This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
In 1904 Henri Poincare tucked a question into the last page of a long paper on topology, the field he had all but founded: is every simply connected, closed, three dimensional shape really the same, topologically, as an ordinary sphere? He did not attempt to prove it. "Cette question nous entrainerait trop loin," he wrote. This question would take us too far.
It took nearly a century. Mathematicians tried and failed for decades, and by the time Grigori Perelman, a mathematician at the Steklov Institute in Saint Petersburg, began circulating three papers on the arXiv preprint server between November 2002 and July 2003, most of the field had come to think of the Poincare conjecture the way number theorists once thought of Fermat's Last Theorem: true, almost certainly, and probably unprovable by any method anyone had yet found. Perelman's route was not new so much as completed. Richard Hamilton had spent twenty-five years building a program around an equation he introduced in 1982, the Ricci flow, which smooths out the curvature of a shape the way heat smooths out temperature differences in a metal plate. Hamilton's program kept running into shapes that pinched off into singularities before it could finish the job. Perelman found the tools to steer around them.
Because Perelman published on the arXiv rather than in a refereed journal, and because the argument was dense even by the standards of differential geometry, the mathematical community did not simply take his word for it. Several independent teams spent years writing up full expositions of the proof: Bruce Kleiner and John Lott, whose notes were finished in 2006; John Morgan and Gang Tian, whose book appeared in 2007; Huai-Dong Cao and Xi-Ping Zhu, whose own account, and the controversy it caused, is its own story; and a European team, Bessieres, Besson, Boileau, Maillot and Porti, in 2009. By the end of that process the Clay Mathematics Institute, which had named the Poincare conjecture one of its seven Millennium Prize Problems in 2000, considered the case closed.
What happened next is the part of the story most people remember. At the end of May 2006 a committee of the International Mathematical Union voted to award Perelman a Fields Medal, mathematics' highest honor, for his work on the Ricci flow, ahead of the public ceremony that August at the IMU's quadrennial congress in Madrid. No mathematician had ever declined one. Perelman did. IMU president Sir John Ball flew to Saint Petersburg to try to change his mind, offering him three options: accept and attend the ceremony in Madrid, accept and receive the medal later, or refuse outright. "From the very beginning," Perelman told him, "I told him I have chosen the third one." Four years later the Clay Mathematics Institute awarded him the Millennium Prize itself, one million dollars, for resolving the conjecture. He declined that too, telling reporters he considered Richard Hamilton's contribution no less than his own. He had already resigned from the Steklov Institute in December 2005 and has lived a private life in Saint Petersburg since, largely out of public view.
Perelman's own account of why he stepped back from a field he had just conquered was less about modesty than about disappointment. "It is not people who break ethical standards who are regarded as aliens," he said. "It is people like me who are isolated." Whatever else the Poincare conjecture proved, it proved that a hard problem can still surprise the people who solve it, and the people who watch them do it.