XII Международная студенческая научная конференция Студенческий научный форум - 2020

Born in Leningrad in the Soviet Union (now St. Petersburg, Russia) on June 13, 1966, Perelman was the son of an electrical engineer father who liked to challenge him with brain teasers. "He gave me logical and other math problems to think about," Perelman told Sylvia Nasar and David Gruber of the New Yorker . "He got a lot of books for me to read. He taught me how to play chess. He was proud of me." Perelman's mother was a math teacher. The family was Jewish, and Perelman had a younger sister, Elena, who also became a mathematician.

At the age of 14 Perelman was noted as the top achiever in his St. Petersburg mathematics club, and Sergey Rukshin, head of the St Petersburg Mathematical Center for Gifted Students, began to nurture his abilities. Two years later he won a gold medal with a perfect score at the International Mathematical Olympiad in Budapest, Hungary. Perelman was something of a loner but was never perceived as hostile or unfriendly by classmates and coworkers. His interests extended beyond math to Italian opera, and he spent his small amounts of pocket money on recordings.

Perelman entered Leningrad State University at age 16 and quickly was placed in advanced geometry courses. He impressed one of his teachers, Yuri Burago, who told Nasar and Gruber, "There are a lot of students of high ability who speak before thinking. Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully. He was not fast. Speed means nothing. Math doesn't depend on speed. It is about deep ." For relaxation, Perelman played table tennis and sometimes played the violin, which was also his mother's instrument.

Perelman continued straight through the programs at Leningrad State, earning the equivalent of a Ph.D. in the late 1980s after writing a dissertation on Euclidean geometry. He worked in the early 1990s at the Steklov Institute of Mathematics, part of the USSR Academy of Sciences. Publishing several papers on topics in geometry, he gained a reputation as a promising young scholar. In 1992 he was invited to spend a year in the United States as a guest scholar at New York University and then the State University of New York at Stony Brook. The timing was lucky, for the Russian economy was contracting rapidly in the aftermath of the fall of Communism and the breakup of the Soviet Union.

Finding the environment in America stimulating, Perelman was well liked even if some found him a bit eccentric. He lived on bread (Russian black bread when he could get it), cheese, and milk, and he let his fingernails grow to a length of several inches. "He looked like Rasputin, with long hair and fingernails," University of California at Los Angeles mathematician Robert Greene later told Dennis Overbye of the New York Times . A hobby from back in Russia that Perelman described to friends was hunting mushrooms on hikes in the woods. A central focus of Perelman's intellectual life was a weekly lecture series at the Institute for Advanced Study at Princeton University in New Jersey, which he and Chinese colleague Gang Tian, later a key explicator of his work, attended in order to interact with the top mathematical minds in the country and the world.

At Princeton Perelman encountered mathematician William Thurston, who had developed a set of generalizations abstracted from the Poincaré conjecture and expounded upon them in lectures. Perelman also met Cornell University mathematician Richard Hamilton, and, realizing the importance of his work, approached him after one talk. "I really wanted to ask him something," he recalled to Nasar and Gruber. "He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton's openness and generosity—it really attracted me. I can't say that most mathematicians act like that."

The Poincaré conjecture was described this way by Overbye: "It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere." The problem of proving the conjecture had implications for, among other things, the study of the shape of the universe. Numerous proofs had been suggested and quickly discarded over the years since 1904, when French mathematician Henri Poincaré first proposed the conjecture. But Thurston's "geometrization conjecture," creating a typology of three-dimensional "manifolds" or abstract surfaces derived from geometrical shapes, was regarded as a promising development. Perelman was granted a two-year fellowship to work at the University of California at Berkeley beginning in 1993, and during one lecture he gave on campus he indicated that he had joined the hunt for a proof of Poincaré's conjecture.

Perelman published several well-regarded papers while at Berkeley and was invited to give a lecture at the International Mathematical Union conference in Zurich, Switzerland, in 1994. Top universities in the United States and Israel then began to court the hot young scholar, but at this point something new—prickliness or perhaps just singlemindedness—began to manifest itself in his personality. He refused to submit a CV (an academic resumé) for a position at Stanford University, arguing that if the committee was familiar with his work, they should not need the document summarizing it. Using similar reasoning, he turned down a prize offered by a European group when he did not feel they were qualified to judge his work.

Perelman was offered several jobs but turned those down as well, opting instead to return in 1995 to St. Petersburg and his old post at the Steklov Institute of Mathematics. He had, he told friends, enough money saved from a few years in America to provide for himself for the rest of his life in Russia. He moved in with his mother—there was space in her small apartment because Perelman's father had departed for Israel. Perelman wrote a letter to Hamilton, proposing that they collaborate, but Hamilton did not reply. So Perelman forged on alone; he had little contact with his colleagues for several years. Though he was isolated from other mathematical thinkers, the rapidly growing Internet medium allowed him to keep abreast of developments in the field.

Then virtually out of the blue, Perelman posted an article on the Internet on November 11, 2002. It appeared on the website arXiv.org, a site devoted to "preprints" or articles ready to be published in mathematical journals. Perelman's article was called "The Entropy Formula for the Ricci Flow and Its Geometric Applications." He did not refer directly to the Poincaré conjecture but rather to Hamilton's concept of the Ricci flow, demonstrating its applicability to the larger Poincaré conjecture. Perelman's article was terse and telegraphic, with large gaps in his reasoning, but after he sent e-mails to a few of his former colleagues they sensed the importance of his discovery.

Perelman seemed uninterested in the prestige or glory his discovery might have brought. His work would have been worthy of a book or a series of articles in top mathematical journals, but he offered nothing other than his three Internet postings. After a series of lectures at American universities in 2003, Perelman essentially withdrew from public communication, although he was friendly enough to reporters intrepid enough to track him down in the labyrinthine streets of his central St. Petersburg neighborhood. "He placed the papers on the web archive and basically said 'that's it,'" Oxford University mathematics professor Nigel Hitchin told James Randerson of the London Guardian .

As other researchers filled in the gaps in Perelman's proof (one explication by two researchers at the University of Michigan ran to 473 pages), Perelman's isolation deepened. He may have been unhappy when a group of Chinese researchers published a set of parallel findings that referred to Perelman's work (and Hamilton's), apparently without fully crediting Perelman for his discoveries. Lingering doubts about Perelman's work may have contributed to his departure from the Steklov Institute in 2006, but those doubts were gradually settled. "I think for many months or even years now people have been saying they were convinced by the argument," Hitchin told the Guardian . "I think it's a done deal."

The International Mathematical Union apparently agreed, for the group prepared to award its prestigious Fields Medal to Perelman in 2006. But Perelman refused to accept the prize or to attend the IMU's 2006 congress in Madrid, despite a personal visit to St. Petersburg from IMU president Sir John M. Ball. His reasons, he told Nasar and Gruber, were simple. "It was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed." Perelman also seemed set to turn down a million-dollar prize offered by the Clay Mathematics Institute in Boston after he solved one of seven "Millennium Problems" of mathematics—the Poincaré conjecture was one. In order to claim the prize, he would have had to publish his proofs in a refereed mathematical journal, and he had shown no signs of doing so. As of late 2006, he was reported to have given up mathematical work. His reputation, however, seemed to be outliving his activity. In Russia Perelman became the subject of popular speculation, jokes, and cartoons. And his solution of the Poincaré problem was, in Overbye's words, "a landmark not just of mathematics, but of human thought."