The Man Who Solved Poincaré

Grigori Yakovlevich Perelman was born in Leningrad in 1966 to Jewish parents, won the International Mathematical Olympiad with a perfect score as a teenager, proved a twenty-year-old conjecture in Riemannian geometry in 1994, and in November 2002 quietly posted three preprints on arXiv that, by 2006, the mathematical community had accepted as a proof of the Poincaré conjecture — one of the seven Clay Millennium Prize problems and the longest-standing open problem in topology. He then declined the Fields Medal in 2006, declined the Clay prize in 2010, walked away from the Steklov Institute where he had worked, and now lives with his mother in the Kupchino district of St Petersburg, largely unreachable. This is a survey of what he did, what it means, and what to make of the man.

The Poincaré conjecture

The conjecture that Henri Poincaré posed in 1904 is, in its modern form, a single sentence: every compact three-dimensional manifold without boundary that is simply connected (every loop in it can be shrunk to a point) is homeomorphic to the three-sphere. In other words, if a 3-manifold has no “essential holes”, it is a sphere.[1, 2]

The statement is a generalisation, in one additional dimension, of the much older observation that a simply connected 2-manifold is a sphere. The two-dimensional case is the classification of surfaces, a theorem of nineteenth-century topology that Riemann, Möbius, and others had in hand by the 1880s. The three-dimensional case is the deep one. It is not a question about how surfaces curve, which has an answer; it is a question about what the three-dimensional shapes of the universe can be, which is the question the conjecture addresses. The conjecture is a foundational claim about the topology of the space we live in.

The conjecture’s history is the history of twentieth-century topology. It was first proved for dimensions other than three in the 1960s: Stephen Smale in 1961 for dimension five and above (the dimension five case earned Smale the Fields Medal in 1966); John Stallings in 1962 for dimension three of a special kind (a sub-case of the Haken conjecture); and William Thurston in 1982 for the case of Haken manifolds, which he packaged into a more general claim that became the Thurston geometrization conjecture: any 3-manifold can be decomposed into pieces, each of which carries one of eight canonical geometries. The Poincaré conjecture is the special case in which the manifold is simply connected; the only admissible piece is the 3-sphere, so the manifold is a sphere.[1, 3]

Geometrization, if true, would solve not only the Poincaré conjecture but a great swathe of three-dimensional topology. The catch — and the reason the problem stayed open for nearly a century — is that geometrization is not a statement one can prove by clever combinatorial or algebraic manipulation alone. The 3-sphere is a particular Riemannian geometry, and the question is whether any simply connected 3-manifold can be smoothly deformed into it.

The soul conjecture

Perelman had already established himself in the late 1980s and early 1990s as one of the most striking geometers of his generation. His work with Yuri Burago and Mikhael Gromov on Alexandrov spaces — generalised metric spaces with a synthetic notion of curvature — became the standard reference for the subject. In 1994, in a paper of four pages in the Journal of Differential Geometry, he proved the soul conjecture: a conjecture of Jeff Cheeger and Detlef Gromoll from 1972, that any complete Riemannian manifold of non-negative sectional curvature that has a single point of strictly positive curvature must be diffeomorphic to Euclidean space. Perelman’s proof ran to four pages where the previous attempts had run to dozens.[4, 5]

The result is significant for two reasons. The first is that it confirmed a structural picture of non-negatively curved manifolds that had been the central open question in Riemannian geometry for twenty years. The second is the kind of result it is: a short, decisive, almost complete argument that uses classical tools with great economy. It is, in retrospect, a foretaste of the work Perelman would do eight years later on a much bigger problem.

After the soul-conjecture paper, Perelman was offered jobs at Princeton, Stanford, and the Courant Institute. He declined them all, returning to the Steklov Institute of Mathematics in St Petersburg (then still called the Steklov Mathematical Institute of the Russian Academy of Sciences, in Leningrad; the city’s name had not yet been restored) for a research-only position, a post he held from 1995 onward.[6]

The Ricci flow and the proof

The decisive technical advance came not from Perelman alone but from the program of Richard S. Hamilton. In 1982, Hamilton introduced the Ricci flow, a partial differential equation defined on a Riemannian metric by ∂g/∂t = −2 Ric(g). It is a continuous deformation of the metric in the direction of its own Ricci curvature. Locally, where curvature is positive, the flow shrinks the metric; where it is negative, the flow expands. The Ricci flow is, in form, analogous to the heat equation of physics, and it deforms a complicated Riemannian metric in the direction of uniformity.[1, 3]

Hamilton’s 1982 paper already established that the Ricci flow, on certain classes of 3-manifolds, smooths initial singularities and produces, in finite time, a geometry that can be classified. In three follow-up papers in the 1980s and 1990s, Hamilton showed that the Ricci flow could resolve the geometry of 3-manifolds of positive Ricci curvature, settling the four-dimensional analogue of the Poincaré conjecture in those cases. The obstacle to the three-dimensional case was a singular behaviour: the Ricci flow can run into infinite curvature in finite time, and no one knew how to continue the flow past a singular point. Hamilton’s “surgery” program — cut out the singular region, glue in a known piece, resume the flow — was a sketch of how to do it; he had not been able to make it work in general.

On 11 November 2002, a paper appeared on the arXiv preprint server, “The entropy formula for the Ricci flow and its geometric applications”, by one author: G. Perelman, Steklov Institute, St Petersburg. The paper’s title, in contrast to its author’s reputation, was understated. The opening section was titled “Ricci Flow as a Gradient Flow”, and it identified a new quantity, which Perelman called entropy, that the Ricci flow always increases. This ruled out one of the kinds of singularities that had stymied Hamilton. Two follow-up papers followed in March and July 2003, filling in the technical machinery: “Ricci flow with surgery on three-manifolds” and “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds”.[1, 3]

The result, in summary, is that Ricci flow with surgery, run on a simply connected closed 3-manifold, eventually rips the manifold apart, leaving a finite collection of spherical pieces. By the geometrization machinery, this means the original manifold was homeomorphic to the 3-sphere, which is the Poincaré conjecture. The result, in more detail, is a hundred pages of intricate analysis, with several new technical ideas that had not appeared in any earlier work in the field. Hamilton’s program was completed. The Poincaré conjecture was proved.[1, 2, 3]

Perelman did not submit the papers to a journal. He posted them on arXiv, the open preprint server, where the proofs could be read by any interested mathematician. The verification took roughly four years. Three independent groups — Cao and Zhu at Lehigh; Morgan and Tian at Columbia and Princeton; Kleiner and Lott at Yale and Michigan — published, between 2006 and 2008, their own expositions filling in the technical details. In 2006, the consensus was that the proof was correct.[2]

The Fields Medal refusal

On 22 August 2006, at the International Congress of Mathematicians in Madrid, the president of the International Mathematical Union announced that one of the four Fields Medal laureates of the year, named for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”, was Grigori Perelman, and that he had declined the prize. He was the first person in the prize’s seventy-year history to do so.[6, 7]

In an interview with Sylvia Nasar and John Miller in The New Yorker, published 19 August 2006, Perelman explained that he did not consider the medal meaningful. “It was completely irrelevant for me,” he said. “Everybody understood that if the proof is correct, then no other recognition is needed.”[7] He also said, in a separate interview with Interfax, that he did not want to appear at the ceremony and did not want to be “treated as a pet”.

Perelman also gave a more general statement of why he was withdrawing from the professional mathematical community. The community, he said, was conformist. “There are many mathematicians who are more or less honest,” he told Interfax. “But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.” He added, again to Interfax, that “it is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated.”[7, 8] He had, in the years before, expressed specific complaints about the credit given to Hamilton’s program and the conduct of the broader field.

The refusals were not isolated gestures. In the same period, Perelman resigned his position at the Steklov Institute and stopped publishing in the professional mathematical literature. The last paper he would write for many years was the July 2003 third preprint on the Ricci flow.[1, 6]

The Clay Millennium Prize

On 18 March 2010, the Clay Mathematics Institute announced that Perelman had met the criteria to receive its first-ever Millennium Prize, the one-million-dollar award for the resolution of one of seven open problems in mathematics. The institute’s usual requirement — peer-reviewed journal publication — was waived; the institute’s Scientific Advisory Board judged that the post-publication verification process had been so extensive that the work counted as more than normally peer-reviewed.[6, 8]

Perelman refused the prize. He gave two reasons. The first, in Interfax, was that the prize was unfair: “the main reason is my disagreement with the organized mathematical community. I don’t like their decisions, I consider them unjust.” The second, in a separate statement, was that his contribution was no greater than Hamilton’s. The Clay Institute, in its public statement, said that Perelman’s decision was “not a complete surprise” given his earlier refusal of the Fields Medal.[6, 8]

Perelman did not attend the ceremony at the Institut Océanographique in Paris on 8 June 2010, at which the prize was to be conferred. The prize, which he had already declined, was never paid out. He remains, in the formal sense, the recipient of a Clay Millennium Prize that he has never accepted, with the one-million-dollar award returned to the institute’s general endowment.[8]

A Russian Jewish life

Grigori Yakovlevich Perelman was born on 13 June 1966 in Leningrad, Soviet Union, to a Jewish family. His father, Yakov Perelman, was an electrical engineer; his mother, Lyubov (Lubov) Lvovna Steingolts, was a teacher of mathematics at a technical college. The surname is sometimes confused with that of the Russian physics populariser Yakov Isidorovich Perelman, but the two are not related. Perelman’s mother gave up her own graduate study in mathematics to raise him. She played the violin, and the household in which he grew up was a Jewish intellectual household of a kind that Leningrad still had in the 1960s and 1970s, though under strain from official Soviet attitudes toward Jewish culture and identity.[6, 9, 10]

His mathematical talent became apparent at the age of ten. His mother enrolled him in the after-school mathematics training programme of Sergei Rukshin, the Soviet Union’s leading trainer of olympiad-level school-age mathematicians, and the same programme that had produced earlier generations of Soviet Fields and Abel laureates. By sixteen, Perelman had won the Soviet Union’s mathematics olympiad, a national honour.[6, 9]

He won the International Mathematical Olympiad in 1982, the year he finished secondary school, with a perfect score. He enrolled that year at the Leningrad State University, the country’s leading mathematics programme, and graduated in 1985 with honours. He took his candidate of sciences (roughly PhD) at the Steklov Institute in 1990 under A. D. Aleksandrov and Yuri Burago, then stayed on at the Steklov as a researcher.[6]

The Steklov Institute, founded in 1934 and named for the Soviet mathematician Vladimir Steklov, is one of the world’s leading centres of pure mathematics. The St Petersburg branch, where Perelman worked, has been since the 1950s the home of the Soviet school of geometry, including Aleksandrov’s work on convex bodies and Burago’s on Alexandrov spaces. It has, over the past half-century, produced a remarkable fraction of the Russian geometry that has made it into the Western literature.[6]

Jewish mathematicians have been a substantial fraction of the Soviet mathematical community — disproportionate to the Jewish share of the Soviet population and disproportionate to the Jewish share of other Soviet scientific establishments, which were subject to far more aggressive antisemitic discrimination than mathematics. The reason, as with mathematics in the German-speaking world earlier in the century, is partly structural: when a field is open to talent and largely free of political supervision, it attracts a representative slice of the population that includes, in many countries, an over-representation of Jewish scholars. The Soviet mathematics tradition, which produced Perelman and his teachers, was a long chapter in the same story.[6, 10]

His father Yakov emigrated to Israel in 1993; his mother remained in St Petersburg. Perelman has lived, since his withdrawal from the Steklov in the mid-2000s, with his mother in the Kupchino district of St Petersburg, a Soviet-era apartment-block neighbourhood in the south of the city. Reports over the years, mostly second-hand, have placed him on the same Kupchino street, walking his mother’s dog, occasionally attending the Mariinsky Theatre (he is a fan of the opera, not the mathematics), playing violin, and reading broadly. He is largely unreachable. Several colleagues and journalists have made the trip to St Petersburg to find him; some have succeeded, some have not.[6]

Where this leaves us

It is hard to know how to characterise Perelman’s place in the recent history of mathematics, because the work is so recent and the man is so much in retreat. A few things can be said.

He is, in mathematical standing, one of the most important mathematicians of the 21st century. The Poincaré conjecture is, with the Riemann hypothesis and the P/NP problem, one of the three most prominent open problems of modern mathematics. The Clay Mathematics Institute’s decision to award him the prize — even with the publication requirement waived — is the formal recognition that the conjecture is solved. Science‘s “Breakthrough of the Year” recognition in 2006 was the first time the award had gone to a mathematical result.[2, 8]

He is also, in his own life, the rarest kind of mathematician: one who, having done the work, refuses the public honours that the work would normally bring. The Fields Medal has been declined once in seventy years; the Clay Millennium Prize, with its one-million-dollar award, has been refused once in its history. Both refusals were by the same person. The refusals are not the protest gestures of a dissident; they are the considered decisions of a man who has decided that the public life of a famous mathematician is not for him.

What he will do next, or whether he will publish again, is not known. He has not, in the years since 2003, indicated that he is working on any particular mathematical problem. He continues to live quietly with his mother in Kupchino. He is sixty years old in 2026. The mathematical community has, by and large, respected his wishes. The Russian press occasionally publishes interviews with neighbours or local merchants; the international press occasionally speculates about a possible return. There is, on the evidence, no return. There is, on the evidence, no great reason to expect one.

The most useful thing to say, in summary, is also the most obvious one. Perelman did what mathematicians had been trying to do for nearly a century. He did it in three short papers, in an unfashionable location, with no co-authors, and with very little fuss. He then, in his own way, walked away. The Poincaré conjecture is now a theorem. The man who proved it is, for the moment, in Kupchino.

Result / moment	Date	Form	Recognition
International Mathematical Olympiad (perfect score)	1982	Gold medal	First Soviet participant to score 42/42 since 1967
Candidate of Sciences (PhD), Steklov Institute, under Aleksandrov and Burago	1990	Thesis	Russian/Soviet academic qualification
Soul conjecture proved (paper of 4 pp., J. Differential Geom.)	1994	Article	One-page Annals announcement; standard reference in Alexandrov geometry
Three preprints on the Ricci flow and the Poincaré conjecture	Nov 2002 – Jul 2003	arXiv preprints (no journal submission)	None from the author; Science “Breakthrough of the Year”, December 2006
Fields Medal, declined	22 Aug 2006	Award (not collected)	First person in the prize’s history to decline
Clay Millennium Prize, declined	18 Mar 2010 (announced), 1 Jul 2010 (refused)	$1,000,000 prize (not collected)	First and only Clay Millennium recipient to refuse the award

References

1. “Poincaré conjecture.” Wikipedia. https://en.wikipedia.org/wiki/Poincaré_conjecture
2. Collins, Graham P. “The Poincaré Conjecture - Proved.” Science 314(5807):1848 - 1849, 22 December 2006. https://www.science.org/doi/10.1126/science.314.5807.1848
3. Morgan, John W.; Tian, Gang. “Ricci Flow and the Poincaré Conjecture.” arXiv:math/0607607, 2006. (Expanded exposition of Perelman’s proof.) https://www.alphaxiv.org/abs/math/0607607
4. Perelman, G. “Proof of the soul conjecture of Cheeger and Gromoll.” Journal of Differential Geometry 40 (1994), 209 - 212. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-40/issue-1/Proof-of-the-soul-conjecture-of-Cheeger-and-Gromoll/
5. Lott, John. “The Work of Grigory Perelman.” (Berkeley preprint, 2006). https://math.berkeley.edu/~lott/preport.pdf
6. “Grigori Perelman.” Wikipedia. https://en.wikipedia.org/wiki/Grigori_Perelman
7. Nasar, Sylvia; Miller, John. “Manifold Destiny.” The New Yorker, 19 August 2006. (Cited here for the Fields Medal refusal statements.) https://www.newyorker.com/magazine/2006/08/21/manifold-destiny
8. “The man who refused the Fields Medal may also…: Dr. Grigoriy Perelman and the Clay Millennium Prize.” International Mathematical Union / Clay Mathematics Institute official notice, ICM 2010. https://www.mathunion.org/fileadmin/IMU/ICM2010/offline/www.icm2010.in/wp-content/icmfiles/uploads/Perelman.pdf
9. “Grigori Yakovlevich Perelman (1966 - ).” Biographical entry, MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Perelman/
10. “Grigori Perelman: A Man Who Solved the World’s Most Difficult Math Problem and Then Walked Away.” Times of India, science feature. https://timesofindia.indiatimes.com/science/meet-grigori-perelman-the-math-genius-who-solved-a-100-year-old-problem-and-rejected-a-1-million-prize-and-the-fields-medal/articleshow/131097202.cms
11. Perelman, G. “The entropy formula for the Ricci flow and its geometric applications.” arXiv:math.DG/0211159, 11 November 2002.
12. Perelman, G. “Ricci flow with surgery on three-manifolds.” arXiv:math.DG/0303109, 10 March 2003.
13. Perelman, G. “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.” arXiv:math.DG/0307245, 17 July 2003.
14. Hamilton, Richard S. “Three-manifolds with positive Ricci curvature.” Journal of Differential Geometry 17 (1982), 255 - 306.
15. Thurston, William P. “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry.” Bulletin of the American Mathematical Society 6(3) (1982), 357 - 381.
16. Cao, Huai-Dong; Zhu, Xi-Ping. “A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman Theory of the Ricci flow.” Asian Journal of Mathematics 10(2) (2006), 165 - 492.
17. Morgan, John W.; Tian, Gang. The Geometrization Conjecture. American Mathematical Society, 2014.
18. Kleiner, Bruce; Lott, John. “Notes on Perelman’s papers.” Geometry & Topology 12 (2008), 2587 - 2855. https://msp.org/gt/2008/12-5/gt-2008-12-005.pdf
19. “Soul conjecture.” Wikipedia. https://en.wikipedia.org/wiki/Soul_conjecture
20. Burago, Yuri; Gromov, Mikhael; Perelman, G. “A. D. Alexandrov spaces with curvature bounded below.” Russian Mathematical Surveys 47(2) (1992), 1 - 58.