Grigori Perelman
Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман, IPA: [ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman] ; born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his research post in Steklov Institute of Mathematics and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006.
In this name that follows Eastern Slavic naming customs, the patronymic is Yakovlevich and the family name is Perelman.
Grigori Perelman
- Proof of the soul conjecture
- Proof of the Poincaré conjecture and geometrization of 3-manifolds
- Saint Petersburg Mathematical Society Prize (1991)
- EMS Prize (1996), declined
- Fields Medal (2006), declined
- Millennium Prize (2010), declined
In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years.
In August 2006, Perelman was offered the Fields Medal[1] for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."[2] On 22 December 2006, the scientific 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.[3]
On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize[4] for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture.[5][6] He had previously rejected the prestigious prize of the European Mathematical Society in 1996.[7]
Early life and education[edit]
Grigori Yakovlevich Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents,[8][9][10] Yakov (who now lives in Israel)[8] and Lyubov (who still lives in Saint Petersburg with Grigori).[8] Grigori's mother Lyubov 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 mathematics training program.[11]
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.[12] In 1982, not long after his sixteenth birthday, he won a gold medal as a member of the Soviet team at the International Mathematical Olympiad, achieving a perfect score.[13] He continued as a student of the School of Mathematics and Mechanics at Leningrad State University, without admission examinations, and enrolled at the university.
After completing his PhD in 1990, Perelman began work at the 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, with a strong recommendation from the geometer Mikhail Gromov,[14] Perelman obtained research positions at several universities in the United States. In 1991, Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below.[15] In 1992, he was invited to spend a semester each at the Courant Institute in New York University, where he began work on manifolds with lower bounds on Ricci curvature. From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley, in 1993. After proving 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.[11]
Early research[edit]
Convex geometry[edit]
In his undergraduate studies, Perelman dealt with issues in the field of convex geometry. His first published article studied the combinatorial structures arising from intersections of convex polyhedra.[P85] With I. V. Polikanova, he established a measure-theoretic formulation of Helly's theorem.[PP86] In 1987, the year he began graduate studies, he published an article controlling the size of circumscribed cylinders by that of inscribed spheres.[P87]
Negatively curved hypersurfaces[edit]
Surfaces of negative curvature were the subject of Perelman's graduate studies. His first result was on the possibility of prescribing the structure of negatively-curved polyhedral surfaces in three-dimensional Euclidean space. He proved that any such metric on the plane which is complete can be continuously immersed as a polyhedral surface.[P88] Later, he constructed an example of a smooth hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature negative and bounded away from zero. Previous examples of such surfaces were known, but Perelman's was the first to exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes.[P89] As such, his construction provided further obstruction to the extension of a well-known theorem of Nikolai Efimov to higher dimensions.[16]
Alexandrov spaces[edit]
Perelman's first works to have a major impact on the mathematical literature were in the field of Alexandrov spaces, the concept of which dates back to the 1950s. In a very well-known paper coauthored with Yuri Burago and Mikhael Gromov, Perelman established the modern foundations of this field, with the notion of Gromov–Hausdorff convergence as an organizing principle.[BGP92] In a followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all Alexandrov spaces with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually homeomorphic.[P91] Vitali Kapovitch, who described Perelman's article as being "very hard to read," later wrote a detailed version of Perelman's proof, making use of some further simplifications.
Perelman developed a version of Morse theory on Alexandrov spaces.[P93] Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the gradient flow of certain functions, in unpublished work.[PP95] They also introduced the notion of an "extremal subset" of Alexandrov spaces, and showed that the interiors of certain extremal subsets define a stratification of the space by topological manifolds.[PP93] In further unpublished work, Perelman studied DC functions (difference of concave functions) on Alexandrov spaces and established that the set of regular points has the structure of a manifold modeled on DC functions.[P95d]
For his work on Alexandrov spaces, Perelman was recognized with an invited lecture at the 1994 International Congress of Mathematicians.[P95a]
Comparison geometry[edit]
In 1972, Jeff Cheeger and Detlef Gromoll established their important soul theorem. It asserts that every complete Riemannian metric of nonnegative sectional curvature has a compact nonnegatively curved submanifold, called a soul, whose normal bundle is diffeomorphic to the original space. From the perspective of homotopy theory, this says in particular that every complete Riemannian metric of nonnegative sectional curvature may be taken to be closed. Cheeger and Gromoll conjectured that if the curvature is strictly positive somewhere, then the soul can be taken to be a single point, and hence that the original space must be diffeomorphic to Euclidean space. In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, Sharafutdinov's retraction is a submersion.[P94b] Perelman's theorem is significant in establishing a topological obstruction to deforming a nonnegatively curved metric to one which is positively curved, even at a single point.
Some of Perelman's work dealt with the construction of various interesting Riemannian manifolds with positive Ricci curvature. He found Riemannian metrics on the connected sum of arbitrarily many complex projective planes with positive Ricci curvature, bounded diameter, and volume bounded away from zero.[P97b] Also, he found an explicit complete metric on four-dimensional Euclidean space with positive Ricci curvature and Euclidean volume growth, and such that the asymptotic cone is nonuniquely defined.[P97c]
Fields Medal and Millennium Prize[edit]
In May 2006, a committee of nine mathematicians voted to award Perelman a Fields Medal for his work on the Ricci flow.[38] 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 attempted 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."[38] He was quoted as saying, "'I'm not interested in money or fame, 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.'"[46]
Nevertheless, on 22 August 2006, at the International Congress of Mathematicians in Madrid, Perelman was offered the Fields Medal "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow".[47] He did not attend the ceremony and the presenter informed the congress that Perelman declined to accept the medal, which made him the only person to have ever declined the prize.[7][48]
He has also rejected a prestigious prize from the European Mathematical Society.[7]
On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem.[49] On 8 June 2010, he did not attend a ceremony in his honor at the Institut Océanographique, Paris to accept his $1 million prize.[50] According to Interfax, Perelman refused to accept the Millennium Prize in July 2010. He considered the decision of the Clay Institute unfair for not sharing the prize with Richard S. Hamilton,[5] and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."[6]
The Clay Institute subsequently used Perelman's prize money to fund the "Poincaré Chair", a temporary position for young promising mathematicians at the Paris Institut Henri Poincaré.[51]
Perelman and the media[edit]
Perelman has avoided journalists and other members of the media. Masha Gessen, author of a biography about Perelman, Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century, was unable to meet him.[58]
A 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" ("Maverick: Perelman's Lesson").
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.[59] Zabrovsky says that in the interview, Perelman explained why he rejected the one million dollar prize.[59]
A number of journalists[60][61][62] believe that Zabrovsky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.
The writer Brett Forrest briefly interacted with Perelman in 2012.[63][64] A reporter who had called him was told: "You are disturbing me. I am picking mushrooms."[65]
Dissertation
Research papers
Unpublished work
Media related to Grigori Perelman at Wikimedia Commons