On 24 May 2000, the Clay Mathematics Institute — a private foundation based in Massachusetts — announced seven problems in mathematics. Each, it said, would carry a reward of one million US dollars for a rigorous solution.
These are the Millennium Prize Problems. They were chosen to be deep, important, and — crucially — open. A quarter-century later, only one has been solved. The other six still wait.
The list
The seven problems, in the order the Clay Institute presented them:
- Birch and Swinnerton-Dyer Conjecture — about elliptic curves and their L-functions
- Hodge Conjecture — about algebraic cycles on complex projective varieties
- Navier-Stokes Existence and Smoothness — about whether fluid equations always have smooth solutions
- P vs NP — about the relationship between solving problems and verifying solutions
- Poincaré Conjecture — about the topology of three-dimensional shapes
- Riemann Hypothesis — about the zeros of a famous complex function
- Yang-Mills Existence and Mass Gap — about the mathematical foundations of quantum field theory
Each problem comes with a detailed formal statement in the Clay Institute’s prize rules, which you can find on their website.
The one solved problem
In 2002 and 2003, the Russian mathematician Grigori Perelman posted three papers on the arXiv preprint server. They claimed a proof of the Poincaré Conjecture — the conjecture that every simply-connected, closed three-dimensional manifold is topologically equivalent to the three-sphere.
The conjecture had stood for 100 years. Perelman’s approach built on Richard Hamilton’s Ricci flow — a technique for deforming manifolds to simplify their geometry — and handled the singularities that Hamilton’s approach had been unable to resolve.
Verifying Perelman’s proof took teams of mathematicians several years and hundreds of additional pages of exposition. The proofs held.
In 2006, Perelman was awarded the Fields Medal. He refused it. In 2010, the Clay Institute awarded him the one million dollars. He refused it too, citing what he felt was unfair allocation of credit.
Poincaré remains the only Millennium problem solved. The other six are still open.
The six that remain
The Riemann Hypothesis
The most famous. Bernhard Riemann conjectured in 1859 that every non-trivial zero of the zeta function has real part . A proof would give the sharpest possible bound in the prime number theorem and unlock hundreds of conditional results in number theory. Despite massive computational evidence — the first zeros verified to lie on the critical line — no proof exists.
P vs NP
The most practically important. Can every problem whose solution is quickly verified also be quickly solved? Almost everyone believes P ≠ NP, but no one can prove it. A proof either way would reshape cryptography, artificial intelligence, and theoretical computer science.
Navier-Stokes
The most applied. The equations of fluid motion work brilliantly in practice, but mathematically we don’t know whether their solutions always stay smooth. Proving global smoothness (or finding a counterexample) would close a major gap in the foundations of continuum mechanics.
Yang-Mills
The most physical. The Standard Model of particle physics is built from Yang-Mills theories, and physicists compute with them routinely. But giving those computations a rigorous mathematical foundation — including a proof of the “mass gap” — remains open.
Hodge Conjecture
The most abstract. A deep conjecture connecting the topology and algebraic structure of complex varieties. Probably the hardest of the seven to explain to non-specialists.
Birch and Swinnerton-Dyer
The most number-theoretic. A conjecture about the rational solutions of elliptic curves — said to have predictive implications for nearly all Diophantine equations.
Why a list like this matters
The Clay list is a deliberate descendant of Hilbert’s 23 problems from 1900. Hilbert used his authority to focus the mathematical community’s attention on what he considered the most important open questions; the Clay Institute does the same, adding money as a signal.
The money isn’t the point — most mathematicians who work on these problems would work on them anyway. The point is symbolic: this list says, “these problems matter.” For early-career mathematicians deciding what to spend the next decade on, the list serves as a curated roadmap.
What to expect in the next decade
Most mathematicians working on the Millennium problems are pessimistic in the short term. P vs NP and the Riemann hypothesis seem to require genuinely new techniques. Navier-Stokes has seen steady partial progress but no breakthrough.
The Hodge and BSD conjectures are deeply connected to other programs in modern number theory and geometry; a solution to one might come as a consequence of broader advances.
Yang-Mills, unusually, is a problem where constructive progress is possible: rigorous foundations for lower-dimensional cases already exist, and techniques from probability and stochastic analysis are slowly advancing into four dimensions.
If a second Millennium problem falls in the next decade, Yang-Mills or Navier-Stokes may be the most likely candidates. But prediction is a bad business — no one saw Perelman’s breakthrough coming either.
Frequently asked
Is the one-million-dollar prize real?
Yes. The Clay Mathematics Institute, founded by Landon Clay, has set aside seven million dollars to be paid out — one million per solved problem. Grigori Perelman was offered one million for his Poincaré proof and famously turned it down.
Which problems have been solved?
One: the Poincaré Conjecture, solved by Grigori Perelman between 2002 and 2003. The other six remain open.