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.