Poincaré conjecture
The conjecture asks whether the same is true for the 3-sphere, which is an object living naturally in four dimensions. This question motivated much of modern mathematics, especially in the field of topology. The question was finally settled in 2002 by Grigori Perelman, a Russian mathematician, with methods from geometry, showing that it is indeed true. He was awarded a Fields Medal and the $1 million Millennium Prize for his work, both of which he declined.