The Poincaré Conjecture
$1,000,000 USD (declined)
1904
Henri Poincaré
Grigori Perelman (2002–2003)
TL;DR
The Poincaré Conjecture asks a question about the shape of the universe — or more precisely, about the shape of any three-dimensional object without boundary.
Imagine a rubber band around an apple: you can always shrink it to a point without cutting the band or lifting it off the surface. That property is called "simply connected." Now imagine the band around a donut, threaded through the hole: no matter what you do, you can't shrink it. The donut is not simply connected.
Poincaré conjectured that in three dimensions, the sphere is the only simply connected, closed manifold — every other such shape must, topologically, be a sphere. The 2D and higher-dimensional versions were proved decades earlier; the 3D case turned out to be the hardest.
Grigori Perelman posted three papers to arXiv in 2002–2003 that proved not just the Poincaré Conjecture but the more general Geometrization Conjecture. He was awarded the Fields Medal in 2006 and the Clay Millennium Prize in 2010. He declined both — declaring the prize belonged to Richard Hamilton, whose Ricci flow program he had completed.
Formal statement
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere .
Why it matters
The Poincaré Conjecture pinned down the topology of three-dimensional space itself. Its proof introduced Ricci flow with surgery as a fundamental tool that has since reshaped geometric analysis, differential geometry, and even parts of mathematical physics.
History
Henri Poincaré formulated the conjecture in 1904, in the last of his six-part series "Analysis Situs" — the founding work of algebraic topology. He initially believed the conjecture obvious, then posed it as an open question in the final installment.
For most of the 20th century, no useful attack existed. The higher-dimensional analogues fell first: Stephen Smale proved the conjecture for dimensions ≥ 5 in 1961 (Fields Medal 1966). Michael Freedman handled dimension 4 in 1982 (Fields Medal 1986). The 3-dimensional case — closest to physical intuition and seemingly simplest — remained.
Richard Hamilton introduced Ricci flow in 1982, a technique that smooths a manifold's geometry by evolving its metric via a heat-like PDE. Hamilton and his students showed Ricci flow works beautifully for certain manifolds but develops singularities in general — the equation blows up in finite time along thin necks.
Grigori Perelman spent nearly a decade in relative isolation at the Steklov Institute in St. Petersburg developing techniques to handle those singularities via "surgery": cutting out the pathological regions and continuing the flow. He posted his proof in three papers to arXiv between November 2002 and July 2003. Multiple independent teams verified the proof over the next three years.
Current status of research
Solved. The proof is complete, verified, and published in expository form by Kleiner–Lott, Cao–Zhu, and Morgan–Tian. Ricci flow has since become a standard tool with applications well beyond the original conjecture.
Notable attempts
Numerous prior attempts by respected mathematicians (Christos Papakyriakopoulos, Colin Rourke, Rob Kirby's students) claimed proofs at various points during the 20th century; all were retracted or found flawed. Perelman's success was possible only after decades of groundwork by Hamilton and others on Ricci flow.
What a resolution would mean
The full Geometrization Conjecture (which subsumes Poincaré) classifies all closed 3-manifolds — a milestone comparable to the classification of surfaces in dimension 2. Ricci flow is now applied to Kähler geometry, general relativity, and mathematical physics.