Millennium Prize Problems

• 1. Introduction

  ›

  The Millennium Prize Problems are seven mathematical puzzles, each carrying a one-million-dollar reward from the Clay Mathematics Institute. They were chosen not because they were new mysteries, but because they had already defeated generations of professional mathematicians. On the 24th of May 2000, mathematicians John Tate and Michael Atiyah stood in the amphitheatre Marguerite de Navarre at the College de France in Paris and announced the list to the world. The questions they named span algebraic geometry, number theory, fluid mechanics, quantum physics, and the theory of computation. Only one has ever been solved. What makes these problems worth a million dollars apiece? Who chose them, why these seven, and what does it mean that a Russian mathematician won the prize and refused to accept it?

• 2. Hilbert's Shadow

  ›

  David Hilbert organized twenty-three unsolved problems at the International Congress of Mathematicians in 1900, and those problems shaped mathematical research throughout the entire twentieth century. The Clay Mathematics Institute drew direct inspiration from that list when assembling its own. The parallel was deliberate. Both lists were meant to serve as compasses, pointing the field toward its hardest open frontiers. There was one key difference. Hilbert's problems were chosen partly to introduce fresh directions. The Clay problems were already famous inside the profession; many researchers were actively working on them before the prize existed. The seven selected problems span algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science.

• 3. The Prize's Purpose

  ›

  Andrew Wiles, who sat on the Clay Institute's scientific advisory board and also wrote the official statement for the Birch and Swinnerton-Dyer conjecture, hoped the one-million-dollar figure would excite a general audience about the thrill of mathematical work. Fields medalist Alain Connes, another board member, had a different concern. He wanted the attention around the problems to push back against a public misconception that mathematics would eventually be overtaken by computers. Not everyone approved. Anatoly Vershik called the monetary prizes "show business" and the "worst manifestations of present-day mass culture." He argued that direct funding of research conferences and young mathematicians was a more meaningful investment, though he also praised exactly those parts of the Clay Institute's work. Fields medalist Shing-Tung Yau later echoed those concerns, adding a separate objection: that a private foundation should not "appropriate" fundamental mathematical questions and attach its name to them.

• 4. The One Solution

  ›

  Henri Poincare posed his conjecture in 1904, asking whether any closed, simply-connected three-dimensional space is topologically equivalent to a three-sphere. Grigori Perelman began working on the problem in the 1990s, building on a program developed by Richard Hamilton over the preceding twenty years. Hamilton's central tool was Ricci flow, a system of partial differential equations in Riemannian geometry that deforms the curvature of a space over time. Perelman released his proof in stages in 2002 and 2003, completing Hamilton's geometrization conjecture along with the Poincare conjecture itself. In 2006, Perelman was awarded the Fields Medal for his contributions to Ricci flow theory. He declined to accept it. On the 18th of March 2010, the Clay Institute formally awarded him the Millennium Prize. He declined that too, stating that Hamilton's contribution was no less than his own. His refusal drew widespread media coverage, which Vershik had anticipated: the prize design, he had argued, would produce exactly that kind of spectacle.

• 5. Six Open Questions

  ›

  Bernhard Riemann first posed his hypothesis in 1859, making it one of the oldest entries on the Clay list. The Riemann hypothesis asserts that all nontrivial zeros of the Riemann zeta function have a real part of one-half. The first nontrivial zeros appear at imaginary parts near 14.135, 21.022, and 25.011. Resolving the hypothesis would carry deep consequences for the distribution of prime numbers; Enrico Bombieri wrote the Clay Institute's exposition of it. The P versus NP problem, whose official statement was prepared by Stephen Cook, asks whether every problem whose solution can be verified quickly can also be solved quickly. Most mathematicians and computer scientists expect the answer is no, but no proof has been found; the question touches cryptography, biology, and philosophy. Charles Fefferman stated the Navier-Stokes existence and smoothness problem, which asks whether smooth solutions to the three-dimensional fluid equations always exist. Pierre Deligne stated the Hodge conjecture, concerning whether Hodge cycles on complex projective varieties can be expressed as rational combinations of algebraic cycles. Arthur Jaffe and Edward Witten stated the Yang-Mills existence and mass gap problem, requiring a rigorous proof that quantum Yang-Mills theory exists and that there is a positive energy gap between the vacuum and the next energy state. Bryan John Birch and Peter Swinnerton-Dyer gave their names to the remaining conjecture, which proposes a way to determine whether an elliptic curve over the rational numbers has finitely or infinitely many rational solutions.

• 6. The Unclaimed Million

  ›

  Vershik's prediction proved accurate. When Perelman refused the Clay prize on the 18th of March 2010, media coverage focused overwhelmingly on the prize value rather than on the mathematics or on Hamilton's foundational role. Perelman's stated reason was precise: Hamilton had developed the Ricci flow program over roughly twenty years before Perelman extended it to a complete proof, and an award given to Perelman alone misrepresented how the result came to exist. The Clay Institute had no mechanism to split or redirect the prize. The one-million-dollar award remains unclaimed. The other six Millennium Prize Problems have attracted large numbers of attempted proofs from both amateur and professional mathematicians, and none of those attempts has been accepted as correct. The question of whether any of the remaining six will fall in the coming decades is, of course, exactly the kind of question no one can answer.