In 1904, Henri Poincaré published a paper in which he asked a question so precise and so simple to state that it sounds like it should have an obvious answer: if a closed three-dimensional space has the property that every loop in it can be continuously shrunk to a point, must it be topologically equivalent to the three-sphere? The three-sphere is the natural three-dimensional analogue of the two-dimensional surface of a ball: a closed, bounded, simply connected space. Poincaré had already proved the analogous result in two dimensions — a closed surface with no holes is a sphere — and he conjectured that the same should hold in three dimensions. A century passed before anyone could prove it.
The condition — every loop can be shrunk to a point — is the definition of simple connectivity. It rules out spaces with handles or holes through which a loop could be threaded and trapped. A torus is not simply connected because a loop going around the tube cannot be shrunk; a sphere is simply connected because every loop on it can slide to a point without leaving the surface. Poincaré's conjecture says that simple connectivity is the defining property of the three-sphere among closed three-manifolds: if your space has no holes of any kind, it must be a sphere. The claim is not obvious. There are three-manifolds that are complicated but also simply connected, and the conjecture says all of them are secretly the same thing.
The analogous result in higher dimensions — for four-sphere, five-sphere, and so on — was proved in the 1960s. Smale proved it for dimensions five and above in 1961, winning a Fields Medal in part for this. Freedman proved the four-dimensional case in 1982, also earning a Fields Medal. The three-dimensional case was the hardest, and it resisted all attempts until 2002 and 2003, when Grigori Perelman posted three papers to arXiv containing a proof.
Perelman's approach was not what anyone had expected. He did not attack the Poincaré conjecture directly. He proved a much more general result — Thurston's geometrization conjecture — of which Poincaré's conjecture is a special case. Thurston had conjectured in the 1970s that every closed three-manifold can be decomposed into pieces, each of which admits one of eight standard geometric structures. This is the three-dimensional version of the uniformization theorem for surfaces, and it classifies all three-dimensional spaces in the same spirit that the classification of surfaces (sphere, torus, double torus, etc.) classifies two-dimensional ones. Perelman proved it using a technique called Ricci flow with surgery, introduced by Richard Hamilton — a process that deforms a manifold's metric in a way that tends to smooth out irregularities, eventually resolving it into geometrically uniform pieces.
Perelman was awarded the Fields Medal in 2006 and declined it. He was also awarded the Clay Millennium Prize — one million dollars for solving one of the seven Millennium Problems — and declined that too. He had already left academic mathematics and, as far as anyone can tell, simply stopped caring about recognition. The proof was later verified by multiple independent teams and is now fully accepted. The theorem is true: a closed, simply connected three-manifold is homeomorphic to the three-sphere.
The cosmological implication is what makes this more than pure mathematics. We live in a three-dimensional space — or, correcting for relativity, a three-dimensional spatial slice of a four-dimensional spacetime — and we do not know its global topology. General relativity tells us about the local curvature of spacetime, not its global shape. A universe could be locally flat and globally a three-torus (finite but unbounded, wrapping around itself), or locally curved and globally a three-sphere. The question of which shape our universe has is observational, not mathematical, and current data is consistent with several possibilities.
What Perelman's proof establishes is one endpoint of the classification: if our universe is simply connected, it is topologically a three-sphere. This does not tell us whether it is — that remains an open empirical question — but it tells us that the category "simply connected closed three-manifold" is not a vast and varied zoo of possibilities. It is exactly one thing. The mathematics has constrained what the universe is allowed to be, even if we cannot yet observe which possibility is actual. There is something I find compelling about that: a theorem that was purely about topology, proved by a person who refused its rewards, has become part of the language in which we describe the shape of everything.