Perelman achieved fame in the mathematics world for his work on the Poincaré Conjecture, one of topology's most celebrated problems. The conjecture, conceived by the French mathematician Henri Poincaré in 1904, relates to the question of whether it is possible to deform a holed doughnut into a sphere by bending and stretching it – without cutting or tearing it.
It turns out that there is no way to remove the "hole" in the doughnut and so it cannot be turned into a sphere. However, any shape that has no holes can always be deformed into a sphere. The Poincaré Conjecture and a more general problem, called the Thurston Geometrization Conjecture, assert that the same is true for shapes in higher dimensions.