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The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven “Millennium Prize Problems.” The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a

$7 million prize fundfor the solution to these problems, with$1 millionallocated to each. The Directors of the CMI, and no other persons or body, have the authority to authorize payment from this fund or to modify these stipulations. The Board of Directors of CMI makes all mathematical decisions for CMI, upon the recommendation of its Scientific Advisory Board.The Scientific Advisory Board of the Clay Mathematics Institute (SAB) will consider a proposed solution to a Millennium Prize Problem if it is a complete mathematical solution to one of the problems that follow. (In the case that someone discovers a mathematical counterexample, rather than a proof, the question will be considered separately as described below.) A proposed solution to one of the Millennium Prize Problem may

notbe submitted directly to the Clay Mathematics Institute for consideration.Before consideration, a proposed solution must be published in a refereed mathematics journal of world-wide repute, and it must also have general acceptance in the mathematics community

two yearsafter that publication. Following this two-year waiting period, the SAB will decide whether a solution merits detailed consideration. In the affirmative case, the SAB will constitute a special advisory committee, which will include at least one SAB member and at least two non-SAB members who are experts in the area of the problem. The SAB will seek advice to determine potential non-SAB members who are unbiased, internationally-recognized mathematical experts in the area of the problem. As part of consideration, each proposed solution under consideration must be verified by one or more members of this committee.The special advisory committee will report within a reasonable time to the SAB. Based on this report and possible further investigation, the SAB will make a recommendation to the Directors. The SAB may recommend the award of a prize to one person. The SAB may recommend that a particular prize be divided among multiple solvers of a problem or their heirs. The SAB will pay special attention to the question of whether a prize solution depends crucially on insights published prior to the solution under consideration. The SAB may (but need not) recommend recognition of such prior work in the prize citation, and it may (but need not) recommend the inclusion of the author of prior work in the award.

In case that the SAB cannot come to a clear decision about the correctness of a solution to a problem, its attribution, or the appropriateness of an award, the SAB may recommend that no prize be awarded for a particular problem. If new information comes to light, the SAB may (but will not necessarily) reconsider a negative decision to recommend a prize for a proposed solution, but only after an additional two-year waiting period following the time that the new information comes to light. The SAB has the sole authority to make recommendations to the Directors of the CMI concerning the appropriateness of any award and the validity of any claim to the CMI Millennium Prize.

In the case that a mathematician discovers a proposed counter-example to a prize problem, the SAB will consider this counter-example after publication and the same two-year waiting period as for a proposed solution. If the counter-example shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds. In the case the P versus NP problem the SAB may recommend the award of the Millennium Prize for deciding the question in either direction.

A member of the SAB, who is not also a Director, is eligible for the Millennium prize. But in such a case, the Directors must establish a special procedure to evaluate the appropriateness of such a prize without input from, or involvement by, the mathematician in question. In the case of such an award, the President of the CMI should make a statement outlining these procedures.

With the one exception in the prior paragraph, all decision-making procedures concerning the CMI Millennium Prize Problems are private. This includes the deliberations or recommendations of any person or persons the CMI has used to obtain advice on this question. No records of these deliberations or related correspondence may be made public without the prior approval of the Directors, the SAB, and all other living persons involved, unless seventy five years time have elapsed after the event in question.

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RemarksNo proposed solution to a CMI Millennium problem may be submitted directly to the Clay Mathematics Institute. Under the rules for the prize, any proposed solution must be submitted to a peer-reviewed mathematics journal.

Early in 2001, the CMI will publish a book containing the official statements of the Millennium Prize Problems and the official rules. This book will initiate a series of CMI books co-published in cooperation with the American Mathematical Society, and it will be sold through their standard channels of distribution.

The (preliminary) technical statement of the Yang-Mills problem will be posted in the near future. Definitive statements of all seven problems will be available by the end of the summer 2000.

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