On Aug. 22, 2006, four Fields Medals, often described as mathematics’ Nobel Prize, were awarded at the International Congress of Mathematicians in Madrid, Spain, to Andrei Okounkov, Terence Tao, Wendelin Werner and Perelman.

Perelman, then 40, was the only winner who did not attend. He had been honored for proving the Poincaré conjecture, a landmark problem about the geometry of multidimensional spaces. Shing-Tung Yau, the first Chinese mathematician to win a Fields Medal, said the understanding of three-dimensional space brought by the Poincaré conjecture could become one of the major pillars of 21st-century mathematics.

But when John Ball, then president of the IMU, traveled to Russia to persuade Perelman to accept the award, he refused.

"It was completely irrelevant for me," he said in a 2006 interview with The New Yorker. "Everybody understood that if the proof is correct then no other recognition is needed."

Russian mathematician Grigori Perelman in 1993. Photo courtesy of the Clay Mathematics Institute

Discovering mathematics

Born in 1966, Perelman grew up in St. Petersburg with his mother, Lyubov, a mathematician, and his father, Yakov, an engineer. He attended a school specializing in advanced mathematics and physics, though he said becoming a mathematician was never a conscious decision.

"There was never a decision point," he said. His father played a major role in shaping that path. "He gave me logical and other math problems to think about," Perelman said. "He got a lot of books for me to read. He taught me how to play chess. He was proud of me."

By 14, he had become the standout student in a local math club. At 16, he won a gold medal with a perfect score at the International Mathematical Olympiad. His former teacher and close friend Sergei Rukshin said Perelman was not entirely consumed by math in his teenage years. He enjoyed table tennis and trips to the opera.

Still, Rukshin believed mathematics was always his deepest attachment. "If Grisha ever looked upon anything with loving eyes," Rukshin told The Telegraph in 2012, using Perelman’s nickname, "it was on the blackboard."

At Leningrad State University, which he entered in 1982 at age 16, Perelman quickly stood out. He solved a problem posed by Yuri Burago of the Steklov Institute of Mathematics, who later became his Ph.D. adviser. "There are a lot of students of high ability who speak before thinking," Burago said. "Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully."

Solving the century-old problem

After completing his Ph.D., Perelman moved to the U.S., where he conducted research at universities including New York University and University of California, Berkeley.

Colleagues remembered him as brilliant, quiet and indifferent to money or status. Robert Greene of the University of California, Los Angeles described him to The New York Times as "a kind of unworldly person" who was friendly but shy. "He looked like Rasputin, with long hair and fingernails."

Despite offers from institutions such as Stanford University and Princeton University, Perelman returned to St. Petersburg in 1995 and resumed work at the Steklov Institute.

By then, he had already begun working on the Poincaré conjecture, proposed in 1904 by French mathematician Henri Poincaré, one of the founders of topology, the study of geometric properties that remain unchanged through stretching or bending. In simple terms, the conjecture says that any three-dimensional space without holes is essentially a sphere.

The problem resisted solution for nearly a century. In 2000, the Clay Mathematics Institute listed it among its Millennium Prize Problems and offered $1 million for a correct proof.

Because the problem had a long history of failed attempts, Perelman told no one about his work. He worked largely in isolation for seven years. In November 2002, he quietly posted his proof online, the first of three papers released over the following eight months.

A mathematical proof follows strict rules. It starts with accepted truths, known as axioms, and builds step by step through logic to reach a conclusion. Once verified, the result becomes a theorem. Unlike proof in law or science, which depends on evidence and can later be challenged, a mathematical proof is considered final.

By those standards, Perelman’s proof was unusual. For such a major result, it was remarkably short, with many difficult arguments compressed into just a few lines instead of being explained in full. He said he was never concerned about claiming sole credit. "My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased," he said. "I never set out to be the sole solver of the Poincaré."

Other teams of mathematicians later expanded his work into manuscripts hundreds of pages long, confirming that he was correct.

Grigori Perelman, solver of Poincaré conjecture, gives a lecture on his solution at New York’s Weaver Hall in 2003. Photo from Facebook

Walking away from mathematics

After giving a series of lectures on his proof in the U.S. in 2003, Perelman returned to St. Petersburg and gradually withdrew from public academic life. In 2005, he resigned from his position, saying he was "disappointed" with mathematics.

A year later, he became the first person in history to reject the Fields Medal. He said the prospect of receiving the award had forced him to make a complete break from his profession. "As long as I was not conspicuous, I had a choice," he said.

"Either to make some ugly thing, or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit."

Manuel de León, chair of the International Congress of Mathematicians, said Perelman "feels isolated from the mathematical community and has no wish to appear as one of its leaders." Ball said Perelman simply viewed life differently and strongly rejected suggestions that he was mentally unstable, according to Science magazine.

In 2010, he again shocked the academic world by refusing the Clay Millennium Prize. "I have all that I need," Perelman reportedly told a colleague. He later told a Russian reporter through the door of the apartment he shared with his mother in St. Petersburg: "I don’t want to be on display like an animal in a zoo."

In 2012, journalist Brett Forrest of The Telegraph traveled to St. Petersburg hoping to interview Grigori Perelman. When Forrest asked what he was working on, Perelman gave a direct answer. "I have left mathematics," he said. "And what I’m doing now, I won’t tell you."

Forrest asked: "Where does your life go from here?" "What?" Perelman replied, then shrugged and said softly: "I don’t know."

Mikhail Gromov, a renowned Russian geometer who had collaborated with Perelman, told The New Yorker that he understood Perelman’s logic. "To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness."

Some may see Perelman’s refusal of the Fields Medal as arrogance, Gromov said, but he viewed it differently.

"The ideal scientist does science and cares about nothing else."

"He wants to live this ideal. Now, I don’t think he really lives on this ideal plane. But he wants to."