While living in an internment camp in Vichy France, Alexander Grothendieck was tutored in mathematics by another prisoner, a girl named Maria. Maria taught Grothendieck, who was twelve, the definition of a circle: all the points that are equidistant from a given point. The definition impressed him with “its simplicity and clarity,” he wrote years later. The property of perfect rotundity had until then appeared to him to be “mysterious beyond words.”

Grothendieck became a revered mathematician. His work involved finding the right vantage point—from there, solutions to problems would follow easily. He rewrote definitions, even of things as basic as a point; his reframings uncovered connections between seemingly unrelated realms of math. He spoke of his mathematical work as the building of houses, contrasting it with that of mathematicians who make improvements on an inherited house or construct a piece of furniture. Colin McLarty, a logician and philosopher of math at Case Western Reserve, told me, “Lots of people today live in Grothendieck’s house, unaware that it’s Grothendieck’s house.” The M.I.T. mathematician Michael Artin, who worked with Grothendieck in the early sixties, laughed when I asked him about Grothendieck’s contributions. “Well, everything changed in the field,” he said. “He came, and it was like night and day. It was a revolution.”

When Grothendieck was forty-two years old, he abruptly left the field of mathematics. For a while, he still did occasional private mathematical work—“to my own surprise, and despite my long-standing conviction,” he later wrote, “that I would never publish a single new line of mathematics in my lifetime.” By the time he was sixty-three, his whereabouts were known by almost no one. Nor was it known whether he was still pursuing solutions to the problems that had obsessed him for decades. Stories circulated of a bearded man wearing a long robe, hermited away somewhere in the Pyrenees.

Grothendieck wrote that his central work had been cruelly abandoned by others—but that wasn’t entirely true. Research was still ongoing in mathematical domains termed “Grothendieck universes,” and although his work wasn’t always cited, his methods were used so often that to cite him would be like citing Leibniz or Newton every time you used calculus. In 1992, two mathematicians, Leila Schneps and Pierre Lochak, decided that they would find Grothendieck.

The mathematical house builder Alexander Grothendieck was born in March, 1928, in Berlin, to Alexander Shapiro and Hanka Grothendieck. Hanka was married to a different man, so the child’s last name at birth was Raddatz. Shapiro, who went by Sascha, came from a middle-class Hasidic family, against whom he had rebelled. Hanka had left behind a well-off Protestant family. Both parents were anarchists. Sascha had been imprisoned in Russia for his involvement in the 1905 revolution; he lost an arm after being shot during one of his attempted escapes.

In 1933, Sascha left Berlin and moved to Paris, and Hanka followed soon afterward. They left Alexander in Hamburg, with a family that took in children. Maidi, his half sister via his mother, was put in an institution for disabled children, though she was not disabled. Sascha and Hanka spent some time in Spain, during the civil war. They wrote only a handful of letters to their children.

By 1939, the family that had taken Grothendieck in had grown concerned. Grothendieck looked Jewish. They located Sascha and Hanka, and the boy was put on a train from Hamburg to Paris. Shortly after Grothendieck’s reunion with his parents, whom he hadn’t seen in six years, Sascha was sent to an internment camp outside the city. (He later died in Auschwitz.) The mother and child were sent to Rieucros, a camp in the south. “The administration of the camp turned a blind eye toward the kids, however undesirable they might be,” Grothendieck writes in “Récoltes et Semailles” (“Harvests and Sowings”)—a manuscript of more than a thousand pages that was recently published, by Gallimard, in France. “We came and went as we pleased. I was the oldest, and the only one to go to school. It was a four- or five-kilometre-long walk, often in rainy and windy weather, wearing makeshift shoes that always got wet.” Grothendieck makes almost no other mention of the camp. He follows its description with a long paragraph about a teacher who unfairly gave him a bad grade for a math proof that he did in his own way, ignoring the textbook. He also decries his textbooks as lacking “serious” definitions of length, area, and volume.

For many years, Grothendieck idealized his parents. He identified closely with his father, with whom he had spent very little time, and whose biography he sometimes conflated with that of another Alexander Shapiro, a famous anarchist of the same era. Grothendieck recalled that as a child he loved rhymes, feeling that their sonic connections pointed to a mystery beyond words. For a time, he spoke exclusively in rhymes, “but fortunately,” he wrote fifty years later, “that period has passed.”

After Grothendieck had spent two years in Rieucros, a Protestant activist organization negotiated with the Vichy government for the release of some of the internees. Grothendieck was separated from his mother and housed as a refugee in Le Chambon-sur-Lignon, an Alpine area famous for centuries of resistance to repressive governments. Many of the local residents were cowherds. There, some five thousand “undesirables,” mostly children, were successfully hidden from the Nazis. The staple food was boiled chestnuts, which was served three times a day. Mushrooms or chicken was added if available. Sometimes the children were sent to the woods to hide for a few days.

If Grothendieck’s childhood was characterized by the fairy-tale aspect of being in a dark wood without parents, then his early adult life was also like a fairy tale, as obstacles were repeatedly overcome with almost magical ease. After the war, Grothendieck reunited with his mother and attended the University of Montpellier. He worked in the vineyards to support himself and Hanka, who was weak from tuberculosis, which she had contracted at Rieucros. While at the university—which was not an important center of mathematics—Grothendieck independently pursued research on ideas having to do with measures, a field that less gifted students might dismiss as obvious. He ended up rediscovering a celebrated problem, Lebesgue’s theorem. From that moment forward, Grothendieck thought of himself as a mathematician.

He went to Paris and studied with the most important French mathematicians of the time, including Laurent Schwartz, who would soon be awarded a Fields Medal, the highest award in mathematics. At the end of a paper co-authored by Schwartz, fourteen questions were listed. “Many of those questions, individually, would have been enough for a Ph.D.,” the mathematician Pierre Cartier said. In a short time, Grothendieck solved them all.

A more pedestrian problem was that Grothendieck was stateless. He had a right to French citizenship but did not avail himself of it, because that would mean he could be conscripted into the military. (When Grothendieck was later invited to visit Harvard, he almost didn’t get a visa, because he refused to pledge not to attempt to overthrow the United States government; he said that he would be fine going to jail in the U.S., so long as he had access to as many books as he wanted.) Without French citizenship, he could not be hired at French universities. He worked in the math department of the University of São Paulo for two years, where he told people that he ate only bananas, bread, and milk, “so as not to lose any time over it.” He then spent a year at the University of Kansas, and while there did work that culminated in a paper now known as the *Tohoku* paper, for the Japanese math journal in which it was published. The paper broadened spectral sequences—a fundamental tool in algebraic topology—and made them more powerful. Grothendieck’s contributions may sound like Martian language to non-mathematicians, but the connections revealed in his work were dramatic. “Spectral sequences wasn’t even seen as a subject on its own two feet,” Barry Mazur, a mathematician at Harvard who was friends with Grothendieck in the nineteen-sixties, told me. “It’s more of a technique. But Grothendieck didn’t approach anything as a mere technique.”

Mazur suggests that it’s possible to glimpse the essence of Grothendieck’s approach to mathematics by looking at two concepts—categories and functors. A category can be thought of almost as a grammar: take triangles, perhaps, and understand them in terms of their relationship to all other triangles. The category consists of objects, and relationships between objects. The objects are nouns and the relationships are verbs, and the category is all the ways in which they can interact. Grothendieck’s discoveries opened up mathematics in a way that was analogous to how Wittgenstein (and Saussure) changed our views of language.

A functor is a kind of translation machine that lets you go from one category to another, while bringing along all the relevant tools. This is more astonishing than it sounds. Imagine if math could be translated into poetry, and somehow it made sense to take the square root of a stanza.

The mathematician Angela Gibney describes Grothendieck’s vantage point in a way that I find particularly approachable: if you want to know about people, you don’t just look at them individually—you look at them at a family reunion. Ravi Vakil, a mathematician at Stanford, said, “He also named things, and there’s a lot of power in naming.” In the forbiddingly complex world of math, sometimes something as simple as new language leads you to discoveries. Vakil said, “It’s like when Newton defined weight and mass. They had not been distinguished before. And suddenly you could understand what was previously muddled.”

As a young man, Léon Motchane studied mathematics and physics in Russia, but after the Revolution he had to give up his studies to help support his family. He worked in insurance and banking, and lived in France. In 1958, he founded the Institut des Hautes Études Scientifiques, in Bures-sur-Yvette, about an hour outside Paris. I.H.E.S. is similar to the Institute for Advanced Study, in Princeton, which Motchane had visited. Part of the guiding principle behind both institutions is that scientific thinking can be nourished in a community, where ideas are worked out through conversations and connections between people. When putting I.H.E.S. together, Motchane contacted the elder statesman of mathematics Jean Dieudonné, who was as revered as his name had destined him to be. Dieudonné had been a founding member of Bourbaki, a group of mathematicians in France who were collectively rewriting the foundations of mathematics, and signing the work N. Bourbaki. (They once sent out invitations for the wedding of N. Bourbaki’s daughter, who was marrying a lion hunter named Hector Pétard.)

Dieudonné agreed to accept a position at the newly formed I.H.E.S., on the condition that Motchane also hire Grothendieck. Initially, the two of them constituted the paid staff of I.H.E.S., and mathematicians came down from Paris to attend a weekly seminar. Grothendieck’s hiring followed the death of his mother, in 1957. By the end of 1959, he was in a relationship with Mireille Dufour, who had cared for his mother. At I.H.E.S., Dieudonné set aside what he was working on in order to be a kind of scribe to Grothendieck. It was as if Matisse had set down his paintbrushes to assist a young Picasso. Nearly twelve golden years of mathematics followed, and thousands of pages of foundational theorems.

Grothendieck’s I.H.E.S. seminar met on Tuesdays. Sometimes he would ask someone else to lecture. “He had this incredible ability to ask the right person to do the right thing,” the mathematician Nick Katz, of Princeton, said. Katz went to I.H.E.S. as a young mathematician in the late sixties. “Grothendieck was engaged in this wonderful project, and to be asked to be a part of it—it was like Jesus asking you to be a disciple.”

The “wonderful project” consisted of looking at algebraic geometry from a new point of view. This was motivated partly by trying to find a solution to the Weil conjectures, an idea that the mathematician André Weil (also a Bourbakist) described in a letter to his sister, the philosopher and mystic Simone Weil, written while he was serving time in a military prison for failing to report for duty in the French Army. (The conjectures were formally introduced in a paper in 1949.) Weil’s conjectures detailed unexpected correspondences between the mathematical fields of number theory and topology. He showed that the number of solutions to certain polynomial equations—you may remember in high school trying to solve for *x* and *y* and coming up with more than one possible solution—was related to the number and kinds of holes in a geometric visualization of the solutions to the equations, and that this seemed to be true for equations in two dimensions or seventeen dimensions or a million dimensions. But Weil’s conjectures were conjectures. Grothendieck saw a way to prove them, using what are called schemes, sheaves, and motives. Sheaves were a mathematical bundling system of sorts, also developed during an incarceration: Jean Leray came up with the system while he was a prisoner of war.

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