Grigori Yakovlevich Perelman (born 1966) is a Russian mathematician who in 2002–2003 proved the Poincaré Conjecture, one of the seven Millennium Prize Problems. In 2006 he turned down the Fields Medal. In 2010 he turned down the Clay Mathematics Institute’s one-million-dollar prize. Both refusals were unprecedented.
Life
Perelman was born in Leningrad (now St. Petersburg) in 1966. His father was an electrical engineer; his mother taught mathematics at a technical college and gave up her own career to support her son’s mathematical development. He attended a specialized school for mathematically gifted children, then Leningrad State University, then the Steklov Institute for his doctorate.
In the 1990s he held positions at Courant Institute (New York), UC Berkeley, and SUNY Stony Brook. His work on Alexandrov geometry and the soul conjecture established him as a rising star. In 1995 he returned to Steklov in St. Petersburg.
Then he went silent. For seven years, he published almost nothing and corresponded with few colleagues.
The Poincaré proof
The Poincaré Conjecture, posed by Henri Poincaré in 1904, states that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Informally: any 3-dimensional shape without holes is a sphere in disguise.
The conjecture resisted almost a century of attacks. It was one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute in 2000, each carrying a one-million-dollar reward.
The Ricci flow approach
The American mathematician Richard Hamilton had, starting in the 1980s, developed a technique called Ricci flow — a way of continuously deforming a manifold to simplify its geometry, analogous to heat flow smoothing out temperature differences. Hamilton conjectured that Ricci flow could prove Poincaré, but he could not handle the singularities that developed during the flow.
Perelman, between 2002 and 2003, posted three papers to the arXiv preprint server that solved the problem. He showed how to understand and manage the singularities of Ricci flow, and in doing so proved not just the Poincaré Conjecture but the far stronger Thurston Geometrization Conjecture, which classifies all closed 3-manifolds.
The papers were only about 70 pages total. Verifying them took teams of mathematicians several years and hundreds of additional pages of exposition. The proofs held.
The refusals
In 2006, the International Mathematical Union awarded Perelman a Fields Medal. He declined, telling the IMU president: “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.”
In 2010, the Clay Mathematics Institute awarded him the one-million-dollar Millennium Prize. He refused this too, citing what he saw as the unfairness of crediting him alone when Richard Hamilton’s work had been essential. “The main reason is my disagreement with the organized mathematical community,” he reportedly said. “I don’t like their decisions, I consider them unjust.”
Life after
Perelman has lived quietly in St. Petersburg with his mother. He rarely speaks to the press, has given no mathematical talks in over a decade, and is reported to have withdrawn from professional mathematics. Rumors have him working on other problems in private. Nothing has been published.
In 2011 he reportedly told an interviewer: “There are many, many people working on mathematics. I don’t need to be in the front.”
Legacy
Perelman’s proof is one of the major mathematical achievements of the 21st century. Together with Hamilton’s groundwork, it solved a 100-year-old problem and validated an entire approach to geometric analysis. Along the way, the Thurston Geometrization Conjecture — a deep classification result — was also settled.
His public refusals raised questions that mathematicians are still thinking about: what does recognition do to a field; what does money do to a field; can a mathematical life be lived entirely for its own sake.
The Poincaré Conjecture is, to date, the only one of the seven Millennium Prize Problems to have been solved.
Known for
- Proof of the Poincaré Conjecture (2002–2003)
- Refusing the Fields Medal (2006)
- Refusing the Clay Mathematics Institute's $1M prize (2010)
Frequently asked
What did Perelman prove?
The Poincaré Conjecture: that every simply-connected, closed 3-manifold is topologically equivalent to the 3-sphere. Informally: any 3-dimensional object with no holes is a sphere in disguise. Posed in 1904, it was one of the seven Millennium Prize Problems.
Why did he refuse the prizes?
Perelman has given different reasons at different times. He has said mathematical prizes are 'completely irrelevant' and objected to what he called injustices in the broader mathematical community. His refusals were unprecedented — no one had ever turned down both the Fields Medal and the Millennium Prize.
What is he doing now?
Essentially unknown. Perelman has lived in St. Petersburg with his mother since roughly 2005, avoids the press, and has largely withdrawn from professional mathematics.