CMI Clay Mathematics Institute AMS American Mathematical Society On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an infl uential role in mathematical research. A century later, on May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute (CMI) announced the creation of a The Millennium Prize Problems US$7 million prize fund for the solution of seven important classic problems which have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution. The Millennium Prize Problems were selected by the founding Scientifi c Advisory Board of CMI—Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten—after consulting with other leading mathematicians. The Millennium Their aim was somewhat different than that of Hilbert: not to defi ne new challenges, but to record some of the most diffi cult issues with which mathematicians were struggling at the turn of the second millennium; to Prize Problems recognize achievement in mathematics of historical dimension; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; and to J. Carlson, A. Jaffe, and A. Wiles, Editors emphasize the importance of working towards a solution of the deepest, most diffi cult problems. The present volume sets forth the offi cial description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics. J. Carlson, A. Jaffe, and A. Wiles, Editors Wiles, A. and Jaffe, A. Carlson, J. For additional information and updates on this book, visit www.ams.org/bookpages/mprize A History of Prizes in Mathematics Jeremy Gray Birch and Swinnerton-Dyer Conjecture Andrew Wiles Hodge Conjecture Pierre Deligne Navier–Stokes Equation Charles L. Fefferman Poincaré Conjecture John Milnor P versus NP Problem Stephen Cook www.claymath.org Riemann Hypothesis Enrico Bombieri www.ams.org Quantum Yang–Mills Theory Arthur Jaffe and Edward Witten MPRIZE 4-color process 184 pages on 70# matte text • Backspace: 1 7/16 inches The Millennium Prize Problems Contents Introduction vii Landon T. Clay xi Statement of the Directors and the Scientific Advisory Board xv A History of Prizes in Mathematics Jeremy Gray 3 The Birch and Swinnerton-Dyer Conjecture Andrew Wiles 31 The Hodge Conjecture Pierre Deligne 45 Existence and Smoothness of the Navier–Stokes Equation Charles L. Fefferman 57 The Poincar´eConjecture John Milnor 71 The P versus NP Problem Stephen Cook 87 The Riemann Hypothesis Enrico Bombieri 107 Quantum Yang–Mills Theory Arthur Jaffe and Edward Witten 129 Rules for the Millennium Prizes 153 Authors’ Biographies 157 Picture Credits 161 v Introduction The Clay Mathematics Institute (CMI) grew out of the longstanding belief of its founder, Mr. Landon T. Clay, in the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life. Discussions over some years with Professor Arthur Jaffe helped shape Mr. Clay’s ideas of how the advancement of mathematics could best be sup- ported. These discussions resulted in the incorporation of the Institute on September 25, 1998, under Professor Jaffe’s leadership. The primary objec- tives and purposes of the Clay Mathematics Institute are “to increase and disseminate mathematical knowledge; to educate mathematicians and other scientists about new discoveries in the field of mathematics; to encourage gifted 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 — Alain Connes, Arthur Jaffe, Edward Witten, and Andrew Wiles — decided to establish a small set of prize problems. The aim was not to define new challenges, as Hilbert had done a century earlier when he announced his list of twenty-three problems at the International Congress of Mathematicians in Paris in the summer of 1900. Rather, it was to record some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium; to recognize achievement in mathematics of historical dimension; to elevate in the consciousness of the general public the fact that, in mathematics, the frontier is still open and abounds in important unsolved problems; and to emphasize the importance of working toward solutions 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 Poincar´eConjecture, the P ver- sus NP problem, the Riemann Hypothesis, and the Mass Gap problem for Quantum Yang–Mills Theory. A set of rules was established, and a prize fund of US$7 million was set up, this sum to be allocated in equal parts to the seven problems. No time limit exists for their solution. vii viii MILLENNIUM PRIZE PROBLEMS The prize was announced at a meeting on May 24, 2000, at the Coll`ege de France. On page xv we reproduce the original statement of the Directors and the Scientific Advisory Board. John Tate and Michael Atiyah each spoke about the Millennium Prize Problems: Tate on the Riemann Hypothesis, the Birch and Swinnerton-Dyer Problem, and the P vs NP problem; Atiyah on the Existence and Uniqueness Problem for the Navier–Stokes Equations, the Poincar´eConjecture, and the Mass Gap problem for Quantum Yang– Mills Theory. In addition, Timothy Gowers gave a public lecture, “On the Importance of Mathematics”. The lectures — audio, video, and slides — can be found on the CMI website: www.claymath.org/millennium. The present volume sets forth the official description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics. The editors gratefully acknowledge the work of Candace Bott (editorial and project management), Sharon Donahue (photo and photo credit re- search), and Alexander Retakh (TEX, technical, and photo editor) for their care and expert craftsmanship in the preparation of this manuscript. James Carlson, Arthur Jaffe, and Andrew Wiles Landon T. Clay Founder Clay Mathematics Institute Landon T. Clay Landon T. Clay Landon T. Clay has played a leadership role in a variety of business, sci- ence, cultural, and philanthropic activities. With his wife, Lavinia D. Clay, he founded the Clay Mathematics Institute and has served as its only Chair- man. His past charitable activities include acting as Overseer of Harvard College, as a member of the National Board of the Smithsonian Institute, and as Trustee of the Middlesex School. He is currently a Great Benefac- tor and Trustee Emeritus of the Museum of Fine Arts in Boston and for 30 years has been Chairman of the Caribbean Conservation Corporation, which operates a turtle nesting station in Costa Rica. He donated the Clay Tele- scope to the Magellan program of Harvard College in Chile. The Clay family built the Clay Science Centers at Dexter School and Middlesex School. He received an A.B. in English, cum laude, from Harvard College. xi Board of Directors and Scientific Advisory Board Board of Directors and Scientific Advisory Board Landon T. Clay, Lavinia D. Clay, Finn M.W. Caspersen, Alain Connes, Edward Witten, Andrew Wiles, Arthur Jaffe (not present: Randolph R. Hearst III and David R. Stone) Statement of the Directors and the Scientific Advisory Board In order to celebrate mathematics in the new millennium, the Clay Math- ematics Institute of Cambridge, Massachusetts, has named seven “Millen- nium Prize Problems”. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solu- tion over the years. The Board of Directors of CMI have designated a US$7 million prize fund for the solution to these problems, with US$1 million allo- cated to each. During the Millennium meeting held on May 24, 2000, at the Coll`egede France, Timothy Gowers presented a lecture entitled “The Impor- tance of Mathematics”, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. 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 Interna- tional 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. Directors: Finn M.W. Caspersen, Landon T. Clay, Lavinia D. Clay, Randolph R. Hearst III, Arthur Jaffe, and David R. Stone Scientific Advisory Board: Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten Paris, May 24, 2000 xv xvi MILLENNIUM PRIZE PROBLEMS Coll`egede France Paris meeting Alain Connes, Coll`egede France, and David Ellwood, Clay Mathematics Institute, undertook the planning and organization of the Paris meeting, assisted by the generous help of the Coll`ege de France and the CMI staff. The videos of the meeting, available at www.claymath.org/millennium, were shot and edited by Fran¸coisTisseyre. MILLENNIUM PRIZE PROBLEMS xvii Poster of the Paris meeting A History of Prizes in Mathematics Jeremy Gray A History of Prizes in Mathematics Jeremy Gray 1.