🇷🇺Grigori
Perelman

In November 2002, a paper appeared on arXiv — the online repository where mathematicians post work before formal publication — under the title "The Entropy Formula for the Ricci Flow and Its Geometric Applications." The author was Grigori Perelman, a Russian mathematician who had been largely absent from the international mathematical community for nearly a decade. The paper was terse, dense, and did not include proofs of several of its central claims. Mathematicians who read it spent days trying to understand what it was saying. Slowly, with growing astonishment, they understood. If the argument worked — and within months it became clear that it did — Perelman had proved the Poincaré Conjecture, one of the seven Millennium Prize Problems, a question that had stood unsolved since Henri Poincaré posed it in 1904. He had done it with tools that nobody had expected to work. He had done it alone. He did not notify anyone in advance. He simply posted the paper and waited.

Grigori Yakovlevich Perelman was born on June 13, 1966, in Leningrad, now St. Petersburg, to a Jewish family. His father was an electrical engineer. His mother, who had given up her own graduate studies to raise him, recognized and cultivated his mathematical ability from the beginning. He was enrolled in a special mathematics and physics school and trained for mathematical competition with an intensity that was, in Soviet educational culture of the era, not unusual for children of exceptional ability. In 1982, at sixteen, he represented the Soviet Union at the International Mathematical Olympiad and achieved a perfect score, winning gold. The achievement placed him among the most promising young mathematicians in the country, and he was admitted to Leningrad State University's Faculty of Mathematics and Mechanics, where the most gifted Soviet students in mathematics received their formation.

After completing his doctorate at the Steklov Institute of Mathematics in Leningrad, Perelman spent several years in the United States, holding positions at institutions including the Courant Institute, Stony Brook, and Berkeley, where his work on Riemannian geometry earned him considerable respect and several prestigious job offers that he declined. In 1995 he returned to Russia and to the Steklov Institute, and largely disappeared from international view. He was, as best anyone could determine, working on the Poincaré Conjecture — a problem so hard that most topologists considered it approachable only through the method of Ricci flow developed by Richard Hamilton, and so intractable even with that method that few believed a proof was imminent.

The Poincaré Conjecture states, roughly, that any closed three-dimensional manifold with the topological properties of a three-sphere is in fact homeomorphic to a three-sphere — that is, any shape that behaves like a sphere in certain precise ways actually is a sphere. The statement is straightforward; the proof, which requires understanding how three-dimensional shapes can be continuously deformed and what obstacles might prevent such deformation, had defied the most powerful tools in topology for nearly a century. Perelman's approach was to take Hamilton's Ricci flow — a process that smooths out irregularities in the curvature of a manifold over time, analogous in some ways to heat diffusing through a material — and handle the "singularities," the points where the flow breaks down, through a technique he called Ricci flow with surgery. The three papers he posted between 2002 and 2003 laid out this approach in full generality, and they proved not just the Poincaré Conjecture but the more general Geometrization Conjecture of William Thurston.

In 2006, the International Mathematical Union awarded Perelman the Fields Medal. He refused it. He gave, through the journalist Masha Gessen who was writing a biography, a brief explanation: he did not consider the mathematical community's judgment fair or honest, particularly with respect to how credit was being assigned for the proof. In 2010, the Clay Mathematics Institute awarded him the Millennium Prize — one million dollars. He refused that too. He said that his contribution was no greater than Hamilton's, and that the prize was unfair if Hamilton was not also recognized. The Clay Institute declined to split the prize. Perelman declined the money. He lives in St. Petersburg with his mother. He has not published mathematics since 2006. He does not give interviews. He walks in the forests outside the city. He is, by every account, content.