The Poincaré Conjecture: The Only Millennium Prize Problem Solved

A Russian mathematician proved it from his apartment in Saint Petersburg, turned down the Fields Medal, and then turned down the million dollars too.

Of the seven Millennium Prize Problems, only one already has a proof accepted by the entire mathematical community and certified by the Clay Mathematics Institute: the Poincaré conjecture. It's also the story with the most human drama of the whole list: a mathematician solves one of the greatest open problems of the century and then turns down both the recognition and the money.

What the conjecture actually states

Formulated by Henri Poincaré in 1904, the conjecture deals with three-dimensional shapes. Put intuitively: if you have a closed three-dimensional space (no boundary, finite size) in which any loop you draw can be shrunk down to a point without leaving the shape — meaning there are no "holes" preventing it — then that shape is, in essence, a three-dimensional sphere.

For a two-dimensional surface, this is easy to picture: the skin of a balloon satisfies the condition and is, indeed, a sphere; the surface of a donut (a torus) doesn't, because a loop going around the central hole can't be shrunk to a point. Poincaré asked whether the same was true one dimension up, for three-dimensional shapes inside a four-dimensional space. That seemingly innocent generalisation turned out to be extraordinarily hard to prove.

"Anything hole-free, in three dimensions, is a sphere. The sentence fits on one line. The proof took ninety-eight years."

Almost a century of attempts

Throughout the 20th century, the Poincaré conjecture became one of the most sought-after problems in topology. Several top-level mathematicians published proofs that later turned out to be flawed, which, far from being a failure, helped develop entirely new mathematical tools. Interestingly, the version of the problem in five or more dimensions was solved earlier (Stephen Smale, 1961) than the four-dimensional version (Michael Freedman, 1982); the original three-dimensional version was the toughest of all.

The piece that changed everything: Ricci flow

In the 1980s, mathematician Richard Hamilton proposed a radically different strategy: instead of attacking the shape directly, use an equation — the Ricci flow — that progressively deforms the space, smoothing out its irregularities in a way similar to how heat diffuses evenly through an object. The idea was that, applied correctly, any shape satisfying the conjecture's conditions would eventually "flow" toward a perfect sphere.

The problem is that Ricci flow, applied on its own, can generate singularities: points where the deformation becomes infinite and the process breaks down. Hamilton developed much of the programme but got stuck exactly there, without a general way to handle those singularities.

Grigori Perelman solves the problem

Between 2002 and 2003, Russian mathematician Grigori Perelman published online (not in a journal, but directly to the arXiv repository) three papers completing Hamilton's programme. His key contribution was a technique called Ricci flow with surgery: when the flow approaches a singularity, the problematic part of the shape is "surgically" cut away, the process continues separately on each piece, and in the end it's shown that the result is consistent with the original conjecture.

The proof was so dense and used such new techniques that several teams of mathematicians took several years to verify it line by line. It wasn't until 2006, with the publication of detailed independent analyses, that the mathematical community broadly accepted the proof as correct.

Two historic refusals

This is where the story steps outside pure mathematics. In 2006, the International Mathematical Union decided to award Perelman the Fields Medal — the closest equivalent to a Nobel Prize in mathematics. Perelman declined to accept it, and didn't even travel to the Madrid congress where it was awarded.

In 2010, when the Clay Mathematics Institute officially confirmed his proof and he was due the million-dollar prize, Perelman turned it down again. In various interviews, he explained that he found it unfair for his work to be recognised above Richard Hamilton's earlier contributions, which he had directly built on, and expressed a broader objection to how the mathematical community decides who deserves credit and who doesn't.

Since then, Perelman has remained largely withdrawn from public academic life. He's probably the most-cited example of someone who solves one of the hardest problems ever posed and decides that recognition simply doesn't interest them.

Why does it matter today?

Beyond the human anecdote, the Poincaré conjecture demonstrates something that goes beyond mathematics: sometimes the key to solving an old problem isn't attacking it head-on, but building an entirely new tool (Ricci flow) designed for another purpose and applying it with the missing piece. The techniques developed by Hamilton and Perelman are still used today in differential geometry, well beyond the original problem.

It's also the benchmark against which any announcement about the remaining problems gets measured: formal publication, years of independent review, and community consensus. As you can see in our article on OpenAI's recent Navier-Stokes announcement, that process is far from complete in most cases.