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Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман; born 13 June 1966) is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture. As a consequence, this 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 August 2006, Perelman was awarded the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." Perelman declined to accept the award or to appear at the congress. On 22 December 2006, the journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year," the first such recognition in the area of mathematics. He has since ceased working on mathematics. On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On July 1, 2010, he turned down the prize, saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of U.S. mathematician Richard Hamilton, who first suggested a program for the solution. Grigori Perelman was born in Leningrad, USSR (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lubov. Grigori's mother Lubov 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 #239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In 1982, as a member of the USSR 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 Science degree (the Soviet equivalent to the Ph.D.) at the Mathematics and Mechanics Faculty 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. In 1991, he spoke at the Geometry Festival on Alexandrov spaces with curvature bounded from below. In 1992, he was invited to spend a semester each at the Courant Institute in New York University and State University of New York at Stony Brook where he began work on manifolds with lower bounds on Ricci curvature. From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. After having proved the soul conjecture in 1994, he was offered jobs at several top universities in the US, including Princeton and Stanford, but he rejected them all and returned to the Steklov Institute in Saint Petersburg in the summer of 1995 for a research only position. He has a younger sister, Elena, who is also a scientist. She received a Ph.D. from Weizmann Institute of Science in Israel and is a biostatistician at Karolinska Institutet, in Stockholm, Sweden. Perelman is also a talented violinist and a strong table tennis player.
Until the autumn of 2002, Perelman was best known for his work in
comparison theorems in Riemannian geometry. Among his notable achievements was a short and elegant proof of the soul conjecture. The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was the most famous open problem in topology. Any loop on a three-dimensional sphere — as exemplified by the set of points at a distance of 1 from the origin in four - dimensional Euclidean space — can be contracted to a point. The Poincaré conjecture asserts that any closed three - dimensional manifold such that any loop can be contracted to a point is topologically a three - dimensional sphere. The analogous result has been known to be true in dimensions greater than or equal to five since 1960 (work of Stephen Smale). The four - dimensional case resisted longer, finally being solved in 1982 by Michael Freedman. But the case of three - manifolds turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three - manifold, there are too few dimensions to move "problematic regions" out of the way without interfering with something else. In 1999, the Clay Mathematics Institute announced the Millennium Prize Problems:
$1,000,000 prizes for the proof of any of seven conjectures, including
the Poincaré conjecture. There was a wide agreement that a
successful proof of any of these would constitute a landmark event in
the history of mathematics. In November 2002, Perelman posted the first of a series of eprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case. Perelman modified Richard Hamilton's program for a proof of the conjecture, in which the central idea is the notion of the Ricci flow. Hamilton's basic idea is to formulate a "dynamical process" in which a
given three - manifold is geometrically distorted, such that this
distortion process is governed by a differential equation analogous to
the heat equation. The heat equation describes the behavior of scalar quantities such as temperature;
it ensures that concentrations of elevated temperature will spread out
until a uniform temperature is achieved throughout an object.
Similarly, the Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor.
Hamilton's hope was that under the Ricci flow, concentrations of large
curvature will spread out until a uniform curvature is achieved over
the entire three - manifold. If so, if one starts with any three - manifold
and lets the Ricci flow occur, eventually one should in principle
obtain a kind of "normal form". According to William Thurston, this normal form must take one of a small number of possibilities, each having a different kind of geometry, called Thurston model geometries. This
is similar to formulating a dynamical process that gradually "perturbs"
a given square matrix, and that is guaranteed to result after a finite
time in its rational canonical form. Hamilton's
idea attracted a great deal of attention, but no one could prove that
the process would not be impeded by developing "singularities", until
Perelman's eprints sketched a program for overcoming these obstacles. According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgery, can systematically excise singular regions as they develop, in a controlled way. We
know that singularities (including those that, roughly speaking, occur
after the flow has continued for an infinite amount of time) must occur
in many cases. However, any singularity that develops in a finite time
is essentially a "pinching" along certain spheres corresponding to the prime decomposition of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the JSJ decomposition. Perelman's work proves this claim and thus proves the geometrization conjecture. Since
2003, Perelman's program has attracted increasing attention from the
mathematical community. In April 2003, he accepted an invitation to
visit Massachusetts Institute of Technology, Princeton University, State University of New York at Stony Brook, Columbia University and New York University, where he gave a series of talks on his work. Three
independent groups of scholars have verified that Perelman's papers
contain all the essentials for a complete proof of the geometrization
conjecture: Nigel Hitchin,
professor of mathematics at Oxford University, has said that "I think
for many months or even years now people have been saying they were
convinced by the argument. I think it's a done deal."
In May 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 in
June 2006 to persuade him to accept the prize. After 10 hours of
persuasion 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." "'I'm
not interested in money or fame,' he is quoted to have said at the
time. 'I don't want to be on display like an animal in a zoo. I'm not a
hero of mathematics. I'm not even that successful; that is why I don't
want to have everybody looking at me.'" Nevertheless, on 22 August 2006, Perelman was publicly offered the medal at the International Congress of Mathematicians in Madrid "for
his contributions to geometry and his revolutionary insights into the
analytical and geometric structure of the Ricci flow." He did not attend the ceremony, and declined to accept the medal, making him the first person to decline this prestigious prize. 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. On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem. On
June 8, 2010, he did not attend a ceremony in his honor at the Institut
Océanographique, Paris, to accept his $1 million prize. According to Interfax,
Perelman refused to accept the Millennium prize in July 2010. 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." Terence Tao spoke about Perelman's work on the Poincaré Conjecture during the 2006 Fields Medal Event: Perelman's proof was rated one of the top cited articles in Math - Physics in 2008.
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." This,
combined with the possibility of being awarded a Fields medal, led him
to quit professional mathematics. He has said that "As long as I was
not conspicuous, I had a choice. 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." (The New Yorker authors
explained Perelman's reference to "some ugly thing" as "a fuss" on
Perelman's part about the ethical breaches he perceived). 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 avoided journalists and other media people.
Masha Gessen, the author of a book about him, found it challenging to write about a person she never met. In
April 2011 Perelman gave his first interview to Aleksandr Zabrovsky,
producer of "President - Film" studio and agreed to shoot a film about
him, under the tentative title The Formula of the Universe. A number of news outlets wrote about this interview, mentioning that Perelman finally explained why he rejected the one million prize: 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? A number of journalists claimed that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.
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