Poincaré Conjecture

In 2002, Perelman proved the Poincaré Conjecture, which had stumped mathematicians since 1904. However, Perelman refused to accept the Fields Medal, the highest honor in mathematics, for the proof. He was the first and, thus far, only person to turn down the award. Four years later, he turned down the awarding of the first Clay Millennium Prize, as well as the accompanying $1 million.