In order to celebrate mathematics in the new millennium, 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 have designated a

$7 million prize fundfor the solution to these problems, with$1 millionallocated to each.During the Millennium meeting held on May 24, 2000 at the Collčge de France, Timothy Gowers presented a lecture entitled “The Importance of Mathematics,” aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem.

One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting.

The rules that follow for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.

Scientific Advisory Board:

Alain Connes --- Arthur Jaffe --- Andrew Wiles --- Edward Witten

Directors:

Finn M. W. Caspersen --- Landon T. Clay --- Lavinia D. Clay

William R. Hearst, III --- Arthur M. Jaffe --- David B. StonePresented - Paris, May 24, 2000

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The Math Problems

Yang-Mills Existence and Mass Gap

Navier-Stokes Existence and Smoothness

The Birch and Swinnerton-Dyer Conjecture

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Remarks

No 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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Historical ContextDavid Hilbert was born in Königsberg, Germany, on January 23, 1862. He obtained his doctorate from the University of Königsberg in 1885, where he remained until 1895 when he took up the professorship at Göttingen he was to hold until his death in 1943.

Hilbert's contributions to mathematics have been vast and far-reaching. His early work was concerned with the theory of invariants, while later he moved into the foundations of geometry and the theory of algebraic number fields. At the turn of the century Hilbert's research efforts broadened yet further, encompassing potential theory, the calculus of variations and various areas of mathematical physics. In his later years Hilbert became primarily involved with the foundations of mathematics, and he is now remembered as one of the greatest mathematicians of the twentieth century. Indeed, it is staggering how many deep results and profound conjectures Hilbert produced across the wide spectrum of his mathematical interests. He also wrote monumental texts on the foundations of mathematics, geometry, logic and algebraic number theory. By the end of the nineteenth century Hilbert's achievements had already lifted him to a point from which he dared to chart out the most promising avenues for research in the twentieth century. In 1900 Hilbert gave life to his vision through the formulation of twenty-three problems that he presented at the International Congress of Mathematicians in Paris. These problems have inspired and guided the minds of mathematicians throughout the last century. Out of the original twenty-three problems eight were of a purely investigative nature. To date twelve of the remaining fifteen have been completely resolved. Quite remarkably, only one problem, the so-called Riemann Hypothesis remains as mysterious and challenging as ever, being now widely regarded as the most important open problem in pure mathematics.

The CMI Millennium Prize Problems are not intended to shape the direction of mathematics in the next century. Rather these problems focus attention on a small set of long-standing mathematical questions, each central to mathematics, that also have resisted many years of serious attempts by experts to solve them. The Riemann hypothesis is one of Hilbert's original questions.

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Information US: 1 (617) 868-8277 (Clay Mathematics Institute)

Information France: +33 (0)1 44 27 12 72 (Véronique Lemaitre)

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