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 The MillenniumPrize ProblemsJ. Carlson, A. Jaffe, and A. Wiles, EditorsClay Mathematics InstituteAmerican Mathematical SocietyA History of Prizes in MathematicsBirch and Swinnerton-Dyer ConjectureHodge ConjectureNavier–Stokes EquationPoincaré ConjectureP versus NP ProblemRiemann HypothesisQuantum Yang–Mills TheoryJeremy GrayAndrew WilesPierre DeligneCharles L. FeffermanJohn MilnorStephen CookEnrico BombieriArthur Jaffe and Edward WittenThe Millennium Prize Problems J. Carlson, A. Jaffe, and A. Wiles, EditorsCMIAMS 

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 The Millennium Prize Problems 

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 ContentsIntroductionviiLandon T. ClayxiStatement of the Directors andthe Scientific Advisory BoardxvA History of Prizes in MathematicsJeremy Gray3The Birch and Swinnerton-Dyer ConjectureAndrew Wiles31The Hodge ConjecturePierre Deligne45Existence and Smoothness of the Navier–Stokes EquationCharles L. Fefferman57The Poincaré ConjectureJohn Milnor71The P versus NP ProblemStephen Cook87The Riemann HypothesisEnrico Bombieri107Quantum Yang–Mills TheoryArthur Jaffe and Edward Witten129Rules for the Millennium Prizes153Authors’ Biographies157Picture Credits161v 

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 IntroductionThe Clay Mathematics Institute (CMI) grew out of the longstandingbelief of its founder, Mr. Landon T. Clay, in the value of mathematicalknowledge and its centrality to human progress, culture, and intellectuallife. Discussions over some years with Professor Arthur Jaffe helped shapeMr. Clay’s ideas of how the advancement of mathematics could best be sup-ported. These discussions resulted in the incorporation of the Institute onSeptember 25, 1998, under Professor Jaffe’s leadership. The primary objec-tives and purposes of the Clay Mathematics Institute are “to increase anddisseminate mathematical knowledge; to educate mathematicians and otherscientists about new discoveries in the field of mathematics; to encouragegifted students to pursue mathematical careers; and to recognize extraordi-nary achievements and advances in mathematical research.” CMI seeks to“further the beauty, power and universality of mathematical thinking.”Very early on, the Institute, led by its founding scientific board — AlainConnes, Arthur Jaffe, Edward Witten, and Andrew Wiles — decided toestablish a small set of prize problems. The aim was not to define newchallenges, as Hilbert had done a century earlier when he announced his listof twenty-three problems at the International Congress of Mathematiciansin Paris in the summer of 1900. Rather, it was to record some of the mostdifficult issues with which mathematicians were struggling at the turn of thesecond millennium; to recognize achievement in mathematics of historicaldimension; to elevate in the consciousness of the general public the factthat, in mathematics, the frontier is still open and abounds in importantunsolved problems; and to emphasize the importance of working towardsolutions of the deepest, most difficult problems.After consulting with leading members of the mathematical community,a final list of seven problems was agreed upon: the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Existence and Uniqueness Prob-lem for the Navier–Stokes Equations, the Poincaré Conjecture, the P ver-sus NP problem, the Riemann Hypothesis, and the Mass Gap problem forQuantum Yang–Mills Theory. A set of rules was established, and a prizefund of US$7 million was set up, this sum to be allocated in equal parts tothe seven problems. No time limit exists for their solution.vii 

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 viiiMILLENNIUM PRIZE PROBLEMSThe prize was announced at a meeting on May 24, 2000, at the Coll`egede France. On page xv we reproduce the original statement of the Directorsand the Scientific Advisory Board. John Tate and Michael Atiyah each spokeabout the Millennium Prize Problems: Tate on the Riemann Hypothesis, theBirch and Swinnerton-Dyer Problem, and the P vs NP problem; Atiyah onthe Existence and Uniqueness Problem for the Navier–Stokes Equat