Short answers, pulled from the story.
Grigori Perelman is famous for proving the Poincaré conjecture, a problem in topology that had been unsolved for roughly a century since Henri Poincaré proposed it in 1904. He posted his proof in three preprints on arXiv in 2002 and 2003. The journal Science named it the scientific Breakthrough of the Year on the 22nd of December 2006, the first such recognition given to a mathematical result.
Perelman refused the Fields Medal in 2006 on the grounds that if the proof was correct, no other recognition was needed. He also expressed broader disillusionment with the ethical standards of the mathematics community. He made him the only person ever to decline the Fields Medal.
Perelman rejected the Millennium Prize in July 2010, stating that the Clay Institute's decision was unfair because it did not recognize Richard Hamilton's foundational contribution to Ricci flow and the conjecture's resolution. He said his main reason was disagreement with the organized mathematical community and its decisions.
Perelman lives in Saint Petersburg, Russia, in seclusion. He has declined requests for interviews since 2006. His mother Lyubov also lives in Saint Petersburg with him.
The Poincaré conjecture, proposed in 1904, asks whether any closed three-dimensional manifold in which every loop can be contracted to a point must be topologically equivalent to a three-sphere. Perelman proved it as a corollary of his broader proof of Thurston's geometrization conjecture, using a technique called Ricci flow with surgery developed on the foundations laid by Richard Hamilton.
Perelman won a gold medal at the International Mathematical Olympiad in 1982, shortly after his sixteenth birthday, as a member of the Soviet team competing in Budapest. He achieved a perfect score.