendieck ving a moped, Grothendieck collided head-on with proposed that Ladegaillerie do a thèse d’état with an automobile. According to Ladegaillerie, he had him in topology and spent a great deal of time ini- turned his eyes from the road to get an apricot out tiating the younger mathematician into his vision of a bag that was behind him. Although he had a and methods. In a brief memoir about Grothen- leg fracture serious enough to require surgery, he dieck, Ladegaillerie wrote: “I had had as professors requested acupuncture as the only anesthetic. He in Paris some of the great mathematicians of the agreed to take antibiotics only when the surgeon day, from Schwartz to Cartan, but Grothendieck was told him that the alternative was to amputate the completely different, an extra-terrestrial. Rather broken leg. than translating things into another language, he At the Université de Montpellier, Grothendieck thought and spoke directly in the language of mod- had a regular faculty position and taught at all lev- ern structural mathematics, to whose creation he els. Although the students were not as strong as the had contributed greatly” [Ladegaillerie]. Once, in ones he had had in Paris, he nevertheless poured a order to verify a certain algebraic computation great deal of energy, enthusiasm, and patience into

1204 NOTICES OF THE AMS VOLUME 51, NUMBER 10 his teaching. He had an unconventional teaching employs mathematicians and scientists to do re- style. For an examination involving polyhedra, he search. Based at universities or research institu- had students submit paper-and-glue models, much tions, CNRS positions usually entail no teaching. In to the dismay of those who had to shepherd the the 1950s, before he went to the IHÉS, Grothendieck exam papers through the grading process. One had held a CNRS position. In the 1970s he applied person who took undergraduate courses from him to reenter the CNRS but was turned down. At that at Montpellier is Susan Holmes, now a statistician time, Michel Raynaud of the Université de Paris- at Stanford University. “I found him very inspiring, Orsay was on the committee of mathematicians that as he was both unconventional and kind to the reviewed CNRS applications. Raynaud said the students, who really didn’t understand at all that CNRS administration had been hesitant to take he was a great mathematician,” she recalled. He Grothendieck on, arguing that it was unclear showed up in the worn-out attire of a hippie and whether he would continue doing mathematics. distributed his homegrown organic apples in class. The committee could not contradict this argument, “He definitely did not explain in a linear fashion and the application was turned down. suited to undergraduates, but his teaching was When Grothendieck reapplied to the CNRS in very inspiring, and one got the impression of some 1984, his application was once again controversial. wonderful mysterious ‘big picture’,” Holmes said. Jean-Pierre Bourguignon, now director of the IHÉS, Grothendieck was never one for reading as a way chaired the committee in charge of reviewing ap- to learn about and understand mathematics. Talk- plications in mathematics, among which was ing to others had always been his primary way of Grothendieck’s. According to Bourguignon, in the finding out what was going on in the field. His de- handwritten letter required for the application, parture from the intense, stimulating atmosphere Grothendieck listed several tasks he would not of the IHÉS, where oral exchanges were his pri- perform, such as supervising research students. Be- mary mode of communication about mathematics, cause CNRS contracts obligate researchers to per- was an enormous change for him. Compared with form some of these tasks, this letter was viewed the pace he kept during the 1960s, Grothendieck’s by the CNRS administration as proof of Grothen- later mathematical work was sporadic. Although he dieck’s ineligibility. Bourguignon said he tried to had several Ph.D. students at Montpellier, he did get Grothendieck to amend his application so that not establish anything like the thriving school he it did not state explicitly all the tasks he refused had headed at the IHÉS. Some of Grothendieck’s for- to carry out, but Grothendieck would not budge. mer students and colleagues from his Paris days After considerable effort on the part of several traveled to Montpellier to visit him. The most fre- people, Grothendieck was eventually put on a spe- quent of these visitors was Deligne, who during the cial kind of position, called a position asterisquée, 1970s was the main person keeping Grothendieck that was acceptable to him and to the CNRS. The aware of new developments. CNRS did not actually hire him but was in charge At Montpellier, Grothendieck did not have a only of paying his salary, and he retained his uni- seminar that met consistently. He formed a small versity affiliation. So for his last few years at Mont- working group with Ladegaillerie, Malgoire, and pellier before his retirement in 1988, Grothendieck some of his other students, but according to Lade- did not teach and spent less and less time at the gaillerie it never really got off the ground. During university. 1980–81, he ran a seminar, whose sole attendee The mathematical part of Grothendieck’s 1984 was Malgoire, on relations between Galois groups application to the CNRS was the now-famous man- and fundamental groups. This is the subject of his uscript Esquisse d’un Programme. In it he outlines, 1,300-page manuscript La Longue Marche à Tra- in a somewhat mysterious but nevertheless pene- vers la Théorie de Galois (The Long March through trating and visionary fashion, a new area that he Galois Theory), completed in 1981. Grothendieck called “anabelian algebraic geometry”. He also did not publish La Longue Marche, but through muses on the inadequacy of general topology and Malgoire’s efforts part of it was published in 1995 presents ideas for a renewal in the form of what by the Université de Montpellier [Marche]. There was he called “tame topology”. The Esquisse also con- also a small working seminar in which Ladegaillerie tains his ideas about dessins d’enfants, which he gave some lectures on William Thurston’s work on originally developed in order to have a simple way Teichmüller spaces, which stimulated Grothen- of explaining to students some notions in alge- dieck’s interest in this subject. braic geometry and which have since spawned a By the 1980s Grothendieck felt he had done all good deal of research. Grothendieck sent the Es- he could in trying to motivate the less-than- quisse to mathematicians who he thought might enthusiastic students at Montpellier and decided take an interest in it, and the manuscript circulated to apply for a position as a researcher in the Cen- unpublished for several years. tre National de la Recherche Scientifique (CNRS). Leila Schneps of the Université de Paris VI read The CNRS, an agency of the French government, the Esquisse in 1991. Before that she had identified

NOVEMBER 2004 NOTICES OF THE AMS 1205 Grothendieck with the foundational works of EGA reproached him for having cut the work into and SGA, and she found that the Esquisse was publishable pieces. As Ladegaillerie put it, Grothen- completely different. “It was a wild expression of dieck tried to enlist him in his “fight against the mathematical imagination,” she recalled. “I loved establishment,” but Ladegaillerie resisted, believ- it. I was bowled over, and I wanted to start work- ing that such a fight was unreasonable and unjus- ing on it right away.” She became an enthusiastic tified. evangelist for the research program described in “Despite such disagreements, we have stayed the Esquisse, and she and others have made a good friends, with highs and lows,” Ladegaillerie said. Of deal of progress on it. She said, “Some of it doesn’t his work with Grothendieck, Ladegaillerie said, “It even seem to make sense at first, but then you was fascinating to work with a genius. I don’t like work for two years, and you go back and look at this word, but for Grothendieck there is no other it, and you say, ‘He knew this’.” She edited a book word possible.…It was fascinating, but it was also on dessins d’enfants, which appeared in 1994 frightening, because the man was not ordinary.” [Schneps1], and in 1995 she and Pierre Lochak, Memories of working on mathematics with Grothen- also of the Université de Paris VI, organized a con- dieck long into the night, by the light of a kerosene ference around the Esquisse. The Esquisse appeared lamp, are “the greatest memories of my life as a for the first time in print in the proceedings of that mathematician.” conference [Schneps2]. Aside from the Esquisse and La Longue Marche, Reaping and Sowing Grothendieck wrote at least one other mathematical Il y a beaucoup de choses dans Récoltes work during the 1980s. À la Poursuite des Champs et Semailles, et les uns et les autres y ver- (Pursuing Stacks), which runs 1,500 pages, began as ront sans doute beaucoup de choses a letter to Daniel Quillen of the University of Oxford. différentes: un voyage à la découverte Completed in 1983, it sketches Grothendieck’s d’un passé; une méditation sur l’exis- vision of a synthesis of homotopical algebra, ho- tence; un tableau de moeurs d’un milieu mological algebra, and topos theory. À la Poursuite et d’une époque (ou le tableau du glisse- des Champs circulated widely among mathemati- ment insidieux et implacable d’une cians but was never published. Although its topic is époque à une autre…); une enquête (qua- mathematics, the style of À la Poursuite des Champs siment policière par moments, et en is completely different from the style of his earlier d’autres frisant le roman de cape et mathematical writings. It was written as a sort of d’épée dans les bas-fonds de la mé- “log book” on a mathematical voyage of discovery, gapolis mathématique…); une vaste di- which includes all the false starts, wrong turns, and vagation mathématique (qui sèmera sudden inspirations that characterize mathemati- plus d’un…); un traité pratique de psy- cal discovery but that are typically omitted from chanalyse appliquée (ou, au choix, un written mathematical works. When nonmathemati- livre de “psychanalyse-fiction”); une cal matters drew his attention, they become part of panégyrique de la connaissance de soi; the log book too: for example, À la Poursuite des “Mes confessions”; un journal intime; Champs contains a digression about the birth of one une psychologie de la découverte et de of his grandchildren. During the 1990s he wrote a la création; un réquisitoire (impitoyable, 2,000-page mathematical work on the foundations comme il se doit…), voire un règlement of homotopy theory called Les Dérivateurs, which de comptes dans “le beau monde math- he gave to Malgoire in 1995 and which is now being ématique” (et sans faire de cadeaux). made available on the Web [Deriv]. While he was at Montpellier, Grothendieck’s un- There are many things in Récoltes et compromising, “anti-establishment” bent seems Semailles, and different people will no to have become more pronounced. After Ladegail- doubt see in it many different things: lerie’s thesis was finished, Grothendieck wrote to a voyage to the discovery of a past; a med- Springer-Verlag to suggest that it be published in itation on existence; a portrait of the morals the Lecture Notes series. He was outraged when he of a milieu and of an era (or the portrait received the reply that the series no longer pub- of an insidious and relentless sliding of lished theses. The thesis was submitted for publi- one era into another…); an inquest (almost cation anyway, with the predictable result that it detective-style at times, and at others was rejected. According to Ladegaillerie, Grothen- bordering on cloak-and-dagger fiction set dieck wrote letters about this to colleagues, in an in the underbelly of the mathematical effort to build a campaign against Springer. Lade- megapolis); a vast mathematical ramble gaillerie decided to publish his thesis in the form (which will leave more than one reader in of several papers rather than as a whole, and the the dust…); a practical treatise on applied main part appeared in Topology. Grothendieck psychology (or, if you like, a book of

1206 NOTICES OF THE AMS VOLUME 51, NUMBER 10 “psychoanalytic-fiction”); a panegyric on One of the reasons for the complexity of the self-knowledge; “My confessions”; a pri- structure of Récoltes et Semailles, and for its spon- vate diary; a psychology of discovery and taneity, is that Grothendieck wrote it without a creation; an indictment (pitiless, as is fit- definite plan in mind. He started writing it as an ting), even a settling of scores in “the world introduction to À la Poursuite des Champs, which of elite mathematics” (and without any was to mark his return to making a serious in- gifts). vestment of time and energy in doing and pub- lishing mathematics. The introduction was intended —Récoltes et Semailles, page L2 to explain the new spirit of his research, which Between June 1983 and February 1986, Grothen- would not focus on the precise and exhaustive dieck wrote Récoltes et Semailles: Réflexions et té- foundation-building of his earlier work, but would moignage sur un passé de mathématicien (Reapings take readers on a “voyage of discovery” of new and Sowings: Reflections and testimony about the mathematical worlds. Grothendieck envisioned Ré- past of a mathematician). It is a work that defies coltes et Semailles as the first volume of a series categorization. The title suggests a memoir, but Ré- called Réflexions, which would contain his thoughts coltes et Semailles is something more and less than and reflections on things mathematical and oth- a memoir. It is more, in that it contains not only erwise. The second volume was to have been À la memories of events in his life but also analyses, Poursuite des Champs, and La Longue Marche à often quite deep and minute, of the moral and psy- Travers la Théorie de Galois and Esquisse d’un chological significance of those events and his at- Programme were also to have been included. tempts to reconcile their meaning with his view of In the first part of Récoltes et Semailles, which he himself and the world. These analyses lead him into called “Fatuité et Renouvellement” (“Complacency philosophical musings about the role of discovery and Renewal”), Grothendieck does a lot of soul- and creativity in mathematics and in life more gen- searching about the mathematical community in erally. At the same time, Récoltes et Semailles is which he worked. The welcoming atmosphere he something less than a memoir, in that it does not encountered upon joining that community as a attempt a systematic and comprehensive account newcomer in 1948 began to disappear, he says, as mathematicians came to use their reputations to set of events in Grothendieck’s life. He is not writing themselves in a superior position. Mathematics be- for future biographers or historians, but primarily came a way to gain power, and the elite mathemati- for himself. Récoltes et Semailles is a probing ex- cians of the day became smug, feared figures who amination of matters closest to his heart. He brings used that power to discourage and disdain when to this work the searching curiosity, the same drive it served their interests. He ruefully recounts some to get to the very bottom of things, that he brought instances in which he himself displayed attitudes of to his mathematics. The result is a dense, multi- conceit and haughtiness and realizes that these layered work that reveals a great and sometimes attitudes had coalesced into a “sportive” or com- terrifying mind carrying out the difficult work of petitive approach to mathematics that had begun trying to understand itself and the world. to hamper his ability to open himself to the beauty Needless to say, Récoltes et Semailles is not an of mathematical things. easy read, and Grothendieck makes a lot of de- It was after writing “Fatuité et Renouvellement” mands on his readers. Much of it has a quotidian that he was suddenly struck by “the insidious re- feel, and in some parts he is obviously setting down ality of a Burial of my oeuvre and at the same time his thoughts as they evolve from one day to the next. of my person, which suddenly imposed itself on me, As a result, within the space of a page there can be with an irresistible force and with this very name, sudden and sometimes disconcerting changes in ‘The Burial’, on [April 19, 1984].” (R&S, page L8). mood and topic. The organization is complex. The On that date he began writing what eventually main text is divided into numbered sections, each became a three-part series called “L’Enterrement” with its own carefully chosen and evocative title. (“The Burial”), comprising more than one thou- Within each section there are cross-references to sand pages. In it he strongly attacks some of his other sections, as well as numerous footnotes, former students and colleagues, whom he believes some quite long and substantial, and sometimes tried to “bury” his work and his style of mathe- even footnotes to the footnotes. The wide-ranging matics by pilfering his ideas and not according vocabulary presents special challenges for those proper credit to him. He also champions the work whose native language is not French, as does his pen- of Zoghman Mebkhout, who during the 1970s de- chant for using colloquialisms, some of them rather veloped some of Grothendieck’s ideas and whose vulgar. Through it all Grothendieck writes with work Grothendieck believes was unfairly margin- great care, insight, and clarity, in a pungent and ar- alized and ignored. “L’Enterrement” presents resting style. He often succeeds at describing things six mathematical areas, or “construction sites” that at first glance would seem quite ineffable. (“chantiers” ), that he says were abandoned when

NOVEMBER 2004 NOTICES OF THE AMS 1207 he left the IHÉS in 1970 and that he believes his mathematical colleagues. Despite Grothendieck’s students should have developed. Throughout “L’En- intention to publish it, the original French-language terrement” he closely analyzes his relationship version of Récoltes et Semailles has never appeared with Pierre Deligne, the most brilliant of all of his in print, as the strong attacks it contains could be students and the one with whom he had the clos- deemed libelous. Nevertheless, it has circulated est mathematical affinity. widely. Copies can be found on bookshelves in math- “L’Enterrement (II) ou La Clef du Yin et du Yang” ematicians’ offices all over the world, especially in (“The Burial (II) or the Key to Yin and France, and in some libraries in uni- Yang”) is rather different from the versities and mathematics institutes. other two parts of “L’Enterrement” Historian of science Alain Herreman in being less directly concerned with of the Université de Rennes has un- the investigation of the “burial”. This dertaken an effort to post on the second part, which Grothendieck Web html files containing the entire notes is the most personal and deep- French original, and partial transla- est part of Récoltes et Semailles, con- tions into English, Russian, and Span- stitutes a wide-ranging meditation ish have appeared there too [R&S]. on diverse themes such as creativity, A Japanese translation of a large por- intuition, violence, conflict, and the tion of Récoltes et Semailles was pre- self. He uses the “yin-yang” dialectic pared by Yu¯ichi Tsuji, who knew to analyze different styles of doing Grothendieck through the Survival mathematics, concluding that his group, and was published in the own style is fundamentally “yin”, or 1990s by Gendaisu¯gakusha, a math- feminine. This style is captured in ematics publisher. According to one especially evocative section called Michel Waldschmidt of the Univer- “La mer qui monte…” (“The rising sité de Paris VI, who was president sea…”). He likens his approach to of the Société Mathématique de mathematics to a sea: “The sea ad- France (SMF) during 2001–04, the vances imperceptibly and without Grothendieck in a society considered, during his pres- sound, nothing seems to happen and photograph from the 1950s. idency, the question of whether to nothing is disturbed, the water is so publish Récoltes et Semailles. The far off one hardly hears it. But it ends up surround- question raised strong opinions both for and against, ing the stubborn substance, which little by little be- Waldschmidt said, and ultimately the SMF decided comes a peninsula, then an island, then an islet, which against publication. itself is submerged, as if dissolved by the ocean stretch- Many mathematicians, especially some of ing away as far as the eye can see” (R&S, page 553). Grothendieck’s former students, were shocked and In “L’Enterrement” he pursues some of the hurt by the accusations in Récoltes et Semailles. One themes established in “Fatuité et Renouvellement” of them, Luc Illusie of the Université de Paris-Orsay, concerning the competitive, snobbish attitudes of recalled that he talked to another former student, the upper crust of the mathematical world. For Jean-Louis Verdier, about whether they should try example, he notes that much of his work in math- to discuss the accusations with Grothendieck. Ac- ematics was marked by an “attitude of service”: cording to Illusie, Verdier, who died in 1989, felt service to the mathematical community of writing that Grothendieck’s state of mind was such that clear and complete expositions that make funda- there was no sound basis for discussion. But, Illusie mental and foundational ideas widely accessible. said, “I thought, ‘It is not possible that Grothendieck Although he candidly admits that his own conceit has become like that. I will try to reason and to dis- sometimes led him into elitist attitudes, he says that cuss with him. Maybe I will agree with him on some he never lost this spontaneous sense of service, “ser- points that he is right and on others he is not vice to all those who leaped with me into a common right.’ Eventually, we settled the material points, but adventure” (R&S, page 630, (*)). He believes that the nothing really emerged, and he remained convinced mathematical community lost this sense of ser- that everyone was against him.” vice as personal aggrandizement and the develop- In Récoltes et Semailles Grothendieck says that, ment of an exclusionary elite became the order of after he left the mathematical world in 1970, his the day. He also decries the devaluation of vision style of doing mathematics was held in contempt and intuition in favor of technical mastery. and that many of the paths he had broken went Apart from “Fatuité et Renouvellement” and the undeveloped. It is true that after that time, re- three parts that make up “L’Enterrement”, Récoltes search in algebraic geometry began to shift, mix- et Semailles has two introductory volumes, as well ing the highly general approach that characterized as an appendix to “La Clef du Yin et du Yang”. About his work with investigation of specific problems. two hundred copies were sent out to his Deligne’s proof of the Weil conjecture, which was

1208 NOTICES OF THE AMS VOLUME 51, NUMBER 10 very much in the spirit of Grothendieck but which also incorporated many new ideas, was one of the Contents of Récoltes et Semailles great advances of the 1970s. Along with develop- Présentation des Themes—ou Prélude en Quatre ments in the theory of D-modules and Deligne’s Mouvements mixed Hodge theory, greater attention began to •En guise d’avant-propos (January 1986: pages A1–A6) be paid to more specific problems, such as the •Promenade à travers une oeuvre—ou l’enfant et la mère (Jan- classification theory of varieties and questions uary 1986: pages P1–P65 ) about low-dimensional varieties. Also, after the •Epilogue en post-scriptum—ou contexte et préalables d’un Antwerp meeting of 1972, collaborations grew débat (February 1986: pages L44–156) between algebraic geometry and representation Lettre—Introduction theory, leading to advances in the theory of auto- •Une lettre, May–June 1985: pages L1–L43 morphic forms and the Langlands program. As •Table des matières (pages T1–T10) Illusie put it, all these developments show that •Introduction (March 1984: sections 1–5, pages i–xi) there has been “quite a natural balance between •Introduction (May–June 1985: sections 6–10, pages xi–xxii) general theory and the study of specific examples Première Partie: Fatuité et Renouvellement at great length, to enrich the theory itself.” (June 1983, February 1984: pages 1–171) Récoltes et Semailles also contains the accusation Deuxième Partie: L’Enterrement I, ou la Robe de l’Empereur that Grothendieck’s work was not always properly de Chine credited. Indeed, his work was so well known and (April–June 1984: pages 173–420) fundamental that credit was not always specifi- Troisième Partie: L’Enterrement II, ou la Clef du Yin et du cally accorded to him. “It is true that everybody Yang knew he had invented motives, for instance, or (September 1984–January 1985: pages 421–774) l-adic cohomology, and so there was no need to quote his name every time one used them,” re- Quatrième Partie: L’Enterrement III, ou les Quatre Opérations marked Jean-Pierre Serre. “His name was rarely (February 1985–June 1985: pages 775–1252) mentioned because of that. On the other hand, it Les Portes sur l’Univers (Appendice à la Clef du Yin et du was well known that it was due to him. Nobody was Yang) saying that it was due to someone else.” Serre (March–April 1986: pages PU1–PU127) noted that Grothendieck’s complaining about lack of credit is in sharp contrast to his behavior dur- Although the accusations of a “burial” have gen- ing the 1960s, when he shared his ideas with great erated a good deal of notoriety, there is much more generosity and in some cases attached other peo- to Récoltes et Semailles. Those who have read ple’s names to ideas he himself had come up with. beyond those parts have been deeply touched by “It was sad to read Récoltes et Semailles because of the work’s beauty and insights. Grothendieck’s that,” Serre said. critique of how the highly competitive atmosphere Even granting that there was a shift away from of the mathematical world stifles creativity and Grothendieck’s style of mathematics and that credit renewal of the field resonated with many. In Récoltes was not always specifically accorded to him, it is et Semailles Grothendieck puts the highest value a long leap from there to the deliberate “burial” on the innocent, childlike curiosity that gives birth that he asserts took place. “In retrospect, very few to the creative impulse, and he mourns the way it mathematical ideas have been as widely used as is trampled on by competitiveness and the desire Grothendieck’s,” said Illusie. “Everybody who is for power and prestige. doing algebraic or arithmetic geometry now uses “I am one of quite probably a minority who Grothendieck’s language, ideas, theorems, and so think that Récoltes et Semailles is a miraculous on. So when you think one second, it is completely document,” said William Messing. “That is not to ridiculous that he suggested that he could have say that there are not parts that are excessive and been buried.” There is no question that mathe- have aspects of what might be referred to as para- matics suffered a great loss when Grothendieck noia. But it’s very striking that the person who halted his research career in 1970. But mathemat- created EGA and SGA would write in such a style. ics did not stop; others continued to work, fol- The systematic and soul-searching aspect is of a lowing their own ideas and interests. In February piece with his approach to mathematics. Those 1986, after receiving a copy of Récoltes et Semailles, who have really read it—as opposed to looking at Serre wrote to Grothendieck: “You are surprised five pages of negative comments—tend to think of and indignant that your former students did not it as an extraordinary document.” continue the work that you had undertaken and largely completed. But you do not ask the most ob- Lightness Descending vious question, the one every reader expects you to answer: and you, why did you abandon the work [A]ujourd’hui je ne suis plus, comme in question?” [Corr]. naguère, le prisonnier de tâches

NOVEMBER 2004 NOTICES OF THE AMS 1209 interminables, qui si souvent m’avaient journal K-Theory dedicated to Grothendieck). The interdit de m’élancer dans l’inconnu, Festschrift seems to have been an attempt to make mathématique ou non. Le temps des amends with Grothendieck and to show that he had tâches pour moi est révolu. Si l’âge m’a not been “buried”, as he asserted in Récoltes et apporté quelque chose, c’est d’être plus Semailles. Some of the people contributing papers léger. were among those he had most heavily criticized. When the Festschrift appeared in 1990, Illusie, who Today I am no longer, as I once was, the was one of the editors, sent a copy to Grothendieck, prisoner of interminable tasks, which so whose reaction was extremely bitter. In a letter to often prevented me from leaping into Illusie, he objected strongly to the brief foreword the unknown, mathematical or other- of the volume and also to the fact that he had wise. The time of tasks for me is over. not been told that the volume would appear. He If age has brought me anything, it is said his work had been used like “confetti,” like lightness. bright, worthless bits one throws into the air to give the pretense of happiness and celebration —Esquisse d’un Programme while ignoring the malaise underneath. Grothen- dieck submitted this letter for publication in the “[T]he ethics of the scientific profession (espe- Bulletin de la Société Mathématique de France. When cially among mathematicians) have degraded to the SMF told him that the Bulletin carries only such a degree that pure and simple theft between mathematics articles but that the letter could colleagues (especially at the expense of those who appear instead in the SMF Gazette, Grothendieck have no position of power to defend themselves) refused. The letter was never published. has almost become the general rule and is in any After he retired, Grothendieck spent little time case tolerated by all, even in the most flagrant and at the Université de Montpellier, though he con- iniquitous cases.” So wrote Grothendieck in an tinued to live in the area, in a village called Les April 19, 1988, letter to the Royal Swedish Acad- Aumettes. At this time, Ladegaillerie said, Grothen- emy of Sciences in which he declined the 1988 dieck seemed to be going through a deep spiritual Crafoord Prize. He also sent to the academy the crisis and wrote strange letters “that made us fear introductory volumes of Récoltes et Semailles. the worst about his condition.” During 1987–88, The academy had awarded the prize of around Grothendieck wrote La Clef des Songes ou Dialogue US$200,000 to him and Pierre Deligne. Grothen- avec le Bon Dieu (The Key to Dreams or Dialogue dieck’s letter became widely known when it was with the Good Lord ), which expresses his convic- published in Le Monde on May 4, 1988 [LeMonde]. tion that God exists and that He speaks to people To play into the game of accepting prizes and hon- through their dreams. It also contains a good deal ors, Grothendieck wrote, would be to validate “a of material about Grothendieck’s early life. La Clef spirit and an evolution in the scientific world that des Songes runs about three hundred pages and is I see as profoundly unhealthy, and condemned to accompanied by another five hundred pages of disappear soon, so suicidal is it, spiritually as well notes. According to a lecture given in the summer as intellectually and materially.” Evidently these sen- of 2004 by Winfried Scharlau of the Universität timents resonated with many readers of Le Monde. Münster, Grothendieck subsumed La Clef des One of the newspaper’s editors told Jean-Pierre Songes under a collection of works that he called Bourguignon that the paper had received more re- Méditations and that included the material making actions to Grothendieck’s letter than to any other up Réflexions, as well as a poetical work called preceding it and that most of the letters registered “Eloge de l’Inceste” (“The Eulogy to Incest”). Nei- approval that finally a scientist had recognized ther that work nor La Clef des Songes was ever how corrupt the scientific milieu had become. News widely distributed. of the letter appeared in other magazines and Many of Grothendieck’s friends and colleagues newspapers, and it was avidly discussed within became aware of his increasing preoccupation with the mathematical community. An English transla- spiritual matters when they received “La Lettre de tion was published in the Mathematical Intelli- la Bonne Nouvelle” (“The Letter of Good News”), gencer [Intell], and a short item appeared in the which is dated January 26, 1990, and which he Notices [Notices]. sent to about two hundred fifty people. The letter The same year in which he turned down the states: “You are part of a group of two to three thou- Crafoord Prize, Grothendieck retired from the Uni- sand people, personally known to me, whom God versité de Montpellier at the age of sixty. Also that destines for a great mission: That of announcing year, six mathematicians decided to assemble and preparing the ‘New Age’ (or Age of Libera- a collection of articles as a “Festschrift” on the tion…), which will commence on the ‘Day of Truth’, occasion of Grothendieck’s sixtieth birthday 14 October 1996.” He says that God manifested [Festschrift] (there was also a special issue of the Himself to him for the first time in 1986 and

1210 NOTICES OF THE AMS VOLUME 51, NUMBER 10 communicated to him through dreams. He also typed into TEX a large chunk of the handwritten La describes encounters with a deity named Flora, Longue Marche à Travers la Théorie de Galois. They who imparts revelations but also cruelly tests his have also started a website, the Grothendieck Cir- faith. Although the content of the letter is baffling, cle, which contains a wealth of material about the way it is written is perfectly lucid. Three months Grothendieck, his life, and his work [Circle]. later Grothendieck sent a “correction”, stating that he was no longer certain of the truth of the The Dancing Star revelations described in “La Lettre de la Bonne Nouvelle”. He writes: “That I was the victim of a Ich sage euch: man muß noch Chaos in mystification by one of more ‘spirits’ (among which sich haben, um einen tanzenden Stern my limited capacity could not distinguish), invested gebären zu können. Ich sage euch: ihr with prodigious powers over my body and in my habt noch Chaos in euch. psyche, I no longer have the least doubt.” Together, the two letters impart an impression of deep dis- I tell you: one must have chaos inside, turbance and suffering. to give birth to a dancing star. I tell you: In July 1990 Grothendieck asked Malgoire to you have yet chaos in you. take possession of all of his mathematical papers, including books, preprints, correspondence, and —Friedrich Nietzsche, Also sprach manuscripts in various states of preparation. Zarathustra Grothendieck wanted to “lighten” himself of many The work of Alexandre Grothendieck has had a things, as Malgoire put it. He burned a huge amount profound impact on modern mathematics and, of material, most of it nonmathematical, including more broadly, ranks among the most important ad- letters that his parents had exchanged in the 1930s. vances in human knowledge during the twentieth He showed Malgoire a 200-liter oil drum filled with century. The stature of Grothendieck can be com- cinders and estimated he had destroyed a total of pared to that of, for example, Albert Einstein. Each 25,000 pages. Grothendieck also left many papers of them opened revolutionary new perspectives and other items, including his mother’s death mask, that transformed the terrain of exploration, and with a friend named Yolande Levine, to whom he each sought fundamental, unifying connections had been very close for the preceding decade. He among phenomena. Grothendieck’s propensity for then disappeared into the Pyrenées to live in com- plete isolation. A small number of people knew investigating how mathematical objects behave where he was, and he instructed them not to for- relative to one another echoes the relativistic view- ward any mail that arrived for him at the univer- point proposed by Einstein. Grothendieck’s work sity. Malgoire said that even today, close to fifteen also has parallels with another great twentieth- years after Grothendieck went into seclusion, the century advance, that of quantum mechanics, which university still gets a great deal of correspondence turned conventional notions upside down by re- addressed to him. In 1995 Grothendieck formally placing point particles by “probability clouds”. conferred the legal rights to his mathematical “[T]hese ‘probability clouds’, replacing the reas- papers to Malgoire. suring material particles of before, remind me Grothendieck has had very little contact with strangely of the elusive ‘open neighborhoods’ that mathematicians in the past fifteen years. Among populate the toposes, like evanescent phantoms, the few who have seen him are Leila Schneps and to surround the imaginary ‘points’,” he wrote (R&S, Pierre Lochak, who met him in the mid-1990s. They page P60). told him about the progress made on the program Yet, as extraordinary as Grothendieck’s achieve- he had outlined in the Esquisse d’un Programme, ments are, he traced his creative capacity to some- and he was surprised to learn that people were still thing rather humble: the naive, avid curiosity of a interested in his work. He had developed a strong child. “Discovery is the privilege of the child,” he interest in physics but expressed frustration with wrote in Récoltes et Semailles (page 1), “the child what he felt was a lack of rigor in that field. Lochak who has no fear of being once again wrong, of and Schneps exchanged some letters with him and looking like an idiot, of not being serious, of also sent him some books on physics that he had not doing things like everyone else.” For the work asked for. In one letter he asked a disarmingly of discovery and creation, Grothendieck saw simple question: What is a meter? His letters began intellectual endowment and technical power as to swing between warm friendliness and cold sus- secondary to the child’s simple thirst to know and picion, and eventually he severed all contact with understand. This child is inside each of us, though them. Although the friendship with Grothendieck it may be marginalized, neglected, or drowned out. could not be sustained, Lochak and Schneps retain “Each of us can rediscover what discovery and a fervent admiration and a deep attachment to the creation are, and no one can invent them” (R&S, man and his work. Together they painstakingly page 2).

NOVEMBER 2004 NOTICES OF THE AMS 1211 One aspect of this childlike curiosity is a scrupu- [LeMonde] Lettre à l’Académie Royale des Sciences de lous fidelity to truth. Grothendieck taught his stu- Suède: Le mathématicien français Alexandre Grothen- dents an important discipline when writing about dieck refuse le prix Crafoord, Le Monde, May 4, 1988. [Marche] A. GROTHENDIECK, La Longue Marche à Travers mathematics: never say anything false. Statements la Théorie de Galois, volume 1, edited and with a that were almost or essentially true were not per- foreword by Jean Malgoire, Université Montpellier II, mitted. It was acceptable to be vague, but when one Département des Sciences Mathématiques, 1995. gives precise details, one must say only things that [Notices] Crafoord Prize recipients named, Notices Amer. are true. Indeed, Grothendieck’s life has been a Math. Soc. (July/August 1988), 811–812. constant search for truth. From his mathematical [R&S] A. GROTHENDIECK, Récoltes et Semailles: Réflexions et work through Récoltes et Semailles and even “La Let- témoignages sur un passé de mathématicien, Univer- sité des Sciences et Techniques du Languedoc, Mont- tre de la Bonne Nouvelle”, Grothendieck wrote with pellier, et Centre National de la Recherche Scientifique, the unblinking honesty of a child. He spoke the 1986. (Parts available in the original French at truth—his truth, as he perceived it. Even when he http:// mapage.noos.fr/recoltesetsemailles/. made factual mistakes or was misled by incorrect Partial translations are available in English at assumptions, he presented candidly what was in http://www.fermentmagazine.org/home5.html, his mind. He has never tried to hide who he is and in Russian at http://elenakosilova.narod.ru/ what he thinks. studia/groth.htm, and in Spanish at http:// kolmogorov.unex.es/navarro/res.) Grothendieck’s search for truth took him to the [Rankin] R. A. RANKIN, Contributions to the theory of Ra- very roots of mathematical ideas and to the far manujan’s function τ(n) and similar arithmetical func- reaches of human psychological perception. He ∞ s tions. I. The zeros of the function n=1 τ(n)/n on the has had a long journey. “In his solitary retirement line Rs =13/2. II. The order of the Fourier coefficients in the Pyrenées, Alexandre Grothendieck has the of integral modular forms, Proc. Cambridge Philos. Soc. right to rest after all he has been through,” wrote 35 (1939), 351–372. Yves Ladegaillerie [Ladegaillerie]. “He deserves our [Schneps1] The Grothendieck Theory of Dessins d’Enfants (L. Schneps, ed.), London Math. Soc. Lecture Note Ser., admiration and our respect but, above all, in think- vol. 200, Cambridge University Press, 1994. ing of what we owe him, we must leave him in [Schneps2] Geometric Galois Actions (L. Schneps and peace.” P. Lochak, eds.), London Math. Soc. Lecture Note Ser., vols. 242 and 243, Cambridge University Press, 1997. References [Vietnam] A. GROTHENDIECK, La vie mathématique en [Circle] The Grothendieck Circle, http://www. République Democratique du Vietnam, text of a grothendieck-circle.org. lecture presented in Paris on December 20, 1967. [Corr] Correspondence Grothendieck-Serre. Société Math- Unpublished. ématique de France, 2001. (Bilingual French-English Acknowledgments edition, AMS, 2003.) The assistance of the following individuals is [Deriv] Les Dérivateurs, by Alexandre Grothendieck, edited gratefully acknowledged: Norbert A’Campo, James by M. Künzer, J. Malgoire, and G. Maltsiniotis. Avail- Arthur, Michael Artin, Hyman Bass, Armand Borel, able at http://www.math.jussieu.fr/~maltsin/ Jean-Pierre Bourguignon, Felix Browder, Justine groth/Derivateurs.html. Bumby, Richard Bumby, Pierre Cartier, Pierre [Aubin] D. AUBIN, A Cultural History of Catastrophes and Deligne, Edward Effros, Gerd Faltings, Momota Chaos: Around the “Institut des Hautes Études Scien- Ganguli, Robin Hartshorne, Alain Herreman, tifiques,” France, doctoral thesis, Princeton Univer- Friedrich Hirzebruch, Susan Holmes, Chaim Honig, sity, 1998. Luc Illusie, Nicholas Katz, Dieter Kotschick, Klaus [Festschrift] The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexan- Künnemann, Yves Ladegaillerie, Pierre Lochak, der Grothendieck, Volumes I–III (P. Cartier, L. Illusie, Andy Magid, Jean Malgoire, Bernard Malgrange, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet, eds.), Barry Mazur, Colin McLarty, Vikram Mehta, William Progress in Mathematics, vol. 87, Birkhäuser Boston, Messing, Shigeyuki Morita, David Mumford, Jose Inc., Boston, MA, 1990. Barros Neto, Arthur Ogus, Michel Raynaud, Paulo [Herreman] A. HERREMAN, Découvrir et transmettre: La Ribenboim, David Ruelle, Winfried Scharlau, Leila dimension collective des mathématiques dans Schneps, Jean-Pierre Serre, John Tate, Karin Tate, Récoltes et Semailles d’Alexandre Grothendieck, Jacques Tits, and Michel Waldschmidt. Prépublications de l’IHÉS, 2000. Available at http:// Photographs used in this article are courtesy of name.math.univ-rennes1.fr/alain.herreman/. Michael Artin, the Tata Institute, Karin Tate, and [Infeld] L. INFELD, Whom the Gods Love. The Story of Évariste the website of the Grothendieck Circle (http:// Galois, Whittlesey House, New York, 1948. www.grothendieck-circle.org). [Intell] English translation of Grothendieck’s letter de- clining the 1988 Crafoord Prize, Math. Intelligencer 11 (1989). [Ladegaillerie] Y. LADEGAILLERIE, Alexandre Grothendieck après 1970. Personal communication.

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