The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged to pay one million US dollars for the first correct solution to each problem.
The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems.
As of 2026, the only Millennium Prize problem to have been officially solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award because it was not also offered to Richard S. Hamilton, upon whose work Perelman built.
In September 2026, the American artificial intelligence company OpenAI presented a proposed counterexample to the Navier–Stokes existence and smoothness problem, saying they would decline the Millennium Prize for the result if offered. However, this result has yet to be verified by the Clay Institute or the independent mathematical community and is the subject of a priority dispute.
Contents
Overview
The Clay Institute was inspired by a set of twenty-three problems organized by the mathematician David Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century. The seven selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution.
The seven problems were officially announced by John Tate and Michael Atiyah during a ceremony held on May 24, 2000 (at the amphithéâtre Marguerite de Navarre) in the Collège de France in Paris.
Grigori Perelman, who had begun work on the Poincaré conjecture in the 1990s, released his proof in 2002 and 2003. His refusal of the Clay Institute's monetary prize in 2010 was widely covered in the media. In 2026, OpenAI claimed to resolve the Navier–Stokes existence and smoothness problem using an internal AI model, but this has yet to be independently verified. The other five Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians.
Andrew Wiles, as part of the Clay Institute's scientific advisory board, hoped that the choice of US$1 million prize money would popularize, among general audiences, both the selected problems as well as the "excitement of mathematical endeavor". Another board member, Fields medalist Alain Connes, hoped that the publicity around the unsolved problems would help to combat the "wrong idea" among the public that mathematics would be "overtaken by computers".
Some mathematicians have been more critical. Anatoly Vershik characterized their monetary prize as "show business" representing the "worst manifestations of present-day mass culture", and thought that there are more meaningful ways to invest in public appreciation of mathematics. He viewed the superficial media treatments of Perelman and his work, with disproportionate attention being placed on the prize value itself, as unsurprising. By contrast, Vershik praised the Clay Institute's direct funding of research conferences and young researchers. Vershik's comments were later echoed by Fields medalist Shing-Tung Yau, who was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them".
Solved problems
Poincaré conjecture
In the field of geometric topology, a two-dimensional sphere is characterized by the fact that it is the only closed and simply connected two-dimensional surface. In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as the Poincaré conjecture, the precise formulation of which states:Any three-dimensional topological manifold which is closed and simply-connected must be homeomorphic to the 3-sphere.
Although the conjecture is usually stated in this form, it is equivalent (as was discovered in the 1950s) to pose it in the context of smooth manifolds and diffeomorphisms.
A proof of this conjecture, together with the more powerful geometrization conjecture, was given by Grigori Perelman in 2002 and 2003. Perelman's solution completed Richard Hamilton's program for the solution of the geometrization conjecture, which he had developed over the course of the preceding twenty years. Hamilton and Perelman's work revolved around Hamilton's Ricci flow, which is a complicated system of partial differential equations defined in the field of Riemannian geometry.
For his contributions to the theory of Ricci flow, Perelman was awarded the Fields Medal in 2006. However, he declined to accept the prize. For his proof of the Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010. However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own.
Unsolved problems
Birch and Swinnerton-Dyer conjecture
This conjecture, named after Bryan John Birch and Peter Swinnerton-Dyer, deals with certain types of equations: those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r, then the L-function L(E, s) associated with it vanishes to order r at s = 1.
The official statement of the Birch and Swinnerton-Dyer conjecture was given by Andrew Wiles.
Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
Hdg
k
(
X
)
=
H
2
k
(
X
,
Q
)
∩
H
k
,
k
(
X
)
.
{\displaystyle \operatorname {Hdg} ^{k}(X)=H^{2k}(X,\mathbb {Q} )\cap H^{k,k}(X).}
We call this the group of Hodge classes of degree 2k on X.
The modern statement of the Hodge conjecture is:
Let X be a non-singular complex projective variety. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
Navier–Stokes existence and smoothness
∂
u
∂
t
⏟
Variation
+
(
u
⋅
∇
)
u
⏟
Convection
⏞
Inertia (per volume)
−
ν
∇
2
u
⏟
Diffusion
=
−
∇
w
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Internal
source
⏞
Divergence of stress
+
g
⏟
P versus NP
The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, to biology, philosophy and to cryptography (see P versus NP problem proof consequences). A common example of an NP problem not known to be in P is the Boolean satisfiability problem.
Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven.
The official statement of the problem was given by Stephen Cook.
Riemann hypothesis
The Riemann zeta function
ζ
(
s
)
{\displaystyle \zeta (s)}
is a function whose arguments may be any complex number other than 1, and whose values are also complex. This function can be expressed as the infinite sum of the power of a unit fraction:
ζ
(
s
)
=
∑
n
=
1
∞
1
n
s
=
1
1
s
+
1
2
s
+
1
3
s
+
⋯
.
Yang–Mills existence and mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
For a given real field
ϕ
(
x
)
{\displaystyle \phi (x)}
, we can say that the theory has a mass gap if the two-point function has the property
⟨
ϕ
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0
,
t
)
ϕ
(
0
,
0
)
⟩
∼
∑
n
A
n
exp
(
−