Published online 6 October 2006 | Nature | doi:10.1038/news061002-14

Corrected online:

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Has $1 million maths problem been cracked using novel techniques?

Turbulent flow: the maths to pin it down are tricky. Getty

A buzz is building that one of mathematics' greatest unsolved problems may have fallen.

Blogs and online discussion groups are spreading news of a paper posted to an online preprint server on 26 September. This paper, authored by Penny Smith of Lehigh University in Bethlehem, Pennsylvania, purports to contain an " immortal smooth solution of the three-space-dimensional Navier-Stokes system".

If the paper proves correct, Smith can lay claim to $1 million in prize money from the Clay Mathematics Institute, based in Cambridge, Massachusetts. In 2000, the institute listed the Navier-Stokes problem among its seven Millennium Prize Problems.

The Navier-Stokes equations describe how a fluid flows. They are derived by applying Newton's laws of motion to the flow of an imcompressible fluid, and adding in a term that accounts for energy lost through the liquid equivalent of friction, viscosity.

What mathematicians want to know is whether these equations always behave themselves, or their solutions sometimes diverge — which would amount to physical impossibilities, such as fluid mass disappearing. Smith claims to show that solutions to the equations never diverge.

Experts in the field say that it is too early to make a call on whether the paper is correct, but they are beginning to comb through the work. Among those looking at the paper is Charles Fefferman, a mathematician at Princeton University, New Jersey, who wrote the description of the Navier-Stokes problem for the Clay Mathematics Institute. "It would be a spectacular achievement of the highest order if it turned out to be right," he says.

The Navier-Stokes equations serve not only as the basis of a challenge to mathematicians but also underpin many practical exercises in physics and engineering, such as the design of chemical plants. Any mathematics developed to tackle the equations could also be put to use in computer simulations or might provide new insights into the nature of complex phenomena such as turbulence.

The challenge

“Somebody asked me why don't I work on one of the Clay problems, so I looked around for one.” Penny Smith

Lehigh University

Smith says that she was prompted to tackle the problem by a colleague, beginning work on it only one month ago.

Her expertise lies in solving differential equations, and she says that she has developed new mathematical tools to do so. She gave a lecture on how these tools could be applied to another set of equations. At the end of the lecture, she says, "somebody asked me why don't I work on one of the Clay problems. So I looked around for one to do with differential equations".

She spotted that the Navier-Stokes equations could be rewritten in the form of the differential equations that she knows how to solve. The method works by setting upper and lower bounds to the solution, then squeezing them together to show that they converge. The paper that she posted online has also been submitted to the Journal of Mathematical Analysis and Applications1, she says.

"I'm pretty confident that my result is right, or I would never have submitted it anywhere," says Smith. "Of course, when there's so much attention being paid it does activate every piece of insecurity one has ever had." This anxiety has led her to revise the paper a number of times since posting it to the arXiv preprint server — but the changes were to correct typographical errors, rather than anything mathematically significant, she says.

The proof

There have been previous claims of solutions to the Navier-Stokes problem, says Fefferman. He recalls seeing maybe half-a-dozen such papers over the past few years, most of which he discovered to have obviously fatal errors within a matter of hours.

He expects that assessing Smith's work will take much longer than that. Although the paper itself is only nine pages long, it relies heavily on her earlier publications, so Fefferman will have to trawl through those too. The earlier papers are in peer-reviewed journals or are listed as "to appear" in such journals. "That increases the probability that they're right, but for something this important I wouldn't trust that," says Fefferman.

Smith used to attend seminars, she says, with Grigory Perelman, the Russian mathematician who is believed to have solved another of the Millennium Prize Problems: the proof of the Poincaré Conjecture. He recently turned down the most prestigious prize in mathematics, a Fields medal (see 'Maths 'Nobel' prize declined by Russian recluse') ; it is rumoured he would not accept a Clay million if it was offered to him.

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Like Perelman, Smith says that she is motivated by the mathematics, not the money. She hopes to serve as a role model for women in mathematics. "On the other hand, I certainly want the prize," she says.

James Carlson, president of the Clay Mathematics Institute, says he is seeking opinions on the paper, but adds, "It is far too early to say whether it is correct or not." To win the prize, Smith's work will have to withstand two years of scrutiny after appearing in a peer-reviewed journal.

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Lehigh University

Corrected: UPDATE 9 Oct 2006: On 8 October, Penny Smith wrote to Nature to say: "I have just found a serious flaw in the paper and have withdrawn it." The work has been pulled from arXiv.