Navier's equations
Claude-Louis Navier derives equations of motion for a viscous fluid.
The Millennium Museum
On 24 May 2000, at the Collège de France, Paris, the Clay Mathematics Institute named seven problems and offered $1,000,000 for each problem. One has been solved; solutions to one more have been claimed, and the Institute has acknowledged none of them.
The seven problems
Open
The Riemann Hypothesis
Do all the non-trivial zeros of the Riemann zeta function lie on the critical line, where the real part is one half?
2026: An AI system reaches 67.2%
Claimed by AI, not acknowledged
Navier–Stokes Existence and Smoothness
In three dimensions, do smooth, finite-energy solutions of the incompressible Navier–Stokes equations exist for all time, or can they break down in finite time?
2026: A Severe Misalignment
Solved
The Poincaré Conjecture
Is every simply connected, closed three-dimensional manifold homeomorphic to the three-sphere?
2010: The prize awarded, and declined
Open
P versus NP
If a solution to a problem can be checked quickly, can it also be found quickly?
2026: OpenAI turns to the Millennium problems
Open
The Hodge Conjecture
On a non-singular complex projective variety, is every rational Hodge class a rational combination of classes of algebraic cycles?
2026: OpenAI turns to the Millennium problems
Open
The Birch and Swinnerton-Dyer Conjecture
Is the rank of an elliptic curve over the rationals equal to the order of vanishing of its L-function at s = 1?
2026: OpenAI turns to the Millennium problems
Open
Yang–Mills Existence and Mass Gap
Does quantum Yang–Mills theory exist rigorously on four-dimensional space-time, with a positive mass gap, for every compact simple gauge group?
2026: OpenAI turns to the Millennium problems
The Clay Mathematics Institute considers a solution only after it has appeared in a qualifying outlet, at least two years have passed, and it has general acceptance in the mathematics community; its advisory board then decides. Source.
The history
68 events- 19th century
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1822
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1845
Stokes
George Gabriel Stokes derives the equations independently, in the form still used.
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Nov 1859
Riemann's paper on the primes
In an eight-page paper on the number of primes below a given size, Riemann extends the zeta function to the complex plane and remarks that it is very probable all its non-trivial zeros have real part one half.
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1896
The prime number theorem
Hadamard and de la Vallée Poussin independently prove the prime number theorem by showing that ζ has no zeros on the line Re s = 1.
- 1900s
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1900
Hilbert's eighth problem
Hilbert includes the Riemann hypothesis in his list of problems for the new century.
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1904
Poincaré's question
Henri Poincaré asks whether a closed three-dimensional manifold whose loops can all be contracted must be the three-sphere.
- 1910s
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1914
Infinitely many zeros on the line
Hardy proves that infinitely many zeros lie on the critical line.
- 1920s
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1924
The Lefschetz (1,1) theorem
Solomon Lefschetz proves the case of divisors, decades before the conjecture is stated.
- 1930s
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1934
Leray's weak solutions
Jean Leray proves that weak solutions exist for all time. Whether they are smooth and unique is left open, and remains so.
- 1940s
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1942
A positive proportion
Selberg proves that a positive proportion of the zeros lie on the critical line.
- 1950s
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1950
Hodge at the ICM
W. V. D. Hodge presents the conjecture at the International Congress of Mathematicians in Cambridge, Massachusetts.
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1951
Hopf
Eberhard Hopf extends Leray's weak solutions to bounded domains; they are now called Leray–Hopf solutions.
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1954
Yang and Mills
Chen-Ning Yang and Robert Mills publish non-abelian gauge theory.
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1956
Gödel's letter
In a letter to John von Neumann, Kurt Gödel asks, in effect, whether proofs can be found as quickly as they can be checked.
- 1960s
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1961
Dimension five and above
Stephen Smale proves the analogue in every dimension from five up.
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1965
Computations on EDSAC
Bryan Birch and Peter Swinnerton-Dyer publish the conjecture, drawn from computations on the EDSAC computer at Cambridge.
- 1970s
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1971
Cook's theorem
Stephen Cook introduces NP-completeness and shows that Boolean satisfiability is NP-complete.
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1972
Karp's 21 problems
Richard Karp shows that 21 natural combinatorial problems are NP-complete.
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1973
Levin
Leonid Levin independently develops the theory of universal search problems in the Soviet Union.
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1973
Asymptotic freedom
David Gross, Frank Wilczek and David Politzer discover that the interaction weakens at short distances.
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1974
More than a third
Levinson shows that more than one third of the zeros lie on the line.
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1974
Lattice gauge theory
Kenneth Wilson formulates gauge theory on a lattice, giving a route to confinement and to numerical computation.
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1977
Coates and Wiles
John Coates and Andrew Wiles prove that a curve with complex multiplication and non-vanishing L-value at 1 has only finitely many rational points.
- 1980s
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1980s
Constructive work in four dimensions
Tadeusz Balaban carries out a renormalisation-group analysis of lattice Yang–Mills, the most substantial constructive progress in four dimensions.
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1982
Partial regularity
Caffarelli, Kohn and Nirenberg show that any singular set of a suitable weak solution is small, of one-dimensional parabolic Hausdorff measure zero.
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1982
Dimension four
Michael Freedman proves the topological analogue in dimension four.
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1982
Ricci flow
Richard Hamilton introduces the Ricci flow, the tool that would eventually settle the conjecture.
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1986
The Gross–Zagier formula
Benedict Gross and Don Zagier relate the derivative of the L-function to the height of a Heegner point.
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1989
Kolyvagin
Victor Kolyvagin's Euler systems, with Gross–Zagier, prove the rank part for curves of analytic rank zero or one.
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1989
More than two fifths
Conrey raises the proportion on the line to more than two fifths.
- 1990s
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1995
Hodge loci are algebraic
Cattani, Deligne and Kaplan prove that Hodge loci are algebraic, as the conjecture predicts.
- 2000s
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize, with an official description by Andrew Wiles.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize, with an official description by Pierre Deligne.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the problem one of seven carrying a $1,000,000 prize, with an official description by Charles Fefferman setting out four alternatives.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names P versus NP one of seven problems carrying a $1,000,000 prize, with an official description by Stephen Cook.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the Riemann hypothesis one of seven problems carrying a $1,000,000 prize, with an official description by Enrico Bombieri.
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24 May 2000
A Millennium Prize Problem
The Clay Mathematics Institute names the problem one of seven carrying a $1,000,000 prize, with an official description by Arthur Jaffe and Edward Witten.
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2001
Every curve is modular
Breuil, Conrad, Diamond and Taylor complete the modularity theorem, so the L-function of every elliptic curve over ℚ extends to the whole plane.
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11 Nov 2002
Perelman's first preprint
Grigori Perelman posts the first of three preprints on arXiv, completing Hamilton's programme.
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2003
The second and third preprints
Perelman posts the remaining two preprints, on Ricci flow with surgery and on finite extinction time.
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2004
Ten trillion zeros checked
Gourdon verifies by computer that the first 10¹³ zeros lie on the critical line.
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2006
A Fields Medal, declined
Perelman is awarded the Fields Medal and declines it, as detailed expositions of his proof appear.
- 2010s
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2010
A claimed proof, withdrawn
Vinay Deolalikar circulates a claimed proof that P differs from NP; experts quickly find fundamental gaps.
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18 Mar 2010
The prize awarded, and declined
The Clay Institute awards the Millennium Prize to Perelman, who declines it.
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2013
Numerical evidence of blowup
Guo Luo and Thomas Hou present numerical evidence that solutions of the 3D Euler equations can blow up in finite time, shifting opinion about the answer.
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2014
Most curves satisfy it
Bhargava, Skinner and Zhang show that more than 66% of elliptic curves over ℚ satisfy the rank part of the conjecture.
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2019
Singularities for Euler
Tarek Elgindi proves finite-time singularity formation for C^(1,α) solutions of the incompressible Euler equations.
- 2020
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2020
More than five twelfths
Pratt, Robles, Zaharescu and Zeindler show that more than five twelfths of the zeros lie on the line.
- 2023
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2023
The Córdoba–Martínez-Zoroa programme
Diego Córdoba and Luis Martínez-Zoroa introduce new constructions of finite-time blowup; with Fan Zheng they reach unforced blowup for 3D Euler and forced blowup for hypodissipative Navier–Stokes.
- 2025
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17 Sep 2025
Unstable singularities
A preprint by 22 authors from universities and DeepMind, including Buckmaster, demonstrates new numerical methods for constructing singularities of the porous media and Boussinesq equations.
- 2026
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10 Aug 2026
An AI system reaches 67.2%
A Claude model, asked to attempt the hypothesis itself, raises the proven proportion of zeros that are simple and on the line from 41.6% to 67.2%. The proof is formalised in Lean, and qed.bot rebuilt it: 23 theorems on the standard axioms.
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15 Aug 2026
Smooth forcing for Euler
Tristan Buckmaster and Levent Alpöge, working with AI models, obtain finite-time blowup with smooth forcing for Euler, porous media and Boussinesq. The results are verified in Lean a week later.
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Sep 2026
A Lean statement in draft
Formal Conjectures begins drafting Lean statements for the Hodge conjecture, Birch and Swinnerton-Dyer, and Yang–Mills.
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Sep 2026
A Lean statement in draft
Formal Conjectures begins drafting Lean statements for Yang–Mills, Hodge, and Birch and Swinnerton-Dyer.
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1 Sep 2026
OpenAI turns to the Millennium problems
OpenAI says it began large-scale work on all the open Millennium problems.
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1 Sep 2026
OpenAI turns to the Millennium problems
OpenAI says it began large-scale work on all the open Millennium problems.
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1 Sep 2026
OpenAI turns to the Millennium problems
OpenAI says it began large-scale work on all the open Millennium problems.
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1 Sep 2026
OpenAI turns to the Millennium problems
OpenAI says it began large-scale work on all the open Millennium problems.
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7 Sep 2026
Buckmaster's statement
Shortly before midnight, Buckmaster announces the pair's results and says news of their progress reached OpenAI before its run began.
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8 Sep 2026
OpenAI claims alternatives (C) and (D)
About twelve hours later, OpenAI announces finite-time blowup for Navier–Stokes with smooth forcing, from an 88-hour run of some 10,000 agents, with a 166-page paper and a Lean formalisation checked against the Formal Conjectures statement.
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11 Sep 2026
A Severe Misalignment
A declaration signed by 26 Fields Medalists warns against AI companies treating unsolved problems as benchmarks without adding to human understanding.
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