Grigori Perelman

• 1. Introduction

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  Grigori Perelman was offered one million dollars, and he said no. He was offered the highest honor in mathematics, the Fields Medal, and he said no to that too. He declined both not out of modesty, but out of conviction. When Sir John Ball, president of the International Mathematical Union, traveled to Saint Petersburg in June 2006 and spent ten hours across two days trying to change his mind, Perelman was unmoved. His response was simple: if the proof is correct, then no other recognition is needed.

  Perelman was born on the 13th of June 1966 in Leningrad, which is now Saint Petersburg. He solved the Poincaré conjecture, a problem that had sat unsolved for a century. The proof was so compressed and difficult to follow that a committee of leading mathematicians spent years checking and expanding his work before the world could be sure he had actually done it. He posted his findings not in a peer-reviewed journal but as informal preprints on the internet, written without the usual academic formalities. Then he stopped going to work, stopped answering the phone, and retreated into his apartment.

  What kind of mind produces a proof like this, and then walks away? And why did a mathematician who revolutionized the understanding of space and shape come to regard his own field as ethically broken?

• 2. A Gifted Childhood

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  Lyubov Perelman gave up her own graduate work in mathematics to raise her son. That sacrifice had consequences. Grigori's talent became apparent by the time he was ten, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program in Leningrad. The program gave him a structure that matched his abilities in a way ordinary school could not.

  At the Leningrad Secondary School 239, a school with advanced mathematics and physics programs, Perelman excelled in every subject except physical education. He was a student who operated at a different register from his peers. In 1982, shortly after his sixteenth birthday, he traveled to Budapest as a member of the Soviet team competing in the International Mathematical Olympiad. He achieved a perfect score and won a gold medal.

  From there he went to the School of Mathematics and Mechanics at Leningrad State University. After completing his PhD in 1990, he joined the Leningrad Department of the Steklov Institute of Mathematics. His doctoral advisors were Aleksandr Aleksandrov and Yuri Burago, two mathematicians whose influence would shape the early arc of his research. In 1991, the Saint Petersburg Mathematical Society awarded him its Young Mathematician Prize for work on Alexandrov spaces. That award pointed toward the territory where his early career would unfold.

• 3. Building Foundations

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  Alexandrov spaces date back conceptually to the 1950s, but it was Perelman's collaboration with Yuri Burago and Mikhael Gromov that gave the field its modern foundations. Their joint paper became one of the best-known works in the area. In a follow-up paper, Perelman proved a stability theorem: within any collection of Alexandrov spaces sharing a fixed curvature bound, all elements close enough to a compact space are topologically equivalent to one another.

  Vitali Kapovitch later wrote a detailed version of Perelman's proof, describing the original as very hard to read. The stability theorem was not easy to access from Perelman's compressed account, but its content was sound. Kapovitch's expansion preserved the result while making it available to more readers.

  Perelman also developed a version of Morse theory for Alexandrov spaces, which normally lack the smoothness that classical Morse theory requires. Working with Anton Petrunin, he studied gradient flows and introduced the notion of extremal subsets, showing that certain subsets stratify the space by topological manifolds. This was unpublished work, which would become a pattern. For this body of contributions to Alexandrov spaces, Perelman was invited to deliver a lecture at the 1994 International Congress of Mathematicians.

  In that same year, he settled a twenty-year-old open question. Jeff Cheeger and Detlef Gromoll had proved in 1972 that every complete Riemannian manifold of nonnegative sectional curvature contains a compact submanifold called a soul. They conjectured that if the curvature is strictly positive somewhere, that soul must collapse to a single point, forcing the original space to be topologically equivalent to Euclidean space. Perelman's proof was short. He showed that under the condition of nonnegative sectional curvature, a map called Sharafutdinov's retraction is a submersion, and the conjecture followed. After settling the soul conjecture, Perelman was offered faculty positions at Princeton and Stanford. He turned them all down and returned to the Steklov Institute in Saint Petersburg in the summer of 1995.

• 4. The Century-Old Problem

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  Henri Poincaré proposed his conjecture in 1904. It asked whether any closed three-dimensional manifold in which every loop can be contracted to a point must be topologically equivalent to a three-sphere. Stephen Smale proved the analogous statement in five or more dimensions in 1961. Michael Freedman handled the four-dimensional case in 1982. The three-dimensional version remained untouched, and the methods that worked in higher dimensions offered no purchase on it. Moving problematic regions out of the way, as those topological arguments required, depended on having room to maneuver that simply does not exist in three dimensions.

  In 1982, William Thurston proposed a broader framework. His geometrization conjecture claimed that any closed three-dimensional manifold can be cut along spheres and tori into pieces, each of which admits a uniform geometric structure. The Poincaré conjecture was a special case. Thurston proved his conjecture under some assumptions but not in full generality.

  Richard Hamilton introduced the Ricci flow in the same year. His prescription, governed by a partial differential equation analogous to the heat equation, deforms the metric on a manifold over time in a way that tends to spread out concentrated curvature. Hamilton showed in three seminal papers in the 1980s that this process could uniformize geometry in certain settings. The difficulty was that in general, singularities develop: curvature concentrates to infinite levels in finite time. Following a suggestion by Shing-Tung Yau, Hamilton spent the 1990s analyzing those singularities, culminating in a 1997 paper on Ricci flow with surgery for four-dimensional spaces. Yau identified that paper as one of the most important in geometric analysis. In 1999, Hamilton published a paper on three-dimensional Ricci flow laying out what would be needed to resolve Thurston's conjecture, a program that became known as the Hamilton program. The key difficulty was that Hamilton had classified singularities in four dimensions but could not do so in three.

• 5. The Proof

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  In November 2002, Perelman posted his first preprint to arXiv. A second followed in March 2003, and a third in July 2003. He did not submit them to a journal. He simply put them online, in a compressed style that omitted many technical details.

  The first preprint contained two major results. Perelman adapted differential Harnack inequalities developed by Peter Li and Shing-Tung Yau to the Ricci flow setting. From that adaptation he derived his noncollapsing theorem, which says that local control of curvature implies control of volume. This was significant because volume control is a precondition for Hamilton's compactness theorem, and with that precondition satisfied, Hamilton's tools could be applied more freely.

  The second major result of the first preprint was the canonical neighborhoods theorem. This gave a precise description of what three-dimensional Ricci flow singularities look like at a microscopic scale: they resemble either a cylinder collapsing to its axis, or a sphere collapsing to its center. This was the quantitative understanding of three-dimensional singularities that Hamilton had not been able to achieve. The proof drew on extensive arguments by contradiction, applying Hamilton's compactness theorem in ways enabled by the noncollapsing theorem.

  The second preprint constructed a Ricci flow with surgery in three dimensions, systematically excising singular regions as they appeared, and proved that the result applied to Thurston's conjecture. The third preprint, along with work by Tobias Colding and William Minicozzi, showed that Ricci flow with surgery on any space satisfying the Poincaré conjecture's conditions exists only for finite time. As a corollary, the Poincaré conjecture followed.

  In April 2003, Perelman traveled to the Massachusetts Institute of Technology, Princeton, Stony Brook, Columbia, and New York University to give short series of lectures clarifying his work for specialists. The mathematical community recognized immediately that he had made major contributions to Ricci flow. Whether those contributions constituted a complete proof of either conjecture took years to determine.

• 6. Verification and Credit

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  Bruce Kleiner and John Lott, then both at the University of Michigan, began posting notes on Lott's website in June 2003 that filled in details of Perelman's first preprint section by section. By September 2004 they had extended their notes to cover the second preprint. They posted a version to arXiv on the 25th of May 2006, and a modified version was published in the journal Geometry and Topology in 2008. At the 2006 International Congress of Mathematicians, Lott said that all indications were that Perelman's arguments were correct. Their written introduction explained that while Perelman's proofs were concise and at times sketchy, they had not found any serious problems, meaning problems that could not be corrected using Perelman's own methods.

  In June 2006, the Asian Journal of Mathematics published a paper by Huai-Dong Cao of Lehigh University and Zhu Xiping of Sun Yat-sen University, giving what they described as a complete proof of both conjectures. Their abstract called it the crowning achievement of the Hamilton-Perelman theory of Ricci flow. Some readers interpreted this framing as a claim to credit at Perelman's expense. When asked, Perelman said he could not see any new contribution by Cao and Zhu and that they had not quite understood certain arguments and had reworked them. A further problem emerged when one page of their article turned out to be essentially identical to a page from Kleiner and Lott's 2003 notes. In a published erratum, Cao and Zhu said this was an oversight from note-taking in 2003.

  John Morgan of Columbia and Gang Tian of MIT posted a detailed presentation of Perelman's proof of the Poincaré conjecture in July 2006. On the 24th of August 2006, Morgan delivered a lecture at the International Congress of Mathematicians in Madrid declaring that Perelman's work had been thoroughly checked. In 2015, Abbas Bahri identified a counterexample to one theorem in Morgan and Tian's work, which they later corrected; the error had been introduced by Morgan and Tian themselves in details not directly addressed by Perelman. Their two articles were eventually published in book form by the Clay Mathematics Institute.

  On the 22nd of December 2006, the journal Science named Perelman's proof of the Poincaré conjecture the scientific Breakthrough of the Year, the first time that recognition had been given to a result in mathematics.

• 7. Declining Every Prize

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  In May 2006, a committee of nine mathematicians voted to award Perelman the Fields Medal. He refused to attend the ceremony. Sir John Ball traveled to Saint Petersburg and spent ten hours over two days attempting to persuade him. Perelman later described Ball offering three options: accept and come, accept and not come and receive the medal by post, or decline. Perelman said he chose the third from the start. He told Ball the prize was completely irrelevant to him.

  His words on the subject were direct. He said he was not interested in money or fame, that he did not want to be on display like an animal in a zoo, and that he was not a hero of mathematics. On the 22nd of August 2006, at the congress in Madrid, the presenter informed the audience that Perelman had declined, making him the only person ever to have refused the Fields Medal.

  He had already declined a prestigious prize from the European Mathematical Society in 1996. The pattern was consistent.

  On the 18th of March 2010, Perelman was announced as the recipient of the first Clay Millennium Prize for resolving the Poincaré conjecture, carrying an award of one million dollars. On the 8th of June 2010, he did not appear at the ceremony held in his honor at the Institut Oceanographique de Paris. He later rejected the prize, stating that he considered the Clay Institute's decision unfair because it did not recognize Richard Hamilton's role. He said his disagreement was with the organized mathematical community and that he considered their decisions unjust. The Clay Institute subsequently used the unclaimed prize money to fund the Poincaré Chair, a temporary position for young promising mathematicians at the Paris Institut Henri Poincaré.

• 8. Withdrawal and Seclusion

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  Perelman quit his position at the Steklov Institute in December 2005. He told interviewers by 2006 that he had left professional mathematics. The reasons he gave were ethical, not mathematical. A 2006 article in The New Yorker quoted him saying that he was disappointed by the ethical standards of the field, that most mathematicians were conformists who tolerated dishonesty without endorsing it, and that it was people like himself, not those who broke ethical standards, who ended up isolated.

  He described his situation before the Poincaré proof made him famous as a choice between making "some ugly thing" or accepting the role of a pet, someone tolerated but not taken seriously. Once he became conspicuous, he said, neither option remained available.

  His friends told journalists that by 2010 he found mathematics a painful subject to discuss and that some believed he had abandoned it entirely. Yakov Eliashberg, a fellow Russian mathematician, said that in 2007 Perelman had confided to him that he was working on other things, though it was too soon to say what. A publication called Le Point reported that he had shown interest in the Navier-Stokes equations, a major set of open problems concerning fluid dynamics. Russian media reported in 2014 that he was working in nanotechnology in Sweden, but he was subsequently seen in Saint Petersburg, and Russian media later speculated that he visited his sister in Sweden while living at home and caring for his elderly mother.

  Masha Gessen, who wrote a biography of Perelman titled Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century, was never able to meet him. A reporter who called him was told: "You are disturbing me. I am picking mushrooms." A Russian documentary titled Maverick: Perelman's Lesson, featuring interviews with Mikhail Gromov, Ludwig Faddeev, Anatoly Vershik, Gang Tian, John Morgan, and others, was released in 2011. Perelman did not participate.