The Poincaré conjecture started out as a question asked by the founder of the subject of topology, Henri Poincaré, in 1904: Is a simply connected, compact, three-dimensional manifold necessarily a hypersphere? Poincaré's intuition that this was one of the most important questions of topology has stood the test of time. In nearly a century since then, topologists have solved a host of related problems. They have invented deep and powerful theories to scale Poincaré's fortress, and these theories have enabled them to gain potentially revolutionary insights into the shape of our universe. Nevertheless, the original question asked by Poincaré remains unanswered.

For non-specialists, the significance of the Poincaré conjecture is obscured by the thicket of terms one has to get through to understand it:

• A hypersphere is the surface of a four-dimensional ball, or the solution set...