It had been expected that Perelman would turn down the $1 million Clay Prize that he won for resolving the Poincaré Conjecture. But the nice part about his refusal is that he gave a good reason. He thinks the prize is unfair on Hamilton who pioneered the Ricci flow method that Perelman then perfected to complete the proof.
I think he is making a good point. The Clay Millenium Prize is given to anyone who can resolve a number of specific mathematical problems. This is unlike other prizes in science and mathematics that are given to people either for some previously undetermined discovery, or even just a life’s work. The millenium prize is supposed to encourage people to take up mathematics, but Perelman has demonstrated that it is not a desire for money that solves problems. It is an inner wish to understand nature or mathematics that drives people.
Furthermore, many problems in mathematics are only resolved after someone makes a breakthrough in a different area that did not at first seem related. Fermat’s Theorem was a typical example of that. Trying to persuade people to tackle a few grand problems head on is probably not the best way to advance mathematics.
But the point made through Perelman’s refusal hits an even stronger point against the prize. If someone makes a discovery that takes a big step towards solving one of the millenium prizes, what are they likely to do? There is a risk that some people may decide to delay publishing the result until they have utterly exhausted their ability to go further. Perhaps in some cases a good discovery could completely unpublished because someone does not want to give away a step that could solve the bigger problem. They no there will be little glory and no money for a partial solution that helps someone else finish it.
Of course this risk exists even without the prize, but the prize raises the status of the millenium problems making other steps seem less important. It is not just the money, the personal significance of solving one of the problems is also heightened. As it happens, the work of Hamilton towards the Poincaré conjecture was carried out before the prize was founded. We don’t know how such a prize would have affected publication of his work. I like to think he worked in the same spirit as Perelman and the prize would not have made much difference to him, but will that be true for everyone?
It is neither fair nor useful to give a prize only to someone who completes a specific problem. It would be like paying only the goal scorers in a football match. The Clay Institute could change the way the prize is awarded but they wont. They want to create individual heros who are associated with their name. Congratulations to Perelman for solving the problem and for taking a stand against such ill-conceived prizes.