A young scholarAlexandre Grothendieck transformed mathematics.

The most extraordinary genius in mathematics turned 80 in March, but no parties were held. His grateful students didn’t give speeches about him. Mathematicians didn’t convene at a grand conference in his honor. No one even lit a candle on his birthday cake. For more than 15 years, Alexandre Grothendieck has lived in self-imposed isolation in a tiny village in the Pyrenees. His rages have discouraged even his most determined visitors.

Nevertheless, he continues to be one of the most revered figures in mathematics. Grothendieck’s work has transformed math the way the Internet has transformed communication: Once you’re used to it, you can’t imagine what life was like before it.

“He was a master of the power of generalization,” says Luc Illusie of Paris-Sud University in France, one of Grothendieck’s students. Much of Grothendieck’s work was a kind of mathematics of mathematics called “category theory.” He divined the essential properties common to many different mathematical objects, laying bare the architecture that underlay the mathematics. The relationships between objects, he argued, were the key to the structure.

Montreal, 1970Like his father, Grothendieck shaved his head for many years.

His abstractions brought concrete results. For example, Grothendieck, together with his student Pierre Deligne and others, proved the Weil conjectures, profound theorems in algebraic geometry. The conjectures are a much more sophisticated version of the startling observation René Descartes made in the 1600s that founded algebraic geometry. Descartes realized that numbers, abstractions from piles of pebbles, aren’t so different from circles or ellipses, abstractions from drawings in the sand. Equations could form a link between them, using numbers to describe curves with perfect precision. The Weil conjectures provide a vastly more complex version of that same link.

“He had an extraordinary sensitivity to the harmony of things,” Illusie says. “It’s not just that he introduced new techniques and proved big theorems. He changed the very way we think about many branches of mathematics.”

Rather than attacking a problem directly, as if pounding on a chisel to crack a nut, Grothendieck built an entire architecture of theory around the problem, so that the solution gradually became easy and natural. He likens his approach to softening the nut in water. “From time to time you rub so the liquid penetrates better, and otherwise you let time pass,” he wrote in his autobiography. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!”

On a different pathMany Buddhists and hippies visited Grothendieck in the Pyrenees after he became politically active.

Grothendieck was legendary for working on mathematics almost every waking moment. But he did have one interest besides mathematics: politics. His pacifism grew out of his childhood experiences during World War II.

His father, Alexander Shapiro, was an anarchist who had rebelled against czarist Russia and spent about 10 years in jail. One story says Shapiro lost an arm in a suicide attempt while trying to avoid capture. When Hitler came to power, Grothendieck’s parents fled from Germany and left him with foster parents. Six years later, he joined his parent in France, but soon the entire family was arrested as “undesirable.” Grothendieck and his mother were sent to an internment camp, and Grothendieck’s father died in Auschwitz.

Grothendieck’s dormant interest in politics rumbled to life in the late ’60s. He was disgusted that the institute where he taught accepted money from the military. Hoping to lead an insurrection of the researchers, he resigned over the matter, but none of his fellow mathematicians joined him. When invited to give math talks, he often spoke about politics instead. He visited Hanoi, Vietnam, and gave a math lecture in support of Vietnamese mathematicians. A few years later, he left his wife and tried unsuccessfully to start a commune.

In his own wordsGrothendieck wrote a 2,000-page memoir called Récoltes et Semailles, or Reapings and Sowings. He typed the entire thing by hand, sometimes with so much force that a letter came through the page. In this excerpt, he says, "What makes the quality of a researcher’s inventiveness and imagination is the quality of his attention to hearing the voices of things." Translation by A. Jackson

He began to have occasional rages, followed by sullen withdrawal. He rejoined the mathematics community at a lesser university in France, without the intense collegiality he was used to, and he became disappointed with the competitiveness of his fellow mathematicians. As his direct influence faded, many mathematicians embraced concrete mathematical questions rather than the intensely abstract approach Grothendieck had favored. Grothendieck felt betrayed and mourned the “burial” of his work. When he and a student of his won the Crafoord Prize, he refused it, citing his disappointment with standards of ethics in the math community.

After he retired, he wrote a letter to 250 people declaring them part of a group chosen by God to prepare for a “New Age,” which he predicted would begin on October 14, 1996. Three months later, he sent a second letter, admitting with pain that he was no longer certain of that revelation.

In 1990, he gave away or destroyed all his papers and disappeared into the Pyrenees Mountains. For some years, no mathematicians knew where he was.

Pierre Lochak and Leila Schneps, two French algebraic geometers, heard from a former neighbor of Grothendieck that the “crazy mathematician” had been spotted in a nearby town. They tracked him down and found him living alone, doing organic farming. They maintained contact with him for several years, but now, Schneps says, “he’s not in a state to be visited. It’s not possible to not quarrel with him – or rather, for him not to quarrel with you.”

She hesitates to call him crazy, though she admits that in a technical sense, that might be true. “His mental state is very, very special.”

Grothendieck’s neighbors keep an eye on him, thwarting his occasional urges to eat nothing but dandelion soup, for example, and keeping Schneps and Lochak informed about his welfare.

Despite this isolation, Grothendieck continues to be a powerful presence through his mathematics. Without his advances, “it would have been impossible to attack any of the great problems in number theory and algebraic geometry that have been solved during the last 30 years,” Illusie says. Among those are Fermat’s last theorem and the Mordell conjecture.

“He had this whole thing about being buried, but it’s so utterly false,” says David Mumford of Brown University. “We would so much like to see him and tell him how much he’s still remembered.”

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