The Millennium Prize Problems: Seven Questions, $7 Million, One Refusal

What the Millennium Prize Problems are

On May 24, 2000, at the Collège de France in Paris, the Clay Mathematics Institute named seven problems and pledged $1 million for the first accepted solution to each. John Tate and Michael Atiyah presented them in lectures that evening.

The setting was deliberate. One hundred years earlier, on August 8, 1900, David Hilbert had presented a list of open problems at the International Congress of Mathematicians in Paris, and that list steered much of 20th-century mathematics. Clay wanted a new list for a new century, with money attached to draw public attention to research that rarely gets any.

The institute commissioned an official statement of each problem from a leading specialist. Andrew Wiles wrote the Birch and Swinnerton-Dyer statement, Stephen Cook wrote P versus NP, Pierre Deligne the Hodge conjecture, Charles Fefferman Navier-Stokes, Enrico Bombieri the Riemann hypothesis, and Arthur Jaffe and Edward Witten the Yang-Mills problem.

The seven problems at a glance

The six that remain open

Riemann hypothesis

The oldest on the list. Bernhard Riemann posed it in 1859, and Hilbert put it eighth on his 1900 list. Computer searches have found no zero off the line, but a search is not a proof. Our Riemann hypothesis guide walks through the statement.

P versus NP

The only problem from computer science. Most experts expect that P does not equal NP, meaning that some problems are easy to check but hard to solve. Much of modern cryptography quietly assumes that expectation is true.

Navier-Stokes

Engineers use these equations every day to model air over wings and water in pipes. Yet no one has proved that smooth solutions always exist in three dimensions, or found a case where they blow up.

Yang-Mills and the mass gap

The theory traces to a 1954 paper by Chen Ning Yang and Robert Mills, and it underpins the Standard Model of particle physics. Physicists use it with great success. Mathematicians have not shown that it exists on rigorous footing.

Hodge conjecture

The most abstract of the seven, and the hardest to state without technical vocabulary. It asks whether a class of objects defined by topology can always be assembled from objects defined by polynomial equations.

Birch and Swinnerton-Dyer

Bryan Birch and Peter Swinnerton-Dyer arrived at the conjecture in the early 1960s from calculations on one of the early EDSAC computers at Cambridge. Andrew Wiles, who proved Fermat's Last Theorem in the 1990s, wrote Clay's statement of it.

Poincaré and Perelman: the one that fell

Henri Poincaré formulated his conjecture in 1904. It asks, roughly, whether a three-dimensional shape that passes a simple test, every loop in it can be shrunk to a point without leaving it, must be a three-dimensional sphere. The question resisted proof for most of a century.

Grigori Perelman, a mathematician in St. Petersburg, settled it in three papers posted to arXiv on November 11, 2002, March 10, 2003 and July 17, 2003. He built on the Ricci flow, a method Richard Hamilton introduced in a 1982 paper, and found ways past the obstacles that had stalled it. Perelman did not submit the papers to a journal. Teams of specialists spent several years checking them.

On March 18, 2010, Clay announced the prize. "It is a major advance in the history of mathematics that will long be remembered," said James Carlson, then Clay's president. On July 1, 2010, Perelman declined it. He told the Russian news agency Interfax that "the main reason is my disagreement with the organized mathematical community," adding: "I don't like their decisions, I consider them unjust." He considered Hamilton's contribution no smaller than his own.

It was his second refusal. In August 2006, at the International Congress of Mathematicians in Madrid, he declined the Fields Medal, the top prize for mathematicians under 40. In a remark widely attributed to him, he said: "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo." Clay and the Institut Henri Poincaré held a conference in Paris in June 2010 to mark the solution anyway.

How a solution wins the money

No one wins by mailing a proof to Clay. The institute does not accept direct submissions. Under its rules, last revised on September 26, 2018, a solution qualifies only when three conditions hold:

• It has been published in a qualifying outlet, as the rules define one.
• At least two years have passed since publication.
• It has won general acceptance in the global mathematics community.

Those conditions explain why claimed proofs, which appear regularly online and in the press, do not move the count. Perelman's case shows some flexibility: he never published in a journal, yet Clay accepted the arXiv papers once experts had verified them.

Do you need a genius IQ to solve one?

No intelligence test can say. Perelman has never published an IQ score, and no verified figure exists for him or for any living mathematician working on these problems. The documented evidence of his ability is his work: a gold medal with a perfect score at the 1982 International Mathematical Olympiad, then the proof itself.

Tests also run out of room. The WAIS-IV reports full-scale scores up to 160, a level that about 1 person in 31,600 reaches, and the number of people above that line is far larger than the number who have advanced a Millennium problem. What separates the two groups is years of specialized knowledge, sustained focus on a single question and, often, a new method. Hamilton's Ricci flow was two decades old before Perelman finished the job.

For the record on historical mathematicians, see mathematicians' IQ ranked. For what high scores do and do not predict, see what counts as a high IQ and signs of high intelligence. Our free IQ test is a practice screener, and the IQ guides explain how professional testing works.

Editorial standard: IQ figures for historical figures are retrospective estimates and are labeled as such; scores for living people appear only when the person disclosed them or a reputable outlet reported them. Corrections: contact the editors.