The Millennium Prize Problems

In May 2000 the Clay Mathematics Institute picked out seven mathematical problems and attached $1,000,000 to each one. The list was meant to mark the questions that the twentieth century had failed to answer. Twenty-six years later, one is solved, the man who solved it turned down the cheque, and the other six are open.

What Clay announced in Paris

The prizes were unveiled at a meeting held at the Collège de France in Paris on 24 May 2000. The institute’s board set aside a $7 million fund and allocated a million to each problem. The venue and the timing were not accidental. In 1900, also in Paris, David Hilbert had put a list of unsolved problems to the International Congress of Mathematicians, and that list ended up steering a good deal of twentieth-century mathematics. Clay was reaching for the same trick a century on.

The money was never really the point. A working mathematician who could prove the Riemann hypothesis would have their pick of jobs anywhere on Earth, and a million dollars is a rounding error against that. What the prize buys is attention: a number a newspaper can print, which gets a problem talked about outside the seminar rooms where it lives. On that measure it worked. “Millennium Prize Problems” is now the phrase most people reach for when they want to name a hard question in maths.

The seven problems, briefly

The board above carries the same seven cards, with the official statement behind each one. In prose:

P vs NP

If you can check an answer quickly, can you find it quickly? Verifying a completed sudoku takes seconds, while filling one in from an empty grid can take an evening. P vs NP asks whether that gap is a fact about the universe or a gap in our cleverness. This is the computer science one, and there is more on it below.

The Poincaré conjecture

Loop a piece of string around an orange, anywhere on the surface, and you can always slide it down to a single point. Try the same on a doughnut and the loop can catch on the hole. Henri Poincaré asked in 1904 whether that test picks out the sphere uniquely in three dimensions. It does. This is the solved one.

The Riemann hypothesis

Primes get rarer as you count upwards, but they arrive in no pattern anyone can predict. In an 1859 paper Riemann found a function whose zeros encode how the primes are distributed, and guessed that every interesting zero lies on a single vertical line. The first ten trillion zeros do. Nobody has proved the rest must.

Navier-Stokes existence and smoothness

The Navier-Stokes equations describe water moving through a pipe and air moving over a wing. Engineers solve them numerically every day, which is why aeroplanes fly and weather forecasts mostly work, but nobody can prove that smooth solutions always exist and never blow up to infinity. The practice has run well ahead of the theory for about a century.

Yang-Mills existence and mass gap

Quantum Yang-Mills theory sits underneath most of particle physics, and both experiment and simulation say its particles have positive mass even though the classical waves travel at light speed. The theory works and has no rigorous mathematical foundation underneath it. Building one, with the mass gap included, takes the prize.

The Hodge conjecture

Mathematicians study complicated shapes by gluing together simpler pieces. The bookkeeping throws off abstract objects, Hodge cycles, that have no obvious geometric meaning. The conjecture says that for a well-behaved class of spaces those abstract objects are always combinations of genuinely geometric ones. It is proved below dimension four and unknown above.

The Birch and Swinnerton-Dyer conjecture

Elliptic curves are the equations behind a lot of modern cryptography and behind Wiles’s proof of Fermat’s last theorem. The conjecture ties the number of rational solutions on a curve to whether an associated function vanishes at a particular point. It has a nice origin for this site: Bryan Birch and Peter Swinnerton-Dyer found the pattern by running experiments on the EDSAC II at Cambridge between 1958 and 1962. A computer noticed it first.

Six of those seven are mathematics rather than computer science, and we are not going to pretend otherwise by writing shallow explainers about Hodge cycles. Each card on the board links to the official Clay statement, written by people who work on these things.

Unsolved math problems: the seven are a selection, not a census

Type “unsolved math problems” into a search box and this list comes back first, which gives a misleading impression. The Millennium seven are seven famous questions chosen by one committee in one year. They are not the complete set of open problems in mathematics, and they are not necessarily the hardest, whatever hardest would mean.

Plenty of the well-known holdouts never made the list. Goldbach’s conjecture, that every even number above two is the sum of two primes, has stood since 1742 and has been checked past 4 x 10^18. The twin prime conjecture asks whether pairs like 11 and 13 keep appearing for ever. Nobody knows whether an odd perfect number exists, a question that has been open for over two thousand years. The Collatz conjecture is the one to try if you want a taste of this: the rule fits on a beer mat, a child can follow it, and Paul Erdős reportedly said that mathematics was not ready for such problems.

Hilbert’s 1900 list is worth a look for perspective too. Most of its problems have been settled, some turned out to be too vague to settle, and one of them, the eighth, is the Riemann hypothesis, which is how it ended up on Clay’s list a century later. So “unsolved” here means something specific. These are the problems that have outlasted everyone who tried, and the queue behind them is long.

P vs NP: the one that belongs to computer science

P vs NP is the only Millennium problem that a programmer meets in the course of ordinary work, usually without noticing. It asks whether every problem whose answer can be verified in polynomial time can also be solved in polynomial time. Stephen Cook set it out formally in 1971, and Leonid Levin reached the same conclusion independently in the Soviet Union.

It is also the only one on the list where a solution would be felt outside mathematics within the week. The other six would rewrite textbooks and redirect careers. A proof of the Riemann hypothesis would confirm what number theorists already assume, since a large body of published work is conditional on it being true. A settled Navier-Stokes would tell fluid dynamicists that the equations they already trust deserve the trust. Important, and mostly internal.

P vs NP is different in kind. If someone found a genuinely practical fast algorithm for an NP-complete problem, public-key cryptography would stop working, because recovering a private key from a public one is exactly a checking problem of that shape. That is not a metaphor. It is the same argument that makes breaking Bitcoin infeasible today, run in reverse. Optimisation would go the same way: airlines and chip designers currently buy approximate answers to problems like the travelling salesman problem, and those problems would suddenly have exact ones worth having. Protein folding and timetabling have the same shape underneath.

Set against that, almost everyone expects the answer to be no, that P is not equal to NP, in which case nothing changes except that we would finally know. And even a proof that P equals NP might be non-constructive, or might hand over an algorithm running in time n to the power 1,000, which is polynomial on paper and useless in a data centre. The full argument, along with a subset-sum puzzle that lets you feel the gap between checking and finding, is on our P vs NP page.

Perelman and the Poincaré conjecture

Grigori Perelman posted three preprints to arXiv between November 2002 and July 2003. They were terse, they did not mention the Poincaré conjecture in the title, and the first ran to 39 pages. They completed a programme Richard Hamilton had begun with Ricci flow, a technique for smoothing out the geometry of a shape until its structure becomes visible, and in doing so they proved not only Poincaré’s 1904 conjecture but Thurston’s more general geometrisation conjecture as well.

Then the verification began, which took several years and several teams of mathematicians filling in steps Perelman had left implicit. The proof held. He was offered the Fields Medal at the 2006 congress in Madrid and did not accept it. The Clay Mathematics Institute named him the winner of the first Millennium Prize on 18 March 2010, held a conference in his honour in Paris that June which he did not attend, and in July he told the institute he would not take the money.

His stated reasons, given to the Russian news agency Interfax, were about the mathematical community rather than the cash: he objected to being singled out when Hamilton’s contribution had been fundamental, and said he disagreed with the community’s judgements. He had by then largely withdrawn from professional mathematics. It is tempting to make him into a fable about purity, and the temptation is worth resisting. He solved a hundred-year-old problem, judged the field’s reward system to be unfair, and declined to take part in it. Clay put the $1 million into the Poincaré Chair at the Institut Henri Poincaré instead, a fellowship for early-career mathematicians, funded for five years out of money nobody collected.

What it actually takes to claim a prize

The rules are stricter than the headlines suggest, and they are worth reading before you email anyone. Clay does not accept submissions. There is no committee waiting to assess your manuscript.

Under the rules adopted by the institute’s board in 2018, a proposed solution has to appear in what Clay calls a qualifying outlet, then at least two years must pass since publication, and in that time the solution must win general acceptance in the global mathematics community. Only then will the institute consider appointing an advisory committee to look at it. The waiting period is the load-bearing part. It exists because proofs of this size fail in ways that take specialists years to find, which is exactly what the verification of Perelman’s work was testing for, except that his survived.

That filter has held so far. The prize fund still stands at $7 million. One problem is off the board with its million untouched, and six remain where the Clay board left them in Paris in 2000.