$500,000 for mathematician who laid Poincaré groundwork

Reclusive mathematician Grigory Perelman proved the Poincaré conjecture. But he refused a $1 million prize for solving this famous puzzle – in part because he believed he wasn't the only one who deserved credit. Now the mathematician who Perelman wanted to see recognised has won a lucrative prize.

On Tuesday, the Shaw Prize Foundation in Hong Kong announced that it would split its annual $1 million prize in the mathematical sciences, with half going to Richard Hamilton at Columbia University in New York, who devised a geometrical process that underpins Perelman's proof of the Poincaré conjecture. The other half goes to Demetrios Christodoulou at the Swiss Federal Institute of Technology Zurich, who works on black hole physics and general relativity.

Henri Poincaré's famous statement, posed in 1904, is one of the oldest and most fundamental conjectures in the field of topology, sometimes called "rubber sheet geometry" – the mathematical study of surfaces and shapes with all possible twists, turns and dimensions. It suggests a way to recognise hard-to-imagine geometrical objects that have more than three dimensions. More specifically, it teaches mathematicians how to recognise higher-dimensional spheres, even if they've been smashed or distorted like Play-Doh.

Beautiful idea

Though Poincaré suspected his test worked, he couldn't prove it – and for a long time, nor could anyone else. In the 1980s, Hamilton devised Ricci flow, a powerful mathematical tool that could be applied to abstract shapes and smooth them out.

"He developed this theory of Ricci flow pretty much from scratch," says Jim Carlson, president of the Clay Mathematics Institute in Cambridge, Massachusetts. "Very early on he had this beautiful idea of what kind of equation could govern the change of a shape. He proceeded to prove a whole series of fantastic results."

But, despite his efforts, these did not include the Poincaré conjecture. Hamilton's idea of Ricci flow didn't work on every different possible case – until Perelman came along and found a way around the roadblocks.

Hamilton's dream

Perelman posted his proof on the arXiv.org site in three parts in 2002 and 2003. In 2006, after other mathematicians had verified the accuracy of his proof, he was awarded the Fields medal, one of the highest honours in mathematics. He refused it – and remained mostly mum about his reasons.

Last July, Perelman also turned down a $1 million prize that the Clay institute awarded him for proving the Poincaré conjecture. On the eve of the Clay celebration, he told the Russian news agency Interfax that he thought the organised mathematical community was "unjust" and that he did not like their decisions. Richard Hamilton, he said, deserved as much credit for the proof as he did.

"The solution to the Poincaré conjecture is an enormous step forward in topology and geometry that we now know is made possible by Hamilton's ideas," adds Carlson. "Perelman should be viewed as achieving the dream that Hamilton had."

Source New Scientist