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Science · Mathematics

The Millennium Prize Problems

2000

Mathematics — open problems and prizes

Seven problems announced by the Clay Mathematics Institute at the Collège de France on 24 May 2000, each carrying a prize of one million US dollars, deliberately echoing the list David Hilbert had set out in Paris a century before. They are the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the Navier–Stokes existence and smoothness problem, P versus NP, the Poincaré conjecture, the Riemann hypothesis, and Yang–Mills existence and mass gap. Only the Poincaré conjecture has been settled: Grigori Perelman proved it in 2002–03 and turned down both the Fields Medal and the million dollars.

How to spot them

  • ▸Seven problems, a million dollars each, announced in Paris in 2000
  • ▸Set by the Clay Mathematics Institute of Cambridge, Massachusetts
  • ▸Consciously modelled on Hilbert's 23 problems of 1900
  • ▸Only one is solved, and its solver refused both the money and the Fields Medal
  • ▸The list includes the two great open problems of number theory and computer science

Key facts

Seven problems announced at the Collège de France on 24 May 2000, worth a million dollars each (2000)

Announced by the Clay Mathematics Institute, founded in 1998 by Landon Clay

Grigori Perelman's 2002–03 preprints, and his refusal of the Fields Medal and the million dollars (2003)

The Poincaré conjecture, the only one of the seven so far resolved

The question of whether every problem whose answer can be checked quickly can also be solved quickly

P versus NP, the central open question of theoretical computer science

The conjecture that every even-dimensional cohomology class of a projective variety is algebraic

The Hodge conjecture, after W. V. D. Hodge of Cambridge

The conjecture relating the rank of an elliptic curve to the order of vanishing of its L-function (1965)

Birch and Swinnerton-Dyer, based on computations on Cambridge's EDSAC

The demand for a mass gap in quantum field theory, and a rigorous construction of the gauge theory

Yang–Mills existence and mass gap — a problem in mathematical physics

The unresolved question of whether the equations of fluid flow can develop a singularity

Navier–Stokes existence and smoothness; OpenAI announced a claimed counterexample in September 2026

A list deliberately echoing the 23 problems set in Paris a hundred years earlier (1900)

David Hilbert's list at the 1900 International Congress of Mathematicians

Key ideas

  • Hilbert's 1900 list shaped twentieth-century mathematics; the 2000 list was an attempt to repeat the trick, with money attached.
  • The rules require a solution to be published in a refereed journal and to survive two years of general acceptance before the prize is considered.

Quick-fire facts

  • •The Clay Mathematics Institute was founded in 1998 by the Boston businessman Landon T. Clay; the problems were announced on 24 May 2000
  • •The seven are: Birch and Swinnerton-Dyer, Hodge, Navier–Stokes, P versus NP, Poincaré, Riemann, and Yang–Mills
  • •The Poincaré conjecture, posed by Henri Poincaré in 1904, is the only one solved
  • •Grigori Perelman posted his proof as three arXiv preprints in 2002–03, declined the Fields Medal in 2006 and the million dollars in 2010
  • •He said the prize should have gone at least equally to Richard Hamilton, whose Ricci flow he used
  • •The Clay Institute used the declined prize money to endow a Poincaré Chair at the Institut Henri Poincaré in Paris
  • •The Riemann hypothesis is the only problem that appears both on Hilbert's 1900 list and on this one
  • •The Birch and Swinnerton-Dyer conjecture grew out of computations run on Cambridge's EDSAC computer in the early 1960s
  • •In September 2026 OpenAI announced that AI agents had produced a Lean-verified singularity for the three-dimensional Navier–Stokes equations, saying it did not intend to claim the prize; the Clay Institute has certified nothing
  • •Michael Atiyah and John Tate presented the problems at the announcement
  • •The rules require publication in a refereed journal and a two-year waiting period of general acceptance

Drill it — 9 cards

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Sources