Mathematics
The Poincaré Conjecture and the Classification of 3-Manifolds
Quick fact
At the moment, no one can fully characterize all three-dimensional shapes, but in 2003, Grigori Perelman proved the Poincaré Conjecture, which says that any closed 3-manifold with the property that every loop can shrink to a point is actually a 3-sphere. This was the first step toward classifying all 3-manifolds.
Why this is interesting
Imagine a loop of string on a rubber ball: you can always shrink it to a point. But what if you lived inside a three-dimensional shape? How do you know it's just a sphere?