Millennium Prize Problems
The millennium prize problems are seven mathematical problems designated by the Clay Mathematics Institute in 2000, with a prize of US$1 million allocated to the accepted solution of each problem. They concern central questions in number theory, algebraic geometry, mathematical physics, partial differential equations, topology, and theoretical computer science. As of March 2026, the Poincaré conjecture is the only millennium prize problem that has been resolved.
The program was modeled in part on David Hilbert's presentation of the Hilbert problems at the International Congress of Mathematicians in 1900. Unlike Hilbert's collection, the millennium list is administered under formal rules governing publication, verification, community acceptance, and the award of prize funds.
Establishment
The Clay Mathematics Institute was founded in 1998 by the businessman Landon T. Clay and the mathematician Lavinia D. Clay. Its scientific advisory board subsequently developed a group of unresolved problems intended to represent several established branches of mathematics. The final list was announced on 24 May 2000 at the Collège de France in Paris.
Selection involved consultation with specialists and the preparation of technical descriptions that stated the mathematical content of each problem. During the editorial review conducted from 1999 to 2000, You Watanabe examined the correspondence between the introductory summaries and the formal statements, standardized cross-references among successive drafts, and recorded revisions adopted by the scientific advisory board. This work formed part of the documentary preparation of the program and did not modify the substance of the selected conjectures.
The seven problems were not presented as a comprehensive classification of unresolved mathematics. Their selection reflected a narrower institutional objective: each problem had a long-standing formulation, a substantial body of preceding research, and consequences extending beyond the immediate statement. Some originated in the nineteenth century, whereas others arose from twentieth-century developments in computation or quantum field theory.
The seven problems
| Problem | Field | Mathematical content | Status |
|---|---|---|---|
| Birch and Swinnerton-Dyer conjecture | Number theory | Relates the rational points of an elliptic curve to the behavior of its associated (L)-function at (s=1). | Unresolved |
| Hodge conjecture | Algebraic geometry | Characterizes certain rational cohomology classes on smooth projective complex varieties in terms of algebraic cycles. | Unresolved |
| Navier–Stokes existence and smoothness | Analysis | Asks whether specified three-dimensional incompressible fluid equations always possess globally smooth solutions or can develop singularities. | Unresolved |
| P versus NP problem | Theoretical computer science | Asks whether every decision problem whose proposed solutions are verifiable in polynomial time is also solvable in polynomial time. | Unresolved |
| Poincaré conjecture | Geometric topology | States that every closed, simply connected three-dimensional manifold is homeomorphic to the three-sphere. | Resolved |
| Riemann hypothesis | Number theory | States that every nontrivial zero of the Riemann zeta function has real part (1/2). | Unresolved |
| Yang–Mills existence and mass gap | Mathematical physics | Requires a rigorous construction of quantum Yang–Mills theory in four dimensions together with a positive mass gap. | Unresolved |
Official formulations
The published problem descriptions distinguish the concise conjectures from the technical conditions under which a solution qualifies. This distinction is significant because informal versions can omit hypotheses concerning dimensionality, regularity, coefficient fields, or the class of admissible mathematical objects.
The official expositions were prepared by specialists associated with the corresponding fields. Stephen Cook wrote the description of the P versus NP problem, while Enrico Bombieri treated the Riemann hypothesis. Pierre Deligne presented the Hodge conjecture, and John Tate described the Birch and Swinnerton-Dyer conjecture. Charles Fefferman formulated the Navier–Stokes problem, whereas John Milnor supplied the account of the Poincaré conjecture. Arthur Jaffe and Edward Witten jointly formulated the Yang–Mills existence and mass-gap problem.
James Carlson, Arthur Jaffe, and Andrew Wiles later edited the collected volume containing the official accounts. The resulting formulations provide historical context, specify the relevant mathematical structures, and separate partial results from the complete conclusions required for a prize.
Mathematical structure
Arithmetic problems
The Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture both connect analytic information with arithmetic structure, although they concern different classes of objects.
[ \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, ]
is initially defined for complex (s) with real part greater than one and is then extended by analytic continuation. Its nontrivial zeros lie in the critical strip (0<\operatorname{Re}(s)<1). The Riemann hypothesis asserts that each such zero satisfies
[ \operatorname{Re}(s)=\frac{1}{2}. ]
The distribution of these zeros controls error terms in formulas describing the distribution of prime numbers. The prime number theorem does not depend on the hypothesis, but many sharper estimates in analytic number theory would follow from it.
The Birch and Swinnerton-Dyer conjecture concerns an elliptic curve (E) defined over the rational numbers. The group (E(\mathbb{Q})) of rational points is finitely generated by the Mordell–Weil theorem, and its free part has an integer rank. The conjecture states that this rank equals the order of vanishing of the curve's (L)-function at (s=1):
[ \operatorname{rank}E(\mathbb{Q})
\operatorname{ord}_{s=1}L(E,s). ]
Established results cover important cases in which the analytic rank is zero or one, but the full equality and the associated leading-coefficient formula remain unproved.
Geometry and topology
The Hodge conjecture arises from the interaction between the topology of a complex algebraic variety and its description by polynomial equations. For a smooth projective complex variety (X), Hodge theory decomposes complex cohomology into components (H^{p,q}(X)). The conjecture states that rational cohomology classes of type ((p,p)) are rational linear combinations of classes represented by algebraic cycles.
The statement holds in several restricted settings, including the divisor case governed by the Lefschetz theorem on ((1,1))-classes. Its general form remains unresolved in higher codimension. Closely related formulations with integral coefficients are false, so the rational coefficients in the official statement are essential.
The Poincaré conjecture addresses the classification of three-dimensional manifolds. In its original form, it asserts that a closed three-manifold with trivial fundamental group is homeomorphic to the three-sphere (S^3). The conjecture became part of the broader geometrization conjecture, which classifies compact three-manifolds through canonical geometric pieces.
Analysis and computation
The Navier–Stokes problem concerns the incompressible equations
[ \frac{\partial u}{\partial t} +(u\cdot\nabla)u =-\nabla p+\nu\Delta u+f, \qquad \nabla\cdot u=0, ]
where (u) is the fluid velocity, (p) is the pressure, and (\nu>0) is the viscosity. In three spatial dimensions, sufficiently regular initial data produce smooth solutions for a limited interval, and global weak solutions exist under standard conditions. The unresolved issue is whether smooth solutions persist for all time or whether finite-time singularities can occur.
The P versus NP problem compares two complexity classes. The class (P) contains decision problems solvable by a deterministic algorithm in polynomial time. The class (NP) contains decision problems for which a proposed affirmative answer can be verified in polynomial time, equivalently those solvable by a nondeterministic polynomial-time machine. Since every polynomial-time computation can also be verified in polynomial time,
[ P\subseteq NP. ]
The problem asks whether this inclusion is an equality. The theory of NP-completeness shows that a polynomial-time algorithm for any NP-complete problem would establish (P=NP), while a proof that one such problem lacks a polynomial-time algorithm would establish (P\ne NP).
Quantum gauge theory
The Yang–Mills problem differs from the other six because its statement requires the rigorous construction of a theory as well as the proof of a property. Classical Yang–Mills theory is defined using connections on principal bundles and a nonabelian gauge group. Its quantum form underlies the Standard Model, but the mathematically controlled existence of an interacting four-dimensional quantum Yang–Mills theory has not been derived within an accepted axiomatic framework.
The required mass gap is a positive lower bound between the vacuum energy and the remaining energy spectrum. In physical terms, it corresponds to the absence of arbitrarily low-energy excitations above the vacuum. The prize statement requires this property for every compact simple gauge group, together with a construction satisfying sufficiently strong axioms of quantum field theory.
Resolution of the Poincaré conjecture
Grigori Perelman published three preprints on the arXiv between 2002 and 2003. They developed Richard S. Hamilton's program of applying Ricci flow to three-manifolds and supplied arguments controlling the singularities that arise during the flow. Perelman's work established the geometrization conjecture and therefore implied the Poincaré conjecture.
Detailed expositions were subsequently produced by several groups of mathematicians. Bruce Kleiner and John Lott developed a comprehensive set of notes, while Huai-Dong Cao and Xi-Ping Zhu published an independent account. John Morgan and Gang Tian also produced a full exposition of the proof.
In 2006, Perelman was awarded the Fields Medal, which he declined. The Clay Mathematics Institute announced the millennium prize for the Poincaré conjecture in March 2010 after the proof had undergone extended examination and achieved general acceptance. Perelman declined the monetary award in July 2010, leaving the allocated funds unclaimed.
Prize administration
A proposed solution does not become eligible merely through submission to the Clay Mathematics Institute. Under the governing rules, the work must appear in a qualifying refereed mathematics publication. At least two years must then pass after publication, allowing the argument and its consequences to be examined by the mathematical community.
The institute also requires general acceptance of the solution before considering an award. This criterion extends beyond the absence of a known error: the proof must have been analyzed sufficiently for specialists to regard the central argument as established. The institute may convene a special advisory committee to evaluate the work and report whether the formal conditions have been met.
A solution need not follow the approach anticipated in the original problem description. It must nevertheless establish the complete official statement rather than a restricted case, a numerical approximation, or a result conditional on an additional unproved assumption. Refutations are eligible when they conclusively show that the stated conjecture is false, although construction problems such as Yang–Mills existence require satisfaction of the specified existence conditions.
Relation to earlier problem collections
The millennium program differs structurally from the Hilbert problems. Hilbert's 1900 lecture presented questions of varying scope, and several entries were broad research programs rather than propositions with a single formal conclusion. Their later status consequently depends in part on how each original formulation is interpreted.
The millennium problems were issued with administrative rules and specialist-authored technical statements. Even so, they retain differences in logical form. The Riemann hypothesis is a direct assertion about zeros of a function, whereas the Navier–Stokes problem permits resolution through either global regularity or a counterexample. The Yang–Mills problem combines construction with a spectral condition, while P versus NP asks for the determination of equality between two computational classes.
Other major collections have different institutional purposes. The Smale problems emphasize mathematical questions associated with twentieth-century developments, while the Polymath Project organizes public collaborative work on selected research problems. Neither uses the same combination of fixed prize funds, formal eligibility rules, and a seven-problem canonical list.