How did a reclusive mathematician solve the Poincaré conjecture and refuse a million dollars?

The Poincaré conjecture asks what a finite three-dimensional space must be like if every loop in it can be tightened to a point. Henri Poincaré posed it around 1904: such a space, he suggested, must be a three-dimensional sphere. In 2002 and 2003 Grigori Perelman posted three papers on arXiv sketching a proof, and then he refused a million dollars for it.

The idea is easiest to see one dimension down. On the surface of a ball, any loop drawn on it can be shrunk to a single point. On the surface of a torus, a doughnut, some loops cannot be, because they go through the hole or around it. So loops can tell a sphere from a torus. The conjecture says the same test works one dimension up.

The twist is that Poincaré himself only posed an open-ended question. He did not venture to guess which way the answer would go. Mathematicians proved the analogous statement in higher dimensions first: Stephen Smale in 1961 for dimensions greater than four, Michael Freedman in 1982 for four. The three-dimensional case, the one Poincaré had asked about, got the reputation of being particularly tricky.

Perelman completed an approach begun by Richard Hamilton, called Ricci flow, an equation formally analogous to the heat equation that gradually reshapes a space's geometry. He sketched a proof of the Poincaré conjecture and of the more general geometrization conjecture of William Thurston. In 2006 he declined the Fields Medal, the only person ever to do so. In 2010 he rejected the Millennium Prize, saying his contribution was no greater than Hamilton's.