Established by the Clay Mathematics Institute in 2000 to herald a new millennium of mathematics, the seven Millennium Prize Problems are among the most challenging and significant in the world of maths. Each problem brings with it a reward of $1 million for the first correct solution, a prize to inspire all kinds of mathematician to solve these notorious problems.
The seven selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science. The Clay Institute was inspired by a set of 23 problems organised by the mathematician David Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century.
Unlike Hilbert’s problems, the problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution. The seven problems were officially announced by John Tate and Michael Taliyah during a ceremony held on May 24, 2000 at the amphitheatre Marguerite de Navarre in the Collège de France in Paris.
The Prize Problems
- Riemann Hypothesis: One of the most famous unsolved problems in mathematics, it suggests that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2.
- P versus NP Problem: A notorious problem in computer science asking whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
- Poincaré Conjecture: This deals with the topology of three-dimensional spaces. Grigori Perelman proved the problem in 2003, and he was awarded the prize in 2010 but declined it.
- Navier–Stokes Existence and Smoothness: This problem concerns the Navier-Stokes equations, which describe the motion of fluid substances like water and air, the aim being to prove or disprove the existence of smooth, globally defined solutions to these equations in three dimensions.
- Birch and Swinnerton-Dyer Conjecture: A conjecture that deals with elliptic curves and their rational points. The conjecture suggests a deep connection between the number of rational points on an elliptic curve and the behaviour of an associated L-function at a specific point.
- Hodge Conjecture: A major unsolved problem in algebraic geometry proposing that certain classes of cohomology classes are algebraic, meaning they can be represented by algebraic cycles.
- Yang–Mills Existence and Mass Gap: This problem is related to quantum field theory and the existence of a mass gap. It seeks to establish the existence of a quantum field theory that satisfies the Yang-Mills equations and has a positive mass gap.
Grigori Perelman, who had begun work on the Poincaré conjecture in the 1990s, released his successful proof in 2002 and 2003, but he refused the Clay Institute’s monetary prize in 2010, stating he was not interested in money or fame and philosophically disagreed with the Prize.
The other six Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians.
Prize money in maths?
Andrew Wiles, as part of the Clay Institute’s scientific advisory board, hoped that the choice of $1 million prize money would popularise, among general audiences, both the selected problems as well as the “excitement of mathematical endeavour”. Another board member, Fields medallist Alain Connes, hoped that the publicity around the unsolved problems would help to combat the “wrong idea” among the public that mathematics would be “overtaken by computers”. Some mathematicians have been more critical. Anatoly Vershik characterized their monetary prize as “show business” representing the “worst manifestations of present-day mass culture”, and thought that there are more meaningful ways to invest in public appreciation of mathematics. He viewed the superficial media treatments of Perelman and his work, with disproportionate attention being placed on the prize value itself, as unsurprising. By contrast, Vershik praised the Clay Institute’s direct funding of research conferences and young researchers. Vershik’s comments were later echoed by Fields medallist Sing-Tung Yau, who was additionally critical of the idea of a foundation taking actions to “appropriate” fundamental mathematical questions and attach its name to them, and they expressed concern that the focus on monetary rewards could detract from the intrinsic value of mathematical exploration.
The Clay Mathematics Institute
The Institute that established the Prize Problems is a private, non-profit foundation “dedicated to furthering the beauty, power and universality of mathematical thought”. While best known for establishing the Millennium Prize Problems, it also supports conferences, workshops, summer school and a postdoctoral program from its headquarters in Denver, Colorado and its President’s office in Oxford.