Millennium Problems

The Millennium Problems are a set of seven problems for which the Clay Mathematics Institute offered a <dollar/>7 million prize fund (one million per problem) to celebrate the new millennium in May 2000. The problems all have significant impacts on their field of mathematics and beyond, and were all unsolved at the time of the offering of the prize.

The seven problems are the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equations, P versus NP, the Poincaré Conjecture, the Riemann Hypothesis, and the Yang-Mills Theory. In 2003, the Poincaré Conjecture was proven by Russian mathematician Grigori Perelman.

History

Announcement

The millennium problems were first announced at Millennium Meeting on May 24, 2000 at the Collège de France. Timothy Gowers first presented a lecture titled The Importance of Mathematics as an introduction. After this, the British mathematician Michael Atiyah and the American John Tate announced the prize: one million dollars to anyone who could solve one of the seven most difficult open problems at the time.

A small committee of mathematicians, selected by the scientific advisory board (SAB) of the Clay Mathematical Institute (which also had organized the meeting), had selected the problems over the previous several months. They were led by Arthur Jaffe, the first director of the CMI, the former director of the American Mathematical Society, and the incumbent of the Landon Clay Chair in Mathematics at Harvard University. This committee included such luminaries as Andrew Wiles, the aforementioned Atiyah and Tate, the American Edward Twitten, and the French Alaine Connes.

Motivation

Partly, the motive of the CMI and its founder (see "Rules and Financing") was the founder's support of mathematical research. However, specifically, the inspiration was a similar prize exactly a hundred years earlier.

Paris had seen a similar event then, at the second International Congress of Mathematicians. The famous German mathematician David Hilbert drew up a list of 23 "Hilbert Problems" on August 8, "setting the agenda for the twentieth century". (Devlin 2003, pp. 2–3) These problems he believed to be the most significant and important unsolved in mathematics.

Some of these problems were either shown to be unsolvable, indefinite, or trivial. However, many were difficult problems, and enormous prestige was given to a mathematician who solved one of them as soon as the mathematical community had pronounced his solution correct. All but one of these problems had been solved by the meeting in 2000, and thus it was natural to create a new set of such problems.

Wiles, however, notes that Hilbert's and the CMI's motivations were slightly different; "Hilbert was trying to guide mathematics by his problems; we're trying to record great unsolved problems. There are big problems in mathematics that are important but where it is very hard to isolate one problem that captures the program." (Devlin 2003, p. 3)

Rules and Financing

The prize fund is financed privately by Landon Clay, the man who had established the CMI one year earlier as a nonprofit organization based in his hometown; Cambridge, MA to aid mathematical research. Clay, a well-off mutual-fund manager, though not a mathematician, was greatly interested in and supportive of the mathematical community.

The rules were laid out as follows: The SAB would consider a proof of one of the problems on several conditions. Firstly, the proof had to be complete. Secondly, it had to "be published in a refereed mathematics publication of worldwide repute and [...] general acceptance in the mathematics community two years after". (Millennium Prize Rules; as of January 19, 2005) If these conditions are met, the SAB will appoint an advisory committee to thoroughly examine the solution. This committee would consist of at least two world-renowned mathematicians and at least one member of the SAB.

After the analysis, the committee would report back to the SAB. The SAB would then report to the directors of the CMI, possibly giving recommendations of whether or not the prize should be awarded and (in the case of a group of mathematicians collaborating on one problem or multiple mathematicians solving a problem near-simultaneously) which person(s) should receive the prize. In the case of multiple prizewinners, the prize would be divided proportionally according to the judgement of the directors. Counterexamples are put through essentially the same process; again, the directors make the final decision, though the SAB can advise them.

In the special case of a problem being shown to be false as stated, but ambiguous with a small adjustment, a small prize may be awarded to the mathematician who discovered this; though the money would be taken from funds other than the Millennium Prize ones. The new problem would then replace the old in the selection of problems, with the same process and conditions.

References

• Devlin, Keith J (2003). Basic Books. The Millennium Problems. ISBN 978-0465017300.

External Links

• Millennium Prize Problems at Clay Mathematics Institute
• Millennium Prize Rules at Clay Mathematics Institute