In November 2002, a paper appeared on the arXiv server under number math/0211159—thirty-nine pages, author Grisha Perelman, title "The entropy formula for the Ricci flow and its geometric applications." No cover letter, no announcements in specialized journals. Just a file in open access. And within a few weeks, the global mathematical community understood: before them was the first part of a proof of the Poincaré conjecture, a problem that had awaited solution for over a century and had been included in the list of seven Millennium Prize Problems by the Clay Mathematics Institute with a prize of one million dollars.
The Poincaré conjecture sounds deceptively simple: any closed three-dimensional surface without holes, on which every loop can be contracted to a point, is topologically equivalent to a sphere. On the two-dimensional surface of a sphere, any rope thrown onto its surface contracts to a single point—unlike a torus, where a rope wrapped around the "donut" won't move anywhere. The three-dimensional analog of this property describes the structure of the space we live in, but mathematicians had been unable to prove this statement since 1904, when French mathematician Henri Poincaré formulated it.
Richard Hamilton in the 1980s developed the method of Ricci flow—a deformation of the geometry of a manifold by analogy with the propagation of heat in physics. Imagine that a three-dimensional surface is a piece of clay that begins to melt according to certain rules: curved areas smooth out, bulges spread, the surface strives for uniformity. The problem is that during deformation, singularities can arise—points where curvature rushes to infinity and the flow breaks down. Hamilton hoped that if one could learn to control these breaks and "surgically" cut them out, the flow would bring the manifold to standard geometric forms—and prove not only the Poincaré conjecture, but also the more general Thurston geometrization conjecture.
In his first preprint, Perelman presented a monotonic function—an expression that decreases along the Ricci flow and allows control over how singularities behave. In the abstract, he stated directly: several assertions related to Hamilton's program for proving the geometrization conjecture for closed three-dimensional manifolds are verified. Perelman did not invent a new method from scratch—he completed a twenty-year program proposed before him, but stuck on the problem of controlling the injectivity radius in regions where singularities form.
November 11, 2002—first preprint. March 2003—second. July 2003—third. Perelman did not submit his work to Annals of Mathematics or Inventiones Mathematicae, where peer review could take years. He posted the proof on arXiv, an open preprint archive, and sent links to several colleagues. The mathematical community gained access to the text instantly—but not to its understanding.
Perelman wrote concisely. He omitted intermediate steps, referenced results by Hamilton and other geometers, not always indicating where exactly in their works to look for the needed statement. For specialists in Ricci flow, this was the language of professionals—but even they could not verify every transition without months of work. Unlike classical publication, where referees check the text in closed mode before release, Perelman's preprints launched a process of public verification: mathematicians around the world read, discussed, searched for gaps.
Bruce Kleiner and John Lott undertook to write a detailed version of the proof—a decryption of Perelman's compressed arguments for a wider audience. John Morgan and Gang Tian worked on an alternative exposition. Huai-Dong Cao and Xi-Ping Zhu published an article in 2006 in the Asian Journal of Mathematics, where they also presented a complete exposition of the proof—and this sparked the loudest scandal around the proof. The journal was headed by Shing-Tung Yau—Hamilton's mentor and the most influential geometer of the era; peer review was completed in record time, and Yau publicly presented the result as a triumph of the Chinese mathematical school. Colleagues saw in this an attempt to redistribute priority. Perelman reacted harshly: "It is not clear to me what new contribution they made. Apparently, Zhu did not quite understand the argument and simply reworked it." And more broadly—about the discipline as a whole: "The strangers here are not those who violate ethical norms—but people like me." Cao and Zhu insisted they had only systematized Perelman's arguments and verified their correctness; Perelman himself later said that the conflict over priority was one of the reasons for his break with the mathematical community.
August 22, 2006, President of the International Mathematical Union John Ball announced at a ceremony in Madrid that Grigori Perelman had been awarded the Fields Medal—the highest award in mathematics—with the citation "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow"—Ball deliberately did not mention the Poincaré conjecture: by July 2006, formal verification had not yet been completed, and he did not want to link the highest award to an unverified proof. Perelman did not come to the ceremony. Did not respond to the invitation to give a plenary lecture. Did not accept the medal.
Ball later linked the refusal to Perelman feeling isolated from the mathematical community and not wanting to become its symbolic figure. Nature News published an article about the refusal, noting that the proof was built on methods for studying deformations of surfaces developed by Hamilton, and that three preprints posted online in 2002 and 2003 became the basis for international verification.
By that time, mathematicians had been dissecting Perelman's texts for several years. Kleiner and Lott were preparing their version of the proof, Morgan and Tian were working on a book, colleagues around the world were conducting seminars and conferences where individual steps were discussed. The process resembled an archaeological expedition: each fragment had to be extracted, cleaned of excess, compared with neighboring finds. Some transitions required weeks of work to make sure that a reference to Hamilton's result was actually applicable in the given context.
In 2010, the Clay Mathematics Institute officially recognized Perelman's proof as complete and awarded him a prize of one million dollars. Perelman refused this too. This was the second public refusal—and this time not of a symbolic award, but of real money. He did not give reasons to institute president James Carlson, but he gave an explanation to the Interfax agency: his contribution is no greater than the contribution of Hamilton, who created the Ricci flow method itself—yet the prize was decreed to be awarded to him alone. "In short, the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unfair." For a man who back in 1996 rejected the prize of the European Mathematical Society on principle, and in 2006 watched as other names accrued his result, this was not a sudden stunt but a consistent position. The mathematical community continued to discuss his motives: some saw in this a protest against the commercialization of science, others—dissatisfaction with how credit was distributed between him and Hamilton; later "extended interviews" about his true motives—like producer Zabrovsky's account in 2011—journalists considered fake.
By the time the prize was awarded, almost eight years had passed since the publication of the first preprint. ArXiv did not cancel scientific verification—it transferred it from editorial procedures to public international work. Instead of two or three referees, closed off from the outside world, the proof was checked by dozens of specialists, exchanging comments at conferences, in correspondence, in their own articles. This was slower than it could have been in an ideal world, but more reliable: each questionable step went through multiple independent checks.
Several dozen pages of preprints required many years of collective analysis before the mathematical community recognized the proof as complete. Hamilton's program for Ricci flow, begun in the 1980s, was brought to completion by Perelman in 2002–2003—but final recognition came only after other mathematicians deciphered the compressed arguments, filled gaps in the exposition, and made sure that surgery on singularities actually worked. Open access did not replace scientific rigor—it distributed it among hundreds of eyes and minds, turning the proof from a closed manuscript into a living organism that grew and tested itself for strength.