Grigori Perelman

Poincaré conjecture

Grigori Perelman (b. 1966) is a Russian mathematician who has made landmark contributions to geometry and topology (the study of geometric deformation). In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston’s geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology.

In 2006, Perelman was awarded the Fields Medal, but declined to accept the award or to appear at the congress, stating: ‘I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo.’ In 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. He turned down the prize ($1 million), saying that he considers his contribution to proving the Poincaré conjecture to be no greater than that of U.S. mathematician Richard Hamilton.

‘I’ve learned how to calculate the voids; along with my colleagues we are getting to know the mechanisms for filling in the social and economic ‘voids.’ Voids are everywhere. They can be calculated, and this gives us great opportunities … I know how to control the Universe. So tell me — why should I chase a million?’

Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) to Jewish parents, Yakov (who now lives in Israel) and Lubov (who gave up graduate work in mathematics to raise him). Grigori’s mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin’s after-school math training program. His mathematical education continued at the Leningrad Secondary School, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education.

In 1982, as a member of the Soviet Union team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score. In the late 1980s, Perelman went on to earn a Candidate of Sciences degree (the Soviet equivalent to the Ph.D.) at the School of Mathematics and Mechanics of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was titled ‘Saddle surfaces in Euclidean spaces.’

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, where his advisors were Aleksandr Aleksandrov and Yuri Burago. In the late 1980s and early 1990s, Perelman held research positions at several universities in the United States. After having proved the soul conjecture in 1994, he was offered jobs at several top universities in the US, but he rejected them all and returned to the Steklov Institute in Saint Petersburg in the summer of 1995 for a research-only position.

In 2006, a committee of nine mathematicians voted to award Perelman a Fields Medal for his work on the Poincaré conjecture. The Fields Medal is the highest award in mathematics; two to four medals are awarded every four years. However, Perelman declined to accept the prize. Sir John Ball, president of the International Mathematical Union, approached Perelman in Saint Petersburg to persuade him to accept the prize. After 10 hours of talking over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: ‘He proposed to me three alternatives: accept and come; accept and don’t come, and we will send you the medal later; third, I don’t accept the prize. From the very beginning, I told him I have chosen the third one… [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed.’ He had previously turned down a prestigious prize from the European Mathematical Society, allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.

In 2010, Perelman was awarded a Millennium Prize for solving the problem. Perelman refused to accept the prize. He considered the decision of Clay Institute unfair for not sharing the prize with Richard Hamilton, and stated that ‘the main reason is my disagreement with the organized mathematical community. I don’t like their decisions, I consider them unjust.’ As of the spring of 2003, Perelman no longer worked at the Steklov Institute. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; some even say that he has abandoned mathematics entirely. According to a 2006 interview, Perelman was then unemployed, living with his mother in Saint Petersburg.

Perelman is quoted in an article in ‘The New Yorker’ saying that he is disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman’s role in the proof and play up the work of Cao and Zhu. Perelman added, ‘I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.’ He has also said that ‘It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.’

It is uncertain whether his resignation from Steklov and subsequent seclusion mean that he has ceased to practice mathematics. Fellow countryman and mathematician Yakov Eliashberg said that, in 2007, Perelman confided to him that he was working on other things but it was too premature to talk about it. He is said to have been interested in the past in the Navier–Stokes equations and the set of problems related to them that also constitutes a Millennium Prize, and there has been speculation that he may be working on them now.

Perelman is also a talented violinist and a strong table tennis player.

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