In 2006, the journal Science named the solution to the century-old Poincaré Conjecture, outlined more than three years earlier by Russian mathematician Grigori Perelman, its "breakthrough of the year." Three years later, the Clay Mathematics Institute, which had pledged to award $1 million to the person who solves this or any of six others of its Millennium Prize Problems, has yet to name a winner. Despite the special circumstances of the Poincaré case, the delay sheds light on the onerous process of vetting solutions to mathematics' toughest problems.
In 2002 and 2003, Perelman, then a researcher at St. Petersburg's Steklov Institute, posted three papers outlining his proof on arXiv.org, a site where scientists make preprints of their work available sooner than traditional journals' peer-review process allows. But unlike other researchers, the eccentric and reclusive Perelman (who later rejected the prestigious Fields Medal for his achievement) never went on to submit his proof to a refereed journal.
He thus skipped the first crucial step for a solution to be considered for a Millennium Prize. Such publication isn't merely standard in scientific work — it's also an easy way for the Clay Institute to delegate the labor-intensive process of prescreening potential candidates, explains Jim Carlson, the mathematician who directs the Clay Institute. Before Andrew Wiles proved Fermat's Last Theorem, the University of Göttingen had put in 90 years sifting through thousands of submissions — many of them cranks — and sending polite rejections. "The Clay Institute didn't want to spend resources on this kind of activity," Carlson says, adding that the Institute prefers to fund research. The many purported solutions to the P versus NP problem (some "proving" that P=NP and others concluding the opposite) bears out Carlson's concern. Since no candidates have appeared in a qualifying journal, the Clay Institute isn't even looking at solutions to P versus NP or any other Millennium Prize Problem — except the Poincaré Conjecture.
For that problem, the Clay Institute paved the way for referred publication, in part by funding other mathematicians to devote themselves to painstakingly explicating Perelman's proof. "His proof was extremely sketchy, even to people who are experts in that particular area," says Stanford researcher Keith Devlin, the author of The Millennium Problems. "But when the experts agreed, it was clear he knew the proof."
Even so, after two pairs of mathematicians separately published their explications in 2007, the clock merely began to tick away at the Institute's required two-year waiting period, time for the rest of the math community to ensure the proof is solid. "For these major conjectures, the proofs are such extraordinary results that they should be subject to extraordinary scrutiny," Carlson says, citing a history of flaws in attempted proofs of difficult theorems, including Fermat's Last.
For the Poincaré, the two-year period is over, yet Perelman's proof must clear another set of hurdles: verification by a committee appointed by the Clay Institute's Scientific Advisory Board and a final decision about whether, when, and how to award the prize. Given that Perelman would almost certainly spurn the honor, these deliberations are less about mathematical rigor than about perceptions of the Clay Institute — and of math itself. "Landon Clay established the prize to share mathematics as this noble pursuit of ultimate truth," says Devlin. "It would be a tragedy if the story were to spin in the opposite direction."
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