Grigori Perelman
Encyclopedia
Grigori Yakovlevich Perelman is a Russia
n mathematician
who has made landmark contributions to Riemannian geometry
and geometric topology
.
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 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
, stating: "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo." 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.
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 considers his contribution to proving the Poincaré conjecture to be no greater than that of U.S. mathematician Richard Hamilton, who introduced the theory of Ricci flow
with the aim of attacking the geometrization conjecture.
, Soviet Union
(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 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 Science 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 surface
s in Euclidean space
s."
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 Perelman won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on Aleksandrov's spaces of 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 Fellows
hip 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 violin
ist and a strong table tennis
player.
s in Riemannian geometry
. Among his notable achievements was a short and elegant proof of the soul conjecture
.
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 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.
, 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:
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."
Perelman's proof was rated one of the top cited articles in Math-Physics in 2008.
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.
, the author of a book about him, found it challenging to write about a person she never met.
A recent Russian documentary about Perelman in which his work is discussed by several leading mathematicians including Mikhail Gromov
was released in 2011 under the title "Иноходец. Урок Перельмана", "A pacer. A lesson from Perelman".
In April 2011 Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe. Zabrovsky says that in the interview, Perelman explained why he rejected the one million prize:
A number of journalists believe that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who has made landmark contributions to Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
and geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
.
In 1992, Perelman proved the soul conjecture
Soul theorem
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...
. In 2002, he proved Thurston's geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
. This consequently solved in the affirmative the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
In August 2006, Perelman was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
." Perelman declined to accept the award or to appear at the congress
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
, stating: "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo." On 22 December 2006, the journal Science
Science (journal)
Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year
Breakthrough of the Year
The Breakthrough of the Year is an annual award made by the journal Science for the most significant development in scientific research. Originating in 1989 as the Molecule of the Year, inspired by Time's Man of the Year, it was renamed the Breakthrough of the Year in 1996...
", the first such recognition in the area of mathematics.
On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
for resolution of the Poincaré conjecture. On July 1, 2010, he turned down the prize, saying that he considers his contribution to proving the Poincaré conjecture to be no greater than that of U.S. mathematician Richard Hamilton, who introduced the theory of Ricci flow
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
with the aim of attacking the geometrization conjecture.
Early life and education
Grigori Perelman was born in LeningradLeningrad
Leningrad is the former name of Saint Petersburg, Russia.Leningrad may also refer to:- Places :* Leningrad Oblast, a federal subject of Russia, around Saint Petersburg* Leningrad, Tajikistan, capital of Muminobod district in Khatlon Province...
, Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
(now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
) 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
Specialized school
Specialized schools are secondary schools with enhanced coverage of certain subjects that constitute the specialization of the school. They should not be confused with vocational schools, whose goal is to deliver skills for a particular type of job....
with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education
Physical education
Physical education or gymnastics is a course taken during primary and secondary education that encourages psychomotor learning in a play or movement exploration setting....
. In 1982, as a member of the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
team competing in the International Mathematical Olympiad
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
, 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 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 surface
Saddle surface
A saddle surface is a smooth surface containing one or more saddle points.The term derives from the peculiar shape of historical horse saddles, which curve both up and down....
s in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s."
After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics
St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences
The St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences for "Петербургское отделение Математического института", Petersburg Department of the Mathematical Institute) is a mathematical research institute in St. Petersburg, part of the Russian Academy of...
of the USSR Academy of Sciences, where his advisors were Aleksandr Aleksandrov
Aleksandr Danilovich Aleksandrov
Aleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...
and Yuri Burago. In the late 1980s and early 1990s, Perelman held research positions at several universities in the United States. In 1991 Perelman won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below . In 1992, he was invited to spend a semester each at the Courant Institute
Courant Institute of Mathematical Sciences
The Courant Institute of Mathematical Sciences is an independent division of New York University under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics...
in New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
and State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
where he began work on manifolds with lower bounds on Ricci curvature
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
. From there, he accepted a two-year Miller Research Fellows
Miller Research Fellows
The Miller Fellow program is the central program of the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science on the University of California Berkeley campus, which constitutes the support of Research Fellows - a group of the world’s most brilliant young scientists...
hip at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in 1993. After having proved the soul conjecture
Soul theorem
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...
in 1994, he was offered jobs at several top universities in the US, including Princeton
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
and Stanford
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
, but he rejected them all and returned to the Steklov Institute in Saint Petersburg
St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences
The St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences for "Петербургское отделение Математического института", Petersburg Department of the Mathematical Institute) is a mathematical research institute in St. Petersburg, part of the Russian Academy of...
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
Weizmann Institute of Science
The Weizmann Institute of Science , known as Machon Weizmann, is a university and research institute in Rehovot, Israel. It differs from other Israeli universities in that it offers only graduate and post-graduate studies in the sciences....
in Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
and is a biostatistician at Karolinska Institutet
Karolinska Institutet
Karolinska institutet is a medical university in Solna within the Stockholm urban area, Sweden, and one of Europe's largest medical universities...
, in Stockholm
Stockholm
Stockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...
, Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
.
Perelman is also a talented violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....
ist and a strong table tennis
Table tennis
Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight, hollow ball back and forth using table tennis rackets. The game takes place on a hard table divided by a net...
player.
Geometrization and Poincaré conjectures
Until the autumn of 2002, Perelman was best known for his work in comparison theoremComparison theorem
A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.-Riemannian geometry:In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.*Rauch...
s in Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
. Among his notable achievements was a short and elegant proof of the soul conjecture
Soul theorem
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...
.
The problem
The Poincaré conjecture, proposed by French mathematician Henri PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
in 1904, was the most famous open problem in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Any loop
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
). The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
. 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
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
announced the Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
: $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.
Perelman's proof
In November 2002, Perelman posted the first of a series of eprints to the arXivArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
, in which he claimed to have outlined a proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
of the geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
, of which the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
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
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
. 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
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
. The heat equation describes the behavior of scalar quantities such as temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
; 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
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
, 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
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, 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
Prime decomposition (3-manifold)
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique collection of prime 3-manifolds....
of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the JSJ decomposition
JSJ decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...
. Perelman's work proves this claim and thus proves the geometrization conjecture.
Verification
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 TechnologyMassachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
, Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
, Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
and New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
, 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:
- On 25 May 2006, Bruce KleinerBruce KleinerBruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory.He received his Ph.D. in 1990 from the University of California, Berkeley. His advisor was Wu-Yi Hsiang. He is now Professor of Mathematics at New York University.Kleiner has...
and John LottJohn Lott (mathematician)John Lott is a Professor of Mathematics at the University of California, Berkeley. He is working on Ricci flow....
, both of the University of MichiganUniversity of MichiganThe University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
, posted a paper on arXiv that fills in the details of Perelman's proof of the Geometrization conjecture. John Lott said in ICM2006, "It has taken us some time to examine Perelman's work. This is partly due to the originality of Perelman's work and partly to the technical sophistication of his arguments. All indications are that his arguments are correct." - In June 2006, the Asian Journal of MathematicsAsian Journal of MathematicsThe Asian Journal of Mathematics is a peer-reviewed scientific journal covering all areas of pure and theoretical applied mathematics. It is published by International Press....
published a paper by Xi-Ping ZhuXi-Ping ZhuZhu Xiping is a Professor of Mathematics at Sun Yat-sen University. He collaborated with Cao Huaidong of Lehigh University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team was one of three teams formed for this purpose...
of Sun Yat-sen UniversitySun Yat-sen UniversitySun Yat-sen University, also unofficially referred to as Zhongshan University , is a prominent university located mainly in Guangzhou, China. The University is named after Dr...
in ChinaChinaChinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
and Huai-Dong CaoHuai-Dong CaoHuai-Dong Cao is A. Everett Pitcher Professor of Mathematics at Lehigh University. He collaborated with Xi-Ping Zhu of Zhongshan University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team is one of three teams formed for this purpose...
of Lehigh UniversityLehigh UniversityLehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
in PennsylvaniaPennsylvaniaThe Commonwealth of Pennsylvania is a U.S. state that is located in the Northeastern and Mid-Atlantic regions of the United States. The state borders Delaware and Maryland to the south, West Virginia to the southwest, Ohio to the west, New York and Ontario, Canada, to the north, and New Jersey to...
, giving a complete description of Perelman's proof of the Poincaré and the geometrization conjectures. The June 2006 paper claimed: "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow."- In November 2006, Cao and Zhu published an erratum disclosing that they had failed to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the AJM editorial board issued an apology for what it called "incautions" in the Cao–Zhu paper.
- On December 3, 2006, Cao and Zhu retracted the original version of their paper, which was titled “A Complete Proof of the Poincaré and Geometrization Conjectures — Application of the Hamilton–Perelman Theory of the Ricci Flow” and posted a revised version, renamed, more modestly, "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". Rather than the grand claim of the original abstract, "we give a complete proof", suggesting the proof is by the authors, the revised abstract states: "we give a detailed exposition of a complete proof". The authors also removed the phrase "crowning achievement" from the abstract.
- In July 2006, John MorganJohn Morgan (mathematician)John Willard Morgan is an American mathematician, well known for his contributions to topology and geometry. He is currently the director of the Simons Center for Geometry and Physics at Stony Brook University.-Life:...
of Columbia University and Gang TianGang TianTian Gang is a Chinese mathematician and an academician of the Chinese Academy of Sciences. He is known for his contributions to geometric analysis and quantum cohomology, among other fields...
of the Massachusetts Institute of Technology posted a paper on the arXiv titled, "Ricci Flow and the Poincaré Conjecture". In this paper, they provide a detailed version of Perelman's proof of the Poincaré conjecture. On 24 August 2006, Morgan delivered a lecture at the ICMInternational Congress of MathematiciansThe International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Madrid on the Poincaré conjecture. This was followed up with the paper on the arXiv, "Completion of the Proof of the Geometrization Conjecture" on 24 September 2008.
The Fields Medal and Millennium Prize
In May 2006, a committee of nine mathematicians voted to award Perelman a Fields MedalFields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
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
John M. Ball
Sir John Macleod Ball, FRSE FRS is Sedleian Professor of Natural Philosophy at the University of Oxford. He was the President of the International Mathematical Union from 2003-06 and a Fellow of Queen's College, Oxford...
, president of the International Mathematical Union
International Mathematical Union
The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
, approached Perelman in Saint Petersburg
Saint Petersburg
Saint Petersburg is a city and a federal subject of Russia located on the Neva River at the head of the Gulf of Finland on the Baltic Sea...
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
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Madrid
Madrid
Madrid is the capital and largest city of Spain. The population of the city is roughly 3.3 million and the entire population of the Madrid metropolitan area is calculated to be 6.271 million. It is the third largest city in the European Union, after London and Berlin, and its metropolitan...
"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
European Mathematical Society
The European Mathematical Society is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians...
, 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
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
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."
Perelman's proof was rated one of the top cited articles in Math-Physics in 2008.
Possible withdrawal from mathematics
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
The New Yorker
The New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons and poetry published by Condé Nast...
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
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
to downplay Perelman's role in the proof and play up the work of Cao
Huai-Dong Cao
Huai-Dong Cao is A. Everett Pitcher Professor of Mathematics at Lehigh University. He collaborated with Xi-Ping Zhu of Zhongshan University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team is one of three teams formed for this purpose...
and Zhu
Xi-Ping Zhu
Zhu Xiping is a Professor of Mathematics at Sun Yat-sen University. He collaborated with Cao Huaidong of Lehigh University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team was one of three teams formed for this purpose...
. 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
Yakov Eliashberg
Yakov Eliashberg is a Russian mathematician. He received his Ph.D. from Leningrad University in 1972 under the direction of Vladimir Rokhlin. From 1972 to 1979 he taught at the Syktyvkar State University of Komi Republic of Russia and from 1980 to 1987 worked in industry as the head of a computer...
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
Navier-Stokes existence and smoothness
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics . These equations describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical...
related to them that also constitutes a Millennium Prize, and there has been speculation that he may be working on them now.
Perelman and media
Perelman avoided journalists and other media people. Masha GessenMasha Gessen
Masha Gessen is an American journalist and author.Born into an Ashkenazi Jewish family in Russia, in 1981 she moved with her family to the United States, returning in 1991 to Moscow where she lives now. Her brother is Keith Gessen....
, the author of a book about him, found it challenging to write about a person she never met.
A recent Russian documentary about Perelman in which his work is discussed by several leading mathematicians including Mikhail Gromov
Mikhail Gromov
Mikhail Leonidovich Gromov, also spelled Mikhael Gromov or Michael Gromov; , , is a Russian mathematician known for important contributions in many different areas of mathematics...
was released in 2011 under the title "Иноходец. Урок Перельмана", "A pacer. A lesson from Perelman".
In April 2011 Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe. Zabrovsky says that in the interview, Perelman 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 believe that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.
See also
- Homology sphereHomology sphereIn algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
- "Manifold DestinyManifold Destiny"Manifold Destiny" is an article in The New Yorker written by Sylvia Nasar and David Gruber and published in the August 28, 2006 issue of the magazine...
" (On The New YorkerThe New YorkerThe New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons and poetry published by Condé Nast...
article) - Soul theoremSoul theoremIn mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...
- Uniformization theoremUniformization theoremIn mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...
- Asteroid 50033 Perelman50033 Perelman50033 Perelman is a main-belt asteroid discovered on January 3, 2000 by S. Sposetti at Gnosca. It is named for Grigori Perelman, the Russian mathematician who proved the Poincaré conjecture.- External links :...
External links
- Fields Medals awarded at the 2006 IMU
- International Congress of Mathematicians 2006
- Perelman's arXiv eprints (link to APS mirror due to server strain on arxiv.org re Perelman)
- Mathematics & Mechanics Faculty of Saint Petersburg State University
- Petersburg Department of Steklov Institute of Mathematics
- Notes and commentary on Perelman's Ricci flow papers
- International Mathematical Olympiad 1982 (Budapest, Hungary)
- Maths solution tops science class (Link to BBC News page)
- Breakthrough Of The Year: The Poincaré Conjecture--Proved
- Perelman refuses a million dollars to live in complete poverty
- ICM 2006 Video Presentation
- He Conquered the Conjecture essay on Perelman by John Allen PaulosJohn Allen PaulosJohn Allen Paulos is a professor of mathematics at Temple University in Philadelphia who has gained fame as a writer and speaker on mathematics and the importance of mathematical literacy...
from The New York Review of BooksThe New York Review of BooksThe New York Review of Books is a fortnightly magazine with articles on literature, culture and current affairs. Published in New York City, it takes as its point of departure that the discussion of important books is itself an indispensable literary activity... - The man who refused a million dollars