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Who Is ? Winfried Scharlau

This article is a translation of the article “Wer ist Alexander Grothendieck?”, originally appeared in German in the Annual Report 2006 of the Research Institute in Oberwolfach, Germany (copyright © Math- ematisches Forschungsinstitut Oberwolfach 2007). The article is based on the Oberwolfach Lecture delivered by Winfried Scharlau in 2006. With the permission of the author and of the institute, the article was translated into English by D. Kotschick, Ludwig-Maximilians-Universität München, with the assistance of Allyn Jackson, Notices deputy editor (translation copyright © D. Kotschick 2008). Scharlau is writing a three-volume biography, Wer ist Alexander Grothendieck?: Anarchie, Mathematik, Spiritual- ität. The first volume, which primarily treats the lives of Grothendieck’s parents, has appeared as a self-published book and is available on the Web http://www.scharlau-online.de/ag_1.html.

or a , it is not hard to give though an exhaustive and satisfactory answer is an answer to the question posed in the surely impossible. title of this lecture: Grothendieck is one of the important of the Grothendieck’s Parents second half of the twentieth century, to One can only understand the life of Grothendieck— F if one can understand it at all—if one knows whom we owe in particular a complete rebuilding of algebraic . This systematic rebuilding about the life of his parents. I report briefly on the permitted the solution of deep number-theoretic life of his father. He was from a Jewish family, was (probably) problems, among them the final step in the proof called Alexander Schapiro, and was born in 1890 of the by Deligne, the proof of in Novozybkov in the border area of , White the Mordell Conjecture by Faltings, and the solu- Russia, and Ukraine. At the age of fifteen he was tion of Fermat’s Last Problem by Wiles. However, recruited by anarchist groups that were fighting this lecture is concerned not with Grothendieck’s against the tsarist regime; in 1905 Russia was in mathematics but with his very unusual life on the uproar. After two years of fierce battles, he and fringes of human society. In particular, there is, all of his comrades were taken as prisoners. All on the one hand, the question of why at the age were sentenced to death, and all but Schapiro were of forty-two Grothendieck first of all resigned his executed; he was led to the execution plaza every professorship at the Institut des Hautes Etudes Sci- day for three weeks before being pardoned because entifiques (IHES); then withdrew from mathematics of his youth and sentenced to life in prison, where completely; and finally broke off all connections he spent the next ten years. In the confusion of to his colleagues, students, acquaintances, friends, the October Revolution and the First World War, as well as his own family, to live as a hermit in he escaped and immediately joined the anarchist an unknown place. On the other hand, one would peasant army of the Ukrainian General Machno. He like to know what has occupied this restless and married a Jewish woman called Rachil and with her creative spirit since his withdrawal from math- fathered a son named Dodek, but carried on a busy ematics. I will try to pursue both questions, even love life outside of marriage. Again, after fierce bat- tles, he was taken prisoner by the Bolsheviks and Winfried Scharlau is emeritus professor of mathematics sentenced to death. Probably during an attempt to at Universität Münster. His email address is scharlau@ escape (or in an assassination attempt?), he lost math.uni-muenster.de. his left arm. With the of various women and

930 Notices of the AMS Volume 55, Number 8 comrades in arms, he managed to flee to western Europe. He went into hiding first in , then in . From this on he lived with forged documents under the name of Alexander Tanaroff. For many years he earned his living as a street photographer. Around the year 1924 he returned to Berlin, where he met Hanka Grothendieck. He introduced himself to her husband, Alf Raddatz, with the words, “I will steal your wife.” And so it happened. In March 1928 Alexander Grothendieck, the son of Alexander Tanaroff and Hanka Grothendieck, was born. For five years the “family”, consisting of these three people, together with Hanka’s daughter Maidi (Frode Raddatz) from her marriage, lived in the so-called “Scheunenvier- tel” in Berlin, where for some time they operated a photography studio. After the National Socialists came to power, the situation in Germany became Scharlau. Winfried of Archive too dangerous for the Jewish Tanaroff, and he Grothendieck around 1936 in the garden of the Heydorn’s moved back to Paris. Hanka Grothendieck decided house in -Blankenese. to follow her companion as soon as possible. Around New Year’s 1933–34 she placed her five- Child to Mathematician to Hermit year-old son in a foster home with the family of the As mentioned before, Alexander Grothendieck was Hamburg pastor Wilhelm Heydorn. (Like all people born on March 28, 1928, as Alexander Raddatz close to Grothendieck, Heydorn was a very remark- in Berlin and lived there with his parents and his able personality about whom a 450-page biography half-sister, Maidi, for the first six years of his life. was published.1) Hanka then went to as well. From early 1934 to the end of April 1939 he lived Both she and Tanaroff took part in the Spanish together with other foster children in the home Civil War, not fighting actively but in supporting of Wilhelm and Dagmar Heydorn in Hamburg- roles. After the defeat of the Republicans, both Blankenese, where he initially attended elementary returned to France. Certainly with the start of school and then the Gymnasium. Except for the the Second World War, Tanaroff was in danger in years at the IHES, this may have been the only France as well—as an alumnus of the Spanish War, period in his life when he lived in “normal circum- as a Jew, and as an illegal alien. He was interned stances”. In early 1939 his situation in Germany in the infamous camp Le Vernet, extradited to the became too dangerous, particularly since his Germans in 1942, and transported to Auschwitz. foster parents opposed the Nazi regime and had Under the name Alexandre Tanaroff, he appears to contend with the possibility that their foster on the list of victims of the Shoah. Throughout children would be taken away from them. In such his adventurous life he had known only one goal: a situation, his Jewish heritage would have come the fight for freedom and self-determination of all to light. So at the end of April 1939, Alexander people. For that he would put his whole existence was sent to his parents in France. It is unknown on the line at any time. where he spent the next few months; he was prob- The life of Hanka Grothendieck was similarly ably with his mother in Nîmes. After the start of dramatic, although the drama is internal the war, Hanka, as a citizen of an enemy nation, than external. Her great goal was to be a writer. was interned together with her son in the camp Although she had remarkable talent, she ultimately Rieucros near Mende. Alexander was able to attend failed. She too lived a life on the fringe. For reasons school there and sometimes had private tutoring of and time, I will not go into any detail in as well. Around 1942 Alexander somehow arrived this lecture. in Le Chambon sur Lignon. This small town in the I now come to Alexander Grothendieck himself. Massif Central was a center of resistance against I first would like to report on the outline of his the Nazis; thousands of refugees were hidden biography. For further information, I refer to the there, given false papers and food vouchers, and very informative article by Allyn Jackson.2 then smuggled across the Swiss border. Thousands 1​Wilhelm Heydorn, Nur Mensch Sein!: Lebenserrinerun- were saved from deportation to German death gen, I. Groschek and R. Hering, editors, Dölling und Galitz camps. The crucial person in this collective resis- Verlag, Hamburg, 1999. tance was the Protestant clergyman André Trocmé, 2 ​Allyn Jackson, “Comme appelé néant—As if systematically traveled to French camps and summoned from the void: The life of Alexandre Gro- tried in particular to get out as many children as thendieck”, Notices, October 2004 and November 2004. possible. Perhaps this is how Grothendieck came to

September 2008 Notices of the AMS 931 Dieudonné, who like Schwartz was teaching in Nancy. From this time on, Grothendieck came into the mathematical mainstream, and it is generally known what he did and achieved during the next twenty years. So that I can keep my account short, I refer for details to Jackson and the literature quoted there. To begin with, Schwartz gave Grothendieck a paper to read that he had just written with Dieudonné, which ended with a list of fourteen unsolved problems. After a few months, Grothen- dieck had solved all of them. Try to visualize the situation: On one side, Schwartz, who had just received a and was at the of his scientific career, and on the other side the unknown student from the provinces, who had a rather inadequate and unorthodox education. Grothendieck was awarded a Ph.D. for his work Archive of Winfried Scharlau, 2006. Scharlau, Winfried of Archive on topological vector spaces and stuck with that Grothendieck’s house in Villecun, where he lived from for a while. He went to for two years 1973 to 1979. and then to Kansas. Largely under the influence of Serre, he turned to beginning in Le Chambon. (The great story of Le Chambon has 1954. The most spectacular new result in the field been the subject of many documentaries, novels, was the theorem of Riemann-Roch-Hirzebruch. and movies.3) In Le Chambon, Grothendieck was Within two years of the awakening of his inter- able to attend the Collège Cévénol, an international est in algebraic geometry, Grothendieck found a private school founded by Trocmé, which from far-reaching generalization and a completely new the beginning was dedicated to nonviolence and proof, which has remained possibly his most sig- the solidarity of all people—not popular ideas in a nificant single achievement in mathematics. time of war. In 1945 Alexander completed there his The next fifteen years of Grothendieck’s sci- rather chaotic schooling with the baccalauréat. entific work were dedicated to the rebuilding of It is probably by accident that Grothendieck algebraic geometry. In 1958 he was appointed ended up in after the war. Perhaps to the IHES, which had just been founded by his mother had found work there. He received the businessman Léon Motchane. Together with a modest scholarship and started his studies of Dieudonné, his former teacher and now colleague mathematics. It soon turned out that the univer- at the IHES, Grothendieck began working on the sity did not have much to offer him, and he had to Eléments de Géométrie Algébrique (EGA) and held rely largely on self-study. Since the time he was in the legendary Séminaire de Géométrie Algébrique school, he had planned to out what concepts (SGA). Many mathematicians who were close to like length and volume really mean, and accord- him in those days emphasize that his way of doing ing to his own reports, he basically developed mathematics was completely singular: He was not the theory of the Lebesgue integral. In the fall of interested in the solving of difficult or famous 1948 he went to Paris for a year, where he met problems, especially if it had to be done “by force”, the most important French mathematicians of the but his goal was to achieve such a deep and com- day, both the active middle generation of Henri plete understanding of the underlying structures Cartan, André Weil, , , that the solutions of such problems would fall out and , as well as the younger “on their own”. generation, his contemporaries Jean-Pierre Serre, During his twelve years at the IHES, Grothen- Pierre Cartier, François Bruhat, and . dieck led an outwardly bourgeois life: He married Originally, Grothendieck had hoped to be able to Mireille Dufour and had three children with her, quickly get a Ph.D. for his work on the “Lebesgue born in 1959, 1961, and 1965. Earlier he had had integral”. Of course, he now found out that to a a son from a previous relationship. However, the large extent he had simply rediscovered known education of his children was unconventional; things. Nevertheless, he wanted to stick with this at least temporarily, they did not attend public subject, so, following the advice of Cartan and schools. Grothendieck thought that finding one’s Weil, on June 20, 1949, he wrote a letter to Jean own way was more important than a formal educa- 3​See for example Philip P. Hallie, Lest Innocent Blood Be tion. His home was hospitable, and he sometimes Shed: The Story of the Village of Le Chambon and How took in people in need for weeks at a time. Goodness Happened There. This book has been published In his IHES seminar, Grothendieck surrounded in several editions by Harper & Row, New York. himself with a of outstanding students to

932 Notices of the AMS Volume 55, Number 8 whom he generously gave his ideas for them to throughout the world. But pursue. At the same time, more and more conflicts his attachment to Buddhism developed with Motchane, the founder and direc- did not last. From around tor of the institute. Grothendieck’s relationship 1980 Grothendieck gravi- with his colleague René Thom was not without tated toward Christian mys- complications either. At the 1966 International tical and esoteric ideas. More Congress of Mathematicians, Grothendieck was and more often there were awarded the Fields Medal. He was at the pinnacle periods of serious psycho- of his fame. In May 1968 the student revolution logical problems; presum- erupted in Paris and made a deep impression on ably things were surfacing Grothendieck that would change his life decisively. in Grothendieck that had al- I will return to this later on. ways been there inside him. In the year 1970 an event occurred that Grothen- For a while he identified with dieck later would often call “the great turning the stigmatized Catholic nun point” (“le grand tournant” ). He gave up his job Marthe Robin, who claimed at the IHES and started turning away from math- to have lived for thirty years ematics, although he did for a few years more on the Eucharist alone. A have positions at the Collège de France and at the kind of angel, whom he calls , Orsay. He turned to the prob- Flora or Lucifera depend- lems of environmental protection and ecology, ing on whether he wants to Scharlau. Winfried of Archive he supported the antinuclear power movement, emphasize her divine or her Grothendieck in the 1980s. and he fought against military buildup, especially devilish side, plays an impor- of nuclear weapons, and the military-industrial tant role in his thoughts. For complex. To pursue these goals actively, he and a nights on end Grothendieck played chorals on the number of comrades founded the group Survivre, piano and sang along. Finally in 1988 a period of which later was also known as Survivre et Vivre. excessive fasting almost cost him his life. Appar- For about three years he devoted all his energy to ently he wanted to force God to reveal Himself. He this movement. wanted to consciously experience the moment of At the same time, his family life dissolved. death and outdo Jesus’s forty-day fast. In 1999 he predicted that the Final Judgment was imminent On a “propaganda trip” for Survivre through and that a golden age would start thereafter. Later America, he met Justine Skalba, with whom he on, these delusions extended to nonreligious areas, lived in France in a commune he founded and with including, for example, questions of cosmology. whom he had a son. For a time, his children from There can be no doubt that, at least since the his first marriage also lived in this commune. In end of the 1980s, his life has been dominated for 1973 there was another decisive change: he left long periods by delusions and that he would have Paris and moved to the tiny village of Villecun, on needed urgent medical and psychiatric help. the southern edge of the Cévennes, about sixty In the summer of 1991 Grothendieck suddenly kilometers northwest of Montpellier. Since that left his residence in Les Aumettes and withdrew time Grothendieck has lived only in small villages to a place that remained unknown for a long or hamlets. More and more he broke off contact time. He refuses almost all contact and seems to with former colleagues, students, acquaintances, be occupied daily with writing down his medita- friends, and his own family; his relationship with tions. Justine Skalba also ended after two years. Soon after his move to Villecun he took a job as a professor at the University of Montpellier, The Great Turning Point although he taught there only irregularly. For Certainly there is a multitude of reasons that con- months or even years on end he stopped doing tributed to the “great turning point” of 1970. They mathematics altogether before starting to are complementary and contradictory. Some seem down obsessively his mathematical “meditations” obvious, while others are buried in the depths of of hundreds or even thousands of pages. In the Grothendieck’s existence and his past and can last few years before his retirement in 1988, he hardly be brought to light. Much—in fact a great again held a research position in the CNRS (Centre deal—remains a riddle. One does not have the National de la Recherche Scientifique), though he impression that one understands or that one can did research only sporadically. understand his radical actions. Grothendieck’s col- From 1974 Grothendieck turned to Buddhism; leagues, pupils, and friends must all have asked several times he was visited by Japanese monks themselves what the causes of this step could from the order Nipponzan Myohoji (in English the have been. I want to emphasize that the following name translates roughly as “Japanese community attempt at an explanation is based on my personal of the wonderful lotus sutra”), which preaches views. Another person might well interpret the strict nonviolence and erects peace pagodas facts at hand in a different way.

September 2008 Notices of the AMS 933 It has often been said that the decisive reason I have the impression that, despite for Grothendieck’s break with the IHES was the fact your well known energy, you were quite that a part of the IHES budget (about 5 percent) simply tired of the enormous work you came from the French defense ministry. This could had undertaken. not be reconciled with Grothendieck’s pacifist, anarchist, and radical leftist political convictions. By letter and in conversations, Serre later con- Grothendieck himself has often confirmed this firmed this view. Considering that, as a colleague version. But I think this explanation is not the put it, Grothendieck had done mathematics twelve whole truth and is not particularly plausible. It is hours a day, seven days a week, and twelve months no doubt correct that the financial support by the a year for twenty years, one can only agree. But a defense ministry was not acceptable to Grothen- question remains. Many scholars (or artists) give dieck. But there had been many discussions of up a project they have started because their cre- this topic between the leadership of the IHES and ativity and strength dissipate. But they do remain the faculty, in which the permanent professors respected members of the community. had largely supported Grothendieck. With good Serre speaks simply of “tiredness”. Similar views will surely the problem could have been solved. have been expressed by others, who, however, see In fact, the relationship between the founder and deeper reasons, including “disappointment”. Ac- director, Motchane, and Grothendieck was already cording to an oral communication from Helmut completely dysfunctional at this time. The reasons Koch, Igor Shafarevich thought that it was a disas- that led to the break with the IHES and in particular trous decision on the part of Grothendieck to begin with Motchane have been analyzed by David Aubin working on the Eléments de Géométrie Algébrique. in his Ph.D. thesis. We refer to his investigation He should have used his creativity on the “great for the details of this conflict.4 In this conflict, it problems” and not on a complete construction of seems much more plausible that cause and effect a gigantic theory. Comments in this direction have were reversed: The dispute over the budget gave also been made by the physicist , a colleague of Grothendieck’s at the IHES: After a Motchane the possibility (finally) to get rid of superhuman effort, Grothendieck had to admit Grothendieck, whom he regarded as a paranoid that he would never be able to complete the oeuvre troublemaker. Perhaps Motchane had no choice, he had begun. It was as if he had set his mind on for if Grothendieck had stayed, probably two of building a cathedral with his own hands. When the the other permanent professors, Thom and Louis walls were two meters high, he had to stop. Michel, would have left. It seems to me that all three—Serre, Shafarev- That the conflict over the IHES budget is insuf- ich, and Ruelle—are making important points but ficient as grounds for Grothendieck’s departure missing the decisive one. They might explain why also follows from the fact that the conflict does not Grothendieck gave up mathematics, but not why explain why he turned away from mathematics and he changed his whole life, why he withdrew from from the mathematical community. Throughout human society. No doubt this event, originating the world he could have found places to work that in the core of his personality, is much more deep- were consonant with his moral and political convic- seated and more emotional than giving up one’s tions. He would have been welcome everywhere; job or withdrawing from scientific research. Again he could have continued his research, and many it seems natural to think that cause and effect of his students would have followed him. might have been reversed: Because, for whatever In his commentary on the meditation Récoltes et reason, Grothendieck could no longer live in the Semailles, Jean-Pierre Serre talks about the decisive society he had lived in from 1950 to 1970, he had point. He says that Grothendieck never had the to leave mathematics as well. urge to do what perhaps the whole world expected Grothendieck’s old friend and colleague Cartier5 of him, namely, to give a coherent explanation in has undertaken a superficial attempt to the 1,600 pages of this treatise. Serre says: explain Grothendieck’s decision. He does not dis- But you do not ask the most obvious count the importance of the financing of the IHES, question, the one every reader expects and he sees the crises in Grothendieck’s math- ematical work, but he also sees that the rupture in you to answer: why did you yourself Grothendieck’s life had deeper reasons: abandon the work in question? I would like to try to analyze the rea- A few lines later Serre attempts to answer his own sons for this abrupt end to a career so question: astonishing and fertile at the age of 42. The reason given was that he had 4David Aubin, “A cultural of catastrophes and chaos: Around the Institut des Hautes Etudes Scientifiques, 5​Pierre Cartier, “A mad day’s work: From Grothendieck France, 1958–1980”, Ph.D. thesis, , to Connes and Kontsevich”, Bulletin of the AMS (N.S.) 38 1998. (2001), 389–408.

934 Notices of the AMS Volume 55, Number 8 discovered that the Ministry of Defense Grothendieck. To explain this, we must recall some had been subsidizing the institute.…In of his political activities. order to understand the vehemence of These activities have to be seen against the Grothendieck’s reaction, one must take backdrop of his own life and the lives of his - account of his past and the political ents. Cartier is certainly right when he emphasizes situation of the time. He is the son of a that Grothendieck was always conscious of the - militant anarchist who had devoted his ample of his parents. His father had fought all his life to revolution. This was a father of life for freedom and self-determination and against whom he had very little direct knowl- the powerful in this world. Grothendieck’s sympa- edge; he knew him mostly through thy was always with the poor, the persecuted, the his mother’s adulation. He lived as an oppressed, those in the shadows, and he always outcast throughout his entire child- held leftist, liberal, and possibly even anarchist hood and was a “displaced person” political convictions. But for many years these for many years.…He had always been convictions were not expressed in political actions. uncomfortable frequenting the “better” In the late 1950s and early 1960s, he opposed the places and felt more at ease among the French war in Algeria as a matter of course, but in poor, even the impoverished. The soli- contrast with many of his closest colleagues such darity of outcasts had created in him a as Schwartz, Chevalley, Samuel, or Cartier, he did strong feeling of compassion. He lived not participate in public protests. At least he took his principles, and his home was always the matter seriously enough to consider emigrating wide open to “stray cats”. In the end he to the . came to consider Bures a gilded cage Grothendieck’s political commitment became that kept him away from real life. To publicly visible in the summer of 1966, when he refused to travel to to receive the Fields this reason he added a failure of nerve, Medal at the International Congress of Mathemati- a doubt as to the value of scientific cians (ICM). This was his protest against the per- activity. Starting in 1957 at a Bourbaki secution and imprisonment of the Russian writers Congress, he confided his doubts to me Yuri Daniel and Andrei Siniavsky. This action at- and told me that he was considering tracted a lot of attention. Some years later it was activities other than mathematics.6 One held very much against Grothendieck by orthodox should perhaps add the effect of a well- communists and socialists who played a big role known “Nobel syndrome”. [After being in the student movement. awarded the Fields Medal in 1966], His next political action was a trip, made at his when he was laboring over the last (de- own initiative, to Hanoi and North during cisive) stages of the proof of the Weil the last three weeks of November 1967 in the mid- conjectures and perhaps beginning to dle of the . He gave a series of lectures perceive that Deligne would be needed about this trip, in Paris on December 20, 1967, to complete in 1974 the program he and later in other places. In addition to reporting had set for himself, and perhaps yield- on scientific and personal contacts and on the ing to the pernicious view that sets 40 lectures he had given, he described the destruction as the age when mathematical creativity the war had wrought, the bomb attacks, material ceases, he may have believed that he deprivations, and the faith of the Vietnamese had passed his peak and that thence- people in their own future. Though he cautiously forth he would be able only to repeat criticized the indoctrination in dialectical material- himself with less effectiveness. ism and the overwhelming regimentation of public life, every sentence of his report spoke of his deep The mood of the time also had a strong sympathy for the struggles of the Vietnamese influence. The disaster that had been people to build a new society under difficult cir- the second Vietnam War from 1963 cumstances and to support public education and to 1972 had awakened many con- scholarship. sciences. Grothendieck’s spontaneous trip to Vietnam was probably typical of him in that it was an “in- When it was said above that the dispute over dividualistic” action. Many French intellectuals, the budget was not the really decisive point, this including well-known figures such as Jean-Paul Sar- was not meant to imply that political, or more tre, had long taken an interest in Indochina. Many precisely socio-political, reasons played no role. mathematicians were also committed to this cause, On the contrary, they were of great importance to none more than Grothendieck’s advisor, Laurent 6​In a short note about the spiritual landmarks of his life, Schwartz. In his autobiography Schwartz talked Grothendieck mentions for 1957 “calling and unfaithful- at length about the fight for an independent Viet- ness” (footnote of . S.). nam and about his love for that country and its

September 2008 Notices of the AMS 935 developed doubts about whether his own scholarly occupation was the right path and even wondered whether it was irresponsible to engage in such activity. This happened to many academics and intellectuals at the time, particularly in France; it was simply the “zeitgeist” (which is stronger than everything else). But Grothendieck reacted to this with his characteristic forcefulness, rigor, and recklessness against others and himself, and perhaps also with obstinacy and with a sense of mission (though maybe he was just more clairvoy- ant than those around him). Archive of Winfried Scharlau. Winfried of Archive Considering this background, there can be no An example of a handwritten mathematical doubt that Grothendieck was incapable of compro- diagram drawn by Grothendieck. mising on the question of the IHES receiving a part of its budget from the French defense ministry. In 7 inhabitants. He negotiated with many influential his meditations (especially in Récoltes et Semailles), politicians, among them the Vietnamese prime he has said repeatedly that this issue had to lead to minister Phan Van Dong and Ho Chi Minh him- a break with the IHES, and his closest colleagues, self. such as Serre and Deligne, have confirmed how Schwartz was also one of the initiators of the decisive this issue was. Nevertheless, it must have Russel tribunals, held in 1967 in and happened as I explained above: Motchane had Roskilde. He mentions as his comrades in arms plenty of reasons to try to get rid of Grothendieck, many well-known French and other mathemati- and Grothendieck had already reached the great cians, among them Jean-Pierre Kahane, Bernard turning point internally, even if he was not yet Malgrange, Pierre Cartier, André Martineau, and conscious of this. , but Grothendieck’s name is men- In trying to understand how the “great turning tioned only in passing. In those years Grothendi- point” happened, one has to take into account eck had no interest in actions organized together Grothendieck’s mental state, which already by then with other people; he did not participate, perhaps must have been unstable and perhaps sometimes he was even indifferent. All those who observed out of control. This would not have been appar- his political actions testified to his good will and ent in his interactions with his colleagues and his serious and honest intentions but at the same students, even though Cartier hints at it. But in time ascribed to him an unbelievable naiveté and closer interaction, a deep personality disorder is even ignorance. (I can barely believe the reports clearly visible. This is not the place to go further that Grothendieck at the time did not know what into this subject. NATO really was.) We now come to the second aspect of the “great In May of 1968 the student revolution that was turning point”, the departure from mathematics. soon to envelop the whole Western world broke It seems to me that this process has a “negative” out in Paris. There were strikes and demonstra- and a “positive” aspect. The “negative” has already tions that sometimes bordered on riots; there were been mentioned: fatigue and disappointment, as demands for radical changes in university cur- seen and described by Serre, Shafarevich, Ruelle, ricula, for abolishment of examinations, for self- and also Cartier. The “positive” aspect is that determined learning, and for equal representation Grothendieck found an occupation that seemed to for faculty, staff, and students; in extreme cases, him to be more important than mathematics and to there were even demands for the destruction of which he devoted himself for the next two or three computing centers and academic departments that years with the same energy and drive that he had were suspected of conducting military research. previously lavished on mathematics. This occupa- It was a true “cultural revolution” (which today tion was environmental protection in the widest seems like a distant past that has already faded). sense of the term, the rising ecological movement In several places in Grothendieck’s meditations (the word “ecology” existed at that time only as written in later times, he alludes to the deep im- the name of a subdiscipline of ), resistance pression these events made on him. He was con- to atomic energy, the struggle against militarism vinced of the seriousness of the young people’s and the arms race, support for a new society and revolution and certain that Western civilization a “cultural revolution”—all in all, a movement that and capitalism were headed for a deep crisis; he followed many ideals of the generation of 1968. 7​Laurent Schwartz, Un Mathématicien aux Prises avec Apparently this movement, the new goals, and the le Siècle, Odile Jacob, Paris, 1997; English translation, A new ideals impressed Grothendieck so much that Mathematician Grappling with His Century, Birkhäuser, he became a committed follower. At this time he Basel, 2001. did not yet consciously abandon mathematics; he

936 Notices of the AMS Volume 55, Number 8 did not yet speak about it being for him like “a three conditions: In addition to his mathematical trip through the desert”, as he would often say lecture, he wanted to give a lecture of equal dura- later on. But he had found something that, at least tion about his ecological goals, and this additional for the moment, was more important to him than lecture was to be advertised and published in the mathematics. same way as the scientific lectures. The organizers Grothendieck’s main activity with regard to of the meeting accepted these conditions, and so these goals was the foundation of a group initially the participants found among the materials that called Survivre and later on Survivre et Vivre: they received at the beginning of the meeting a text based on the earlier lecture in Orsay. Apparently SURVIVRE Grothendieck’s charismatic personality impressed Mouvement international pour la survie a whole group of mostly young mathematicians so de l’espèce humaine much that Survivre was founded spontaneously by An international and interprofessional the group. One of the most active members was movement for the survival of humanity Gordon Edwards, who was then a Ph.D. student under Grothendieck’s friend from student days, The goals of this movement are summarized in , and who later became a leader its first bulletin of August 1970 (this quotation is in the antinuclear movement in Canada. from the English edition of the bulletin, as is the The first bulletin of the group contains a list of slogan just above): members running to twenty-five people, eighteen of whom were mathematicians. One has to suspect To fight for the survival of the human that most of them were “recruited” by Grothen- species and of life in general, threat- dieck. His mother-in-law, Julienne Dufour, was ened as they are by the ecological dis- among them, as was his son Serge, then seventeen equilibrium created by contemporary years old. Grothendieck soon succeeded in winning industrial society (pollution, waste, over other prominent mathematicians, particularly devastation of natural resources), as those who had always been part of leftist move- well as by military conflicts and the ments. For the second issue of the bulletin, Claude threat of military conflicts. Chevalley was the directeur de publication and a On balance, it seems that Grothendieck’s in- member of the editorial committee. About a year volvement in this movement, in complete contrast later, joined the French editorial with his work in mathematics, has been without committee. lasting effect and ended in defeat. That is how it As far as I know, a total of nineteen issues of the would have seemed to him, but perhaps that point bulletin of the movement, totaling approximately of view is too superficial: It is certain that during seven hundred pages, were published between his Survivre et Vivre period, Grothendieck deeply 1970 and 1975. There is no doubt that in the early impressed some young people and completely years the main burden of the editorial work rested changed their lives. And maybe the group did on Grothendieck, who surely wrote many of the a contribution to the establishment of the “green” unsigned articles. After his move to Villecun in movement that has firmly taken root in society and 1973, he must have been less involved, if at all. As politics in Europe. is usual with such groups, a tendency to disinte- Shortly after his official letter of resignation, grate started after only a short time; for example, on June 26, 1970, Grothendieck gave a lecture to Samuel left the group in 1973. hundreds of listeners at the University of Paris in It is important to note that at the beginning of Orsay in which he talked about all that had become his period of ecological and antimilitarist activities, important to him: the spread of nuclear weapons, Grothendieck consciously tried to profit from his the arms race, the threat to humanity posed by reputation as a scholar. He was deeply imbued technological progress. He went so far as to call with the truth of the goals of Survivre et Vivre, and mathematical research dangerous because it is without doubt he thought that anyone would have part of this technological progress. The content of to come to the same conclusions when presented this lecture was later circulated in various unof- with appropriate enlightenment and information. ficial manuscripts under titles such as “Respon- He took it for granted that any rational, sensible sabilité du savant dans le monde d’aujourd’hui: Le person would have to agree with the views of the savant et l’appareil militaire” (“The Responsibility group Survivre, so it was natural for him to first of in Today’s World: The and try to convince other mathematicians. And at the the Military Establishment”). beginning, he believed that his efforts would be The foundation of the group Survivre came about successful. during a summer school on algebraic geometry in In Montreal he had convinced some participants July and August 1970 in Montreal. Grothendieck just through the momentum of the early enthu- had been invited to this meeting to deliver lectures siasm. Perhaps it was not difficult to win over on crystalline . He agreed to go under in personal conversations some acquaintances

September 2008 Notices of the AMS 937 who had always been left-wing activists, such as be to the credit of any writer. He does of course Chevalley or Samuel. Others, like Serre or Deligne, have a “genetic predisposition”. His mother had must have been more cautious. But the moment the ambition to be a writer and left behind an of truth came when he tried to attract new con- important literary work, the autobiographical verts through public activism, for example at the novel Eine Frau. His father too saw literature as 1970 ICM in . There he set up an information his real calling, although his lifelong struggle booth and attracted attention through spectacular for the anarchist movement prevented him from actions in the expectation that mathematicians pursuing this vocation. Grothendieck himself first would the group Survivre in droves. As he played with the idea of turning to poetry after the himself summarized, this attempt was an utter proof of the Riemann-Roch theorem. He wrote a failure and surely contributed to his estrangement good deal of occasional verse (in German, French, from the community of mathematicians. After a and English; most of it has been lost), and he has few years of toiling in vain, he must have reached translated poetic texts from German into French. the conclusion that mathematicians and scientists He undertook his first serious poetic attempt in are blind to the dangers threatening human society 1979, when he wrote down the Eloge, about which and do not think and behave rationally. And so more will be said later, and he had further plans Grothendieck withdrew more and more, not just at that time. from mathematics but from the community of To create an overview, we now give a chronologi- mathematicians. survey of Grothendieck’s known “Meditations”, At this point I want to conclude the discussion followed by a few comments about their content. of the reasons that might have led to the “great 1979: L’Eloge de l’Inceste (In Praise of Incest) turning point”. But it seems to me that the deci- (January to July 1979, perhaps lost) sive point has not been touched upon yet: Why 1981: La Longue Marche à Travers la Théorie has Grothendieck withdrawn from human society de Galois (The Long March through ) itself? Thinking about his whole personal life, one (January to June 1981, about 1,600 pages, plus has the impression that, for whatever reason, it has about the same amount of commentary and sup- been impossible for him to maintain a long-term plementary material; unpublished, but since 2004 personal relationship with anybody. Whenever parts have been available on the Internet) such a relationship did not come to an end for 1983: A la Poursuite des Champs (Pursuing purely external reasons, it inevitably led to deep Stacks) (approximately 650 pages, started as a “let- conflicts and usually terrible reproaches, even im- ter” to D. Quillen, unpublished). Associated with precations, on the part of Grothendieck. He could this is an extensive correspondence with Ronnie not live long-term in human society as it exists, Brown and Tim Porter. and therefore he also had to give up mathematics 1984: Esquisse d’un Programme (Sketch of Pro- and activities connected to it. gram) (January 1984) No doubt this is a depressing tally for the life 1983–1985: Récoltes et Semailles: Réflexions of a unique scientist and man. et Témoignage sur un Passé de Mathématicien The Meditations of Grothendieck (Reapings and Sowings: Reflections and Testimony on the Past of a Mathematician) (1,252 pages plus We now turn to the question of what Grothendi- approximately 200 pages of introduction, com- eck did after his withdrawal from society. It goes mentary, and summaries; produced as photocop- without saying that such an active and creative ies; available on the Internet) mind could not remain idle. His main intellectual 1987: La Clef des Songes (The Key to Dreams) occupation clearly was, and still is, writing down (315 pages, unpublished) his “Meditations”, which, as far as is known, cover 1987–88: Notes pour la Clef des Songes (Notes biographical, religious, esoteric, and philosophi- on the Key to Dreams) (691 pages, unpublished); cal themes. I use here Grothendieck’s own word, includes a freestanding work, Les Mutants “Meditations” (sometimes also “Reflections”), 1990: Développements sur la Lettre de la Bonne whereby he means, as he says on many occasions, Nouvelle (Developments on the Letter of Good “meditating” as well as writing. Since the 1960s, News) (82 + 2 pages, unpublished; written Febru- when he spent many hours daily at the typewriter, ary 18–March 15, 1990) he has been used to putting down his thoughts in 1990: Les Dérivateurs (about 2,000 pages, un- writing. (It is natural to conjecture that, from the published, but parts available on the Internet) 1980s onwards, this habit practically became a compulsion.) Surely this considerable number of manuscripts is To begin with, one has to observe that all his not all that Grothendieck wrote during those years. life Grothendieck felt a calling to be a writer and Various eyewitnesses have reported that one day that he is without doubt a master of written ex- (perhaps in 1990 or in early 1991) he burned many pression. His linguistic and stylistic prowess, and manuscripts and perhaps other documents, such above all his creativity in inventing words, would as correspondence, in an old oil barrel. His only

938 Notices of the AMS Volume 55, Number 8 work of fiction, L’Eloge de l’Inceste, possibly fell and clarified and that their form will victim to this deed; it is also possible that a copy have emerged, at least in outline. survives somewhere. It seems best to discuss La Longue Marche à We now make a few comments on some of these Travers la Théorie de Galois (LM) and Esquisse d’un “Meditations” only to show that a more extensive Programme (EP) together, because EP is in a certain analysis and interpretation in a wider context sense a summary of LM. With the EP, Grothendi- would be desirable. eck applied for a position in the CNRS. The text Without doubt the best known of his “Medi- contains a summary of his mathematical thoughts tations” is Récoltes et Semailles, which contains since the early 1970s. It has now been published above all his reckoning with mathematics and the together with an English translation.8 The central mathematical community. Mostly because of his objects of the investigations are the moduli spaces attacks on many colleagues and former pupils, M​(g​, ​n​) of compact Riemann surfaces of genus g​ which seem more or less unjustified, this text has with n​ marked points that had been studied earlier achieved a certain notoriety. The widespread idea by and . Grothendi- that he is “crazy” and “paranoid” is based mostly eck makes a connection to arithmetic objects, in on this text. By now there is an extensive “unof- particular the of the field of ficial” literature about this work (which is easy to rational numbers Q​. To come to grips with the el- research on the Internet), so we will not comment ementary geometric and combinatorial aspects of further on it here. It is hard to say what this text these questions, Grothendieck designs his theory really is: not an autobiography, not a work of fic- of “children’s drawings” (dessins d’enfants). He also tion, but not a scientific work either; in a letter speculates on an “anabelian geometry”. Of all of to German friends, Grothendieck once called it a Grothendieck’s mathematical “meditations”, these “mathematical phantasmagoria”. have certainly had the largest . There are many Grothendieck has said about L’Eloge de L’Inceste papers concerned with these questions, and in the that it is on the one hand his first systematic re- 1990s several workshops were organized around flection of a philosophical nature and on the other these topics. hand a work of fiction—he calls it a “song”. He mentions it occasionally in his other meditations La Clef des Songes and apologizes for the somewhat flamboyant (un I now come to the meditation La Clef des Songes, peu tapageur) title. In his correspondence with which is still largely unknown. As with the others, his German friends, Grothendieck mentions the it is easier to say what it is not than to say what it is. Eloge on several occasions, for the first time on It is not a scholarly work, because it has no clearly August 17, 1979: defined subject and the considerations follow no scholarly methodology. It is not an autobiography, Since the beginning of June, I have although Grothendieck occasionally recounts epi- withdrawn to a solitary hermitage in sodes from his life. And it is certainly not a work the Vaucluse, where nobody knows of fiction in any form, for it has no narrative, no me—maybe I will stay here for a whole plot line, and no characters that could have carried year to “turn in” quietly. At the end of a plot. However, at many points Grothendieck uses July, I finished the first version of the poetic language, and much of it can be understood first song “In Praise of Incest”. At the only in the way in which one understands—or beginning of September, I want to go “absorbs”—poetry: not rationally but emotionally. over it with a friend and then slowly (An example: The only God is silent. And when He up a clean version—it will prob- speaks, it is in such a low voice that nobody ever ably take two to three months, a few understands.) And the text is not a systematic pages a day, after all I have other things analysis of, say, the phenomenon of dreams, for it to do as well.…And so in November or is not about concrete dreams. It is perhaps some- December, I will have it photocopied— thing like a confessional—but what would he be but not at the university,…Two hun- confessing? It is best to let Grothendieck speak for dred copies to begin with—it’s going to himself: “It is the record of a long meditation. A be about 170 pages all in all.… I have meditation that has no aim, in which the thoughts not yet decided 100 percent whether I are left largely to themselves.” will publish the song. Probably . It is 8 surely the most meaningful thing I have The following conference proceedings concern the circle ever done—but then there isn’t much of ideas around LM and EP, in particular the letter to Faltings and the text of EP: The Grothendieck Theory of except mathematics. In any case, I will Dessins d’Enfants, , editor, London Math- at least until the spring before I ematical Society Lecture Notes 200 (1994); Geometric entrust a publisher with the first song. Galois Actions 1 and 2, Leila Schneps and Pierre Lochak, By then I assume that the substance of editors, London Mathematical Society Lecture Notes 242 the next two songs will have ripened and 243 (1997).

September 2008 Notices of the AMS 939 Many people who which prevent their unfolding. He discusses the know Grothendieck important role of “eros” as a decisive creative report that he has power. Furthermore, he discusses three levels “always” been very at which a person develops both in general and interested in dreams. with regard to creative powers: the physical, the But they became the mental-intellectual, and the spiritual level. Spiritu- central theme of his ality is a key concept in Grothendieck’s thoughts, thinking only after not only in La Clef des Songes. He measures all the “great turn- people according to how far they have come in ing point” of 1970. attaining a spiritual life. He also discusses the It appears that he spiritual aspect of his mathematical work. Finally, thoroughly worked he speaks about the many deformations of human- through, for exam- ity that go hand in hand with a loss of spirituality ple, Freud’s Traum- and manifest themselves, for example, in a loss of deutung (The Inter- the feeling for beauty in all areas of life. pretation of Dreams) I personally consider the Notes pour la Clef des and also read other Songes to be the most interesting of Grothendieck’s relevant literature.9 meditations known so far. Originally it was really On the other hand, meant to be remarks about La Clef des Songes. But he does not describe soon an independent text called Les Mutants de- even a single one of his own dreams that veloped out of it. The somewhat strange title “The Archive of Winfried Scharlau. Winfried of Archive had great importance Mutants” (a word that in French too comes from Grothendieck in May 1988. for him, and he says the vocabulary of science fiction) refers to people nothing about how he who differ from “mere mortals” in a spiritual way; analyzed them. in particular, they are ahead of their times. At one Instead of attempting a summary or a table place in the text Grothendieck gives the following of contents of La Clef des Songes, we will con- explanation of this concept (slightly shortened in fine ourselves to mentioning a few key ideas. translation): Grothendieck starts with the statement that there There have been in this century (as is an external being, the “dreamer”, who knows doubtless in other centuries past) a the people and sends them dreams so that they certain number of isolated men who will recognize themselves. Among these dreams seem to my eyes to be “new men”—men there are some that carry particularly important who appear to be “mutants” and who messages. Because of torpor and fear of change, already today, in one way or another, many people do not understand these messages. The dreams are not the result of mental processes prefigure the “man of tomorrow” em- of people; rather, they come from outside. Next bodied in the present; the man in the Grothendieck analyzes the nature of the “dreamer” full sense of the word, who undoubt- and comes to the conclusion that God exists and edly will emerge in the generations to is the dreamer (Le reveur n’est autre que Dieu). He come, in the course of the “post-herd” then discusses the question of how he himself - age, of which the dawn is very close and rived at his belief in God. There is a rather detailed which they tacitly herald. description of the biography of his parents—both For hundreds of pages Grothendieck describes were convinced atheists and anarchists—and he and discusses the lives and works of a total of talks a bit about his childhood and youth. Surely eighteen mutants. It becomes that he sees he is trying to express that, given his biography, it a personal connection between these mutants was by no means obvious that he would find the and himself; for example, he occasionally calls “path to God”, but that this required an impulse himself their heir, or he calls them his elders. from outside (namely, from God himself). We now give the list of these mutants, as he as- Grothendieck is convinced that each person sembled it himself. No doubt their selection is has a “mission” and that an important part of this mission consists of finding one’s self, of recogniz- rather arbitrary. A central (and not very original) ing one’s own self. Only through this search are theme in Grothendieck’s thinking is the spiritual a person’s creative powers liberated, for they are decline of humanity, necessarily followed by an ordinarily suppressed in many ways through the apocalypse and soon thereafter by the “new age”, constraints of society and through inner torpor, the age of freedom and self-determination and of life in harmony with one’s own “soul”. The mutants 9​This is apparent from, for example, marginal notes in are people who announce and anticipate this new Grothendieck’s copy of Freud, Die Traumdeutung, Fischer age. This is the criterion by which he selected Studienausgabe, Band II. them. The list comprises the following people, all

940 Notices of the AMS Volume 55, Number 8 men. The remarks on their works are taken from To complete the picture, I would like to mention Grothendieck: that Felix Carrasquer and his wife, Matilde Escuder, were close friends of the Grothendieck family (the C. F. S. Hahnemann: German medical doctor and original acquaintance was through Grothendieck’s scholar, renewed the medicine of his time wife, Mireille Dufour) and that the writing of the C. Darwin: English natural scientist, scholar Notes pour la Clef des Songes, including Les Mu- W. Whitman: Journalist, American writer, poet tants, was substantially inspired by Grothendieck’s and teacher reading of the books of M. Legaut. A more detailed B. Riemann: German mathematician, scholar discussion must be left for another occasion. Râmakrishna: Indian (Hindu) preacher, Among these texts, the philosophical ones (and teacher in a certain way the mathematical ones as well) R. M. Bucke: American doctor and psychiatrist, all follow a common presentation. Grothendieck scholar and annonciateur (herald) recorded his thoughts section by section, as if in P. A. Kropotkine: Russian geographer and a diary, and later edited these sections very little scholar, anarchistic revolutionary if at all. When he had more to say on one of his E. Carpenter: Priest, farmer, English thinker and sections, he usually did so through footnotes or writer, teacher addenda, which sometimes led to whole new sec- S. Freud: Austrian doctor and psychiatrist; tions. It also happened that he would meditate on scholar and creator of psychoanalysis, key to a parts of the written text; this generated remarks new scientific humanism on remarks on…as well as numerous footnotes. R. Steiner: German scholar, philosopher, writer, Surely this presentation does not facilitate read- orator, pedagogue...; visionary teacher, creator of ing, but in my view the more important criticism anthroposophy is that many of these long manuscripts do not M. K. Gandhi: Indian lawyer and politician, have a clear aim. Both in Récoltes et Semailles and teacher, worked for the spread of ahimsa (non- in La Clef des Songes it is obvious that new points violence) of view appeared after the writing had already P. Teilhard de Chardin: French (Jesuit) priest begun. As the texts do not seem to have any clear and paleontologist, Christian religious ecumeni- aim, there is no clear structure either. They me- cal thinker, mystical visionary, worked toward the ander, unstreamlined and unchanneled, in loops reconciliation of religion and science that change direction through a wide swath of A. S. Neill: English teacher and pedagogue who landscape, as if through the valley of a primor- championed freedom in education dial stream. The author drifts without any will to N. Fujii (called Fujii Guruji): Japanese Buddhist focus. It is completely different from his earlier monk, teacher writings on mathematics: Both EGA and SGA go J. Krishnamurti: Orator, Indian religious thinker into the breadth and the details, but there is a very and writer, teacher clear aim, the “correct” development of algebraic M. Legaut: University professor, farmer, French geometry or the “correct” cohomology theory in Christian religious thinker and writer, disciple of algebraic geometry. Jesus of Nazareth, worked for a spiritual renewal Since his disappearance, Grothendieck has of Christianity written down tens of thousands of pages of his F. Carrasquer: Spanish elementary school- meditations. Printed in their entirety, they would teacher and pedagogue; militant anarchist for a surely fill dozens of volumes. It seems almost “self-determined” school and society impossible that all of this, or even a large part of Slovik: American worker and low-ranking it, could be of importance. One cannot imagine employee apparently without any particular voca- that truly important writings can come about in tion complete intellectual and human isolation. I cannot do much more here than give the names But one has to remember that Grothendieck is a and list the aspects under which these people true master of language, that he certainly has un- are discussed; they are sex (sexe), war (guerre), conventional ideas and thoughts, and that he sees self-knowledge (connaissance de soi), religion the world in an unusual and even singular way. So (followed by an extensive explanation of what is one can imagine that in the countless pages of his meant—certainly not the church as an institution manuscripts there might often be finished texts: and not liturgy either), science (science), culture poems, biographical episodes from his life and the (la civilisation actuelle et ses valeurs, “culture”), lives of people who were close to him, commentar- eschatology (la question des destinées de l’humanité ies on books he read, perhaps lyric texts that go dans son ensemble, “eschatologie”), social justice beyond poetry, philosophical thoughts, apocalyp- (justice sociale), education (education), spirituality tic visions. It seems necessary to begin before it is (“science de demain” ou “science spirituelle”). too late to preserve for future generations those Perhaps this small bit of information gives a parts of Grothendieck’s life’s work that may be vague impression of what this meditation is about. important beyond his mathematics.

September 2008 Notices of the AMS 941