Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман), born 13 June 1966 in Leningrad, USSR (now St. Petersburg, Russia), sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he has proved Thurston's geometrization conjecture. This solves in the affirmative the famous Poincaré conjecture, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics. June 13 is the 164th day of the year in the Gregorian calendar (165th in leap years), with 201 days remaining. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... Leningrad (Russian: Ленинград) may mean: St. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... The Fields Medal 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 years. ... June 13 is the 164th day of the year in the Gregorian calendar (165th in leap years), with 201 days remaining. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... Saint Petersburg (Russian: Санкт-Петербу́рг, English transliteration: Sankt-Peterburg), colloquially known as Питер (transliterated Piter), formerly known as Leningrad (Ленингра́д, 1924–1991) and... Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ...

In August 2006, Perelman was awarded the Fields Medal,^[1] for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow". The Fields Medal is widely considered to be the top honor a mathematician can receive. However, he declined to accept the award or appear at the congress. For the Manfred Mann album, see 2006 (album). ... The Fields Medal 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 years. ... In differential geometry, Ricci flow is the flow of Riemannian metrics given by the equation where g is the metric and Ric is the Ricci curvature. ... The Fields Medal 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 years. ... The International Congress of Mathematicians (ICM) is the biggest congress in mathematics. ...

On December 22, 2006, Science Magazine recognized Perelman's proof of the Poincaré Conjecture as the scientific "Breakthrough of the Year," the first such recognition in the area of mathematics.^[2] December 22 is the 356th day of the year (357th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... Science is the journal of the American Association for the Advancement of Science (AAAS). ... The Breakthrough of the Year is an annual award made by the journal Science for the most significant development in scientific research. ...

Early life and education

Grigori Perelman was born in Leningrad (now St. Petersburg) to a Jewish family on June 13, 1966. His early mathematical education occurred at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. 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.^[3] In the late 1980s, Perelman went on to earn a Candidate of Science degree (the Russian 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 entitled "Saddle surfaces in Euclidean spaces". He is also a talented violinist and plays table tennis.^[4] The word Jew ( Hebrew: יהודי) is used in a wide number of ways, but generally refers to a follower of the Jewish faith, a child of a Jewish mother, or someone of Jewish descent with a connection to Jewish culture or ethnicity and often a combination... June 13 is the 164th day of the year in the Gregorian calendar (165th in leap years), with 201 days remaining. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... Saint Petersburg Lyceum 239 (Russian: ), is a public high school in Saint Petersburg, Russia that specializes in mathematics and physics. ... Specialized schools are secondary schools with enhanced coverage of certain subjects that constitute the specialization of the school. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... The International Mathematical Olympiad (IMO) is an annual mathematical olympiad for high school students. ... Kandidat (Russian: кандидат) or Candidate of Science (кандидат наук) is a holder of first post-graduate scientific degree in former USSR since 1934 and in some post-Soviet states, awarded on dissertation (the Doctor of Science is the next higher degree). ... Categories: Russia-related stubs | Universities and colleges in Russia | Saint Petersburg ... This article is about the thesis in dialectics and academia. ... Hyperbolic parabloid A model of an ellyptic hyperboloid of one sheet A saddle surface is a smooth surface all points of which are saddle points. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences. His advisors at the Steklov Institute were Aleksandr Danilovich Aleksandrov and Yuri Dmitrievich Burago. In the late 80s and early 90s, Perelman held posts at several universities in the United States. In 1992, he was invited to spend a semester each at New York University and Stony Brook University. From there, he accepted a two-year fellowship at the University of California, Berkeley in 1993. He returned to the Steklov Institute in the summer of 1995. Steklov Institute of Mathematics or Steklov Mathematical Institute (Russian: Математический институт имени Ð’.А.Стеклова) is a research institute specialized in Mathematics. ... Russian Academy of Sciences (Росси́йская Акаде́мия Нау́к) is the national academy of Russia. ... Aleksandr Danilovich Aleksandrov (Russian: Александр Данилович Александров, alternative transliterations: Alexandr or Alexander (first name), and Alexandrov (last name)) (August 4, 1912–July 27... Yuri Dmitrievich Burago (Russian: ) is a Russian mathematician. ... New York University (NYU) is a private, nonsectarian, coeducational institution in New York City. ... Stony Brook University The State University of New York at Stony Brook (SUNYSB), or Stony Brook University (SBU) (a marketing ploy used to woo students [1] ), located in Stony Brook, New York, USA, is a public research university in the United States, is a public research university located in Stony... Sather tower (the Campanile) looking out over the San Francisco Bay and Mount Tamalpais. ...

Geometrization and Poincaré conjectures

Until the autumn of 2002, Perelman was best known for his work in comparison theorems in Riemannian geometry. Among his notable achievements was the proof of the soul conjecture. Comparison theorem is a popular name for 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. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, the soul theorem is the following theorem of Riemannian geometry: If (M,g) is a complete non-compact Riemannian manifold with sectional curvature K ≥ 0, then (M,g) has a compact totally convex, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S...

The problem

Main article: Poincaré conjecture

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was the most famous open problem in topology. Loosely speaking, the conjecture surmises that if a closed three-dimensional manifold is sufficiently like a sphere in that each loop in the manifold can be tightened to a point, then it is really just a three-dimensional sphere. The analogous result has been known to be true in higher dimensions for some time, but the case of three-manifolds had 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 mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...

In 1999, the Clay Mathematics Institute announced the Millennium Prize Problems – a one million dollar prize for the proof of several conjectures, including the Poincaré conjecture. There is universal agreement that a successful proof would constitute a landmark event in the history of mathematics, fully comparable with the proof by Andrew Wiles of Fermat's Last Theorem, but possibly even more far-reaching. The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Pierre de Fermats informal conjecture written in the margin of his book proved to be one of the most intriguing and enigmatic math problems ever devised. ...

Perelman's proof

In November 2002, Perelman posted to the arXiv the first of a series of eprints in which he claimed to have outlined a proof of the geometrization conjecture, a result that includes the Poincaré conjecture as a particular case. arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Eprints is free, open source software for generating an Open Access (OA) Institutional Repository (IR) that is compliant with the Open Archives Initiative Protocol for Metadata Harvesting (OAI-PMH). ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ...

Perelman modifies 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 work its magic, 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 flavor of geometry, called Thurston model geometries. Richard S. Hamilton (b. ... In differential geometry, Ricci flow is the flow of Riemannian metrics given by the equation where g is the metric and Ric is the Ricci curvature. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Fig. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ...

This is similar to formulating a dynamical process which gradually "perturbs" a given square matrix, and which is guaranteed to result after a finite time in its rational canonical form. In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix. ...

Hamilton's idea had attracted a great deal of attention, but no-one could prove that the process would not "hang up" 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. Eprints is free, open source software for generating an Open Access (OA) Institutional Repository (IR) that is compliant with the Open Archives Initiative Protocol for Metadata Harvesting (OAI-PMH). ...

It is known that singularities (including those which occur, roughly speaking, after the flow has continued for an infinite amount of time) must occur in many cases. However, mathematicians expect that, assuming that the geometrization conjecture is true, any singularity which develops in a finite time is essentially a "pinching" along certain spheres corresponding to the prime decomposition of the 3-manifold. If so, any "infinite time" singularities should result from certain collapsing pieces of the JSJ decomposition. Perelman's work apparently proves this claim and thus proves the geometrization conjecture. In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds. ... The JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: Irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible torii such that each component of the 3-manifold obtained by cutting along the tori...

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 Technology, Princeton University, State University of New York at Stony Brook, Columbia University and Harvard University, where he gave a series of talks on his work.^[3] The Massachusetts Institute of Technology (MIT) is a private, coeducational research university located in Cambridge, Massachusetts. ... Princeton University is a private coeducational research university located in Princeton, New Jersey, in the United States of America. ... State University of New York at Stony Brook (SUNYSB), also known as Stony Brook University (SBU) [1] , is a public research university located in Stony Brook, New York (on the north side of Long Island, about 55 miles east of Manhattan, New York). ... Columbia University is a private research university in the United States. ... Harvard University (incorporated as The President and Fellows of Harvard College) is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League. ...

On 25 May 2006, Bruce Kleiner and John Lott, both of the University of Michigan, posted a paper on arXiv that fills in the details of Perelman's proof of the Geometrization conjecture.^[5] is the 145th day of the year (146th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... The University of Michigan, Ann Arbor (UM, U of M or U-M) is a coeducational public research university in the state of Michigan. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

In June 2006, the Asian Journal of Mathematics published a paper by Xi-Ping Zhu of Sun Yat-sen University in China and Huai-Dong Cao of Lehigh University in Pennsylvania, giving a complete description of Perelman's proof of the Poincaré and the geometrization conjectures.^[6] According to the Fields medalist Shing-Tung Yau "Cao and Zhu put the finishing touches to the complete proof of the Poincaré Conjecture"^[7]. Cao has stated, "Hamilton and Perelman have done the most important fundamental works. They are the giants and our heroes. In my mind there is no question at all that Perelman deserves the Fields Medal. We just follow the footsteps of Hamilton and Perelman and explain the details. I hope everyone who read our paper would agree that we have given a rather fair account." ^[8] Xi-Ping Zhu is a Professor of Mathematics in Zhongshan University. ... Sun Yat-sen University or Zhongshan University (Traditional Chinese: 中山大學; Simplified Chinese : 中山大学; pinyin: Zhōngshān Dàxué) is a prominent university in the southern part of China, located mainly in Guangzhou. ... Huai-Dong Cao is a Professor of Mathematics in Lehigh University. ... Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. ... Capital Harrisburg Largest city Philadelphia Area  Ranked 33rd  - Total 46,055 sq mi (119,283 km²)  - Width 280 miles (455 km)  - Length 160 miles (255 km)  - % water 2. ... The Fields Medal 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 years. ... Shing-Tung Yau (Chinese: ; pinyin: ; born April 4, 1949) is a prominent mathematician working in differential geometry, and involved in the theory of Calabi-Yau manifolds. ...

On December 3, 2006, in response to plagiarism charges, Cao and Zhu retracted their original paper titled, “A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow” and renamed it more modestly, "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture." ^[9]. They also took the phrase "crowning achievement" out of the abstract.^[9] December 3 is the 337th (in leap years the 338th) day of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ...

In July 2006, John Morgan of Columbia University and Gang Tian 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.^[10] On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture.^[11] John Willard Morgan is a Professor of Mathematics at Columbia University. ... Gang Tian (Chinese: ; pinyin: Tián Gāng; 1958 -) is a Chinese mathematician and an academician of the Chinese Academy of Sciences. ... August 24 is the 236th day of the year in the Gregorian calendar (237th in leap years), with 129 days remaining. ... For the Manfred Mann album, see 2006 (album). ...

The above work demonstrates that Perelman's outline can indeed be expanded into 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."^[12]

The Fields Medal and Millennium Prize

In May 2006, a committee of nine mathematicians voted to award Perelman a Fields Medal for his work on the Poincaré conjecture.^[3] The Fields Medal is the highest award in mathematics; two to four medals are awarded every four years. The Fields Medal 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 years. ...

Sir John Ball, president of the International Mathematical Union, approached Perelman in St. Petersburg in June 2006 to persuade him to accept the prize. After 10 hours of persuading over two days, he gave up. Two weeks later, Perelman summed up the conversation as: "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." He went on to say that the prize "was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed."^[3] Prof Sir John Macleod Ball FRS (born 1948) is Sedleian Professor of Natural Philosophy at the University of Oxford. ... The International Mathematical Union is an international non-governmental organization devoted to international cooperation in the field of mathematics. ... Saint Petersburg (Russian: Санкт-Петербу́рг, English transliteration: Sankt-Peterburg), colloquially known as Питер (transliterated Piter), formerly known as Leningrad (Ленингра́д, 1924–1991) and...

On August 22, 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".^[13] He did not attend the ceremony, and declined to accept the medal, making him the first person in history to decline this prestigious prize.^[14][15] August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... The International Congress of Mathematicians (ICM) is the biggest congress in mathematics. ... Motto: De Madrid al Cielo (From Madrid to Heaven) Location Coordinates: Country Spain Autonomous Community Comunidad Autónoma de Madrid Province Madrid Administrative Divisions 21 Neighborhoods 127 Founded 9th century Government  - Mayor Alberto Ruiz-Gallardón (PP) Area  - Land 607 km² (234. ...

He had previously turned down a prestigious prize from the European Mathematical Society,^[15] allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.^[12] The European Mathematical Society (EMS) is an european organization dedicated to the development of mathematics in Europe. ...

Perelman may also be due to receive a share of a Millennium Prize. The rules for this prize require his proof to be published in a peer-reviewed mathematics journal. While Perelman has not pursued publication himself, other mathematicians have published papers about the proof. This may make Perelman eligible to receive a share of the prize. Perelman has stated that "I’m not going to decide whether to accept the prize until it is offered."^[3] The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... Peer review (known as refereeing in some academic fields) is a scholarly process used in the publication of manuscripts and in the awarding of funding for research. ...

Terence Tao spoke about Perelman's work on the Poincare Conjecture during the 2007 Fields Medal Event [1]: Terence Chi-Shen Tao ( 陶哲軒)(born 1975) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. ... In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ...

In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ...

Withdrawal from mathematics

As of the spring of 2003 Perelman no longer works in the Steklov Institute.^[4] 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.^[16] According to a recent interview, Perelman is currently jobless, living with his mother in St Petersburg.^[4]

Although Perelman says in The New Yorker article that he is disappointed with the ethical standards of the field of mathematics, the article implies that Perelman refers particularly to Yau's efforts to downplay his role in the proof and play up the work of Cao and Zhu. Perelman has said that "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."^[3] 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."^[3] The New Yorker is an American magazine that publishes reportage, criticism, essays, cartoons, poetry and fiction. ...

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" (a fuss about the mathematics community's lack of integrity) "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.”^[3]

Bibliography

• Перельман, Григорий Яковлевич (1990). Седловые поверхности в евклидовых пространствах:Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук (in Russian). Ленинградский Государственный Университет.  (Perelman's dissertation)

• Perelman, G.; Yu. Burago, M. Gromov (1992). "Aleksandrov spaces with curvatures bounded below". Russian Math Surveys 47 (2): 1-58. 

• Perelman, G. (1993). "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers" (PDF). Comparison Geometry 30: 157-163. Retrieved on 2006-08-23. 

• Perelman, G. (1994). "Proof of the soul conjecture of Cheeger and Gromoll". J. Differential Geom. 40: 209-212. 

• Perelman, G. (1994). "Elements of Morse theory on Aleksandrov spaces". St. Petersbg. Math. J. 5 (1): 205-213. 

• Perelman, G.Ya.; Petrunin, A.M. (1994). "Extremal subsets in Alexandrov spaces and the generalized Liberman theorem". St. Petersburg Math. J. 5 (1): 215-227. 

Perelman's proof of the geometrization conjecture: PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... For the Manfred Mann album, see 2006 (album). ... August 23 is the 235th day of the year (236th in leap years) in the Gregorian calendar. ...

• Perelman, Grisha (November 11, 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159. 
• Perelman, Grisha (March 10, 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109. 
• Perelman, Grisha (July 17, 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245. 

November 11 is the 315th day of the year (316th in leap years) in the Gregorian calendar, with 50 days remaining. ... For album titles with the same name, see 2002 (album). ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... March 10 is the 69th day of the year (70th in leap years) in the Gregorian calendar. ... 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... July 17 is the 198th day of the year (199th in leap years) in the Gregorian calendar. ... 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

Notes

For the Manfred Mann album, see 2006 (album). ... is the 120th day of the year (121st in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... December 29 is the 363rd day of the year (364th in leap years) in the Gregorian calendar, with 2 days remaining. ... June 3 is the 154th day of the year (155th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... August 29 is the 241st day of the year (242nd in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... December 3 is the 337th (in leap years the 338th) day of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Portable Document Format (PDF) is a file format created by Adobe Systems in 1993 for desktop publishing use. ... August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... The current BBC News logo BBC News and Current Affairs is a major arm of the BBC responsible for the corporations newsgathering and production of news programmes on BBC television, radio and online. ... August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ...

References

The Cable News Network, commonly known as CNN, is a major cable television network founded in 1980 by Ted Turner. ... July 1 is the 182nd day of the year (183rd in leap years) in the Gregorian calendar. ... shelby was here 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... August 15 is the 227th day of the year in the Gregorian calendar (228th in leap years), with 138 days remaining. ... The Wall Street Journal is an influential international daily newspaper published in New York City, New York with an average daily circulation of 1,800,607 (2002). ... July 21 is the 202nd day of the year (203rd in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ... PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... August 23 is the 235th day of the year in the Gregorian Calendar (236th in leap years), with 130 days remaining. ... For the Manfred Mann album, see 2006 (album). ... is the 145th day of the year (146th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... For the Manfred Mann album, see 2006 (album). ... August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ... This article concerns the British newspaper. ... For the Manfred Mann album, see 2006 (album). ... is the 232nd day of the year (233rd in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... August 24 is the 236th day of the year in the Gregorian calendar (237th in leap years), with 129 days remaining. ... July 25 is the 206th day of the year (207th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... New Scientist is a weekly international science magazine covering recent developments in science and technology for a general English-speaking audience. ... August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... The New Yorker is an American magazine that publishes reportage, criticism, essays, cartoons, poetry and fiction. ... August 21 is the 233rd day of the year (234th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... For the Manfred Mann album, see 2006 (album). ... August 24 is the 236th day of the year in the Gregorian calendar (237th in leap years), with 129 days remaining. ... The New York Times is an internationally known daily newspaper published in New York City and distributed in the United States and many other nations worldwide. ... For the Manfred Mann album, see 2006 (album). ... August 15 is the 227th day of the year in the Gregorian calendar (228th in leap years), with 138 days remaining. ... For the Manfred Mann album, see 2006 (album). ... August 15 is the 227th day of the year in the Gregorian calendar (228th in leap years), with 138 days remaining. ... The Guardian is a British newspaper owned by the Guardian Media Group. ... is the 228th day of the year (229th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... The New York Times is a daily newspaper published in New York City by Arthur Ochs Sulzberger Jr. ... 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... April 15 is the 105th day of the year in the Gregorian calendar (106th in leap years). ... For the Manfred Mann album, see 2006 (album). ... is the 232nd day of the year (233rd in leap years) in the Gregorian calendar. ... July 17 is the 198th day of the year (199th in leap years) in the Gregorian calendar. ... shelby was here 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... New Scientist is a weekly international science magazine covering recent developments in science and technology for a general English-speaking audience. ... shelby was here 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... April 15 is the 105th day of the year in the Gregorian calendar (106th in leap years). ... For the Manfred Mann album, see 2006 (album). ... August 22 is the 234th day of the year (235th in leap years) in the Gregorian calendar. ...

See also

• Geometrization conjecture
• Homology sphere
• Manifold Destiny (On The New Yorker article)
• Soul theorem
• Uniformization theorem

The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. ... Manifold Destiny[1] is an article in The New Yorker written by Sylvia Nasar and David Gruber, published in the August 28, 2006 issue of the magazine, but made available online around August 21, 2006. ... The New Yorker is an American magazine that publishes reportage, criticism, essays, cartoons, poetry and fiction. ... In mathematics, the soul theorem is the following theorem of Riemannian geometry: If (M,g) is a complete non-compact Riemannian manifold with sectional curvature K ≥ 0, then (M,g) has a compact totally convex, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...

External links

• Grigori Perelman at the Mathematics Genealogy Project
• 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)
• Staff listing for Perelman at Petersburg Department of Steklov Institute of Mathematics
• Mathematics & Mechanics Faculty of St. 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