The Poincaré Conjecture
Imaginestretching a rubber band around the surface of an apple, then shrinking it down slowly. This shrinking could occur without tearing the rubber band or breaking the apple - and the band would never have to leave the surface. However, if this rubber band were to be stretched across, say, a tire - there is no way to shrink to a point without breaking one or the other. The surface of such an apple is “simply connected,” but the tire is not. Henri Poincaré (shown below), during the early twentieth century - knew that two dimensional spheres had this ‘connected’ property - and he asked if the same applied for three dimensional spheres.
The conjecture turned out to be immensely difficult to prove. After more than a century, Grigori Perelman finally devised a solution. In 2006, Perelman was awarded the Fields Medal for this contribution, but he decided to turn it down, stating that:
“I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo.”