Clay currently lists only the Poincaré Conjecture as solved. Grigoriy Perelman’s proof was posted in 2002 and 2003, and Clay awarded the Millennium Prize in 2010. Navier–Stokes has a separate, unsettled status: in a September 11, 2026 announcement, Clay said the problem had “apparently been settled,” but also said its evaluation and credit process was still under way. Clay’s main status page continues to label Navier–Stokes an active problem.
What counts as “solved” for the Millennium Problems?
The Clay Mathematics Institute created seven Millennium Prize Problems and announced them at a Paris meeting on May 24, 2000. Each problem carries an allocation of $1 million, making the stated fund $7 million in total. Those figures describe the prize structure, not proof that every allocation has been paid.
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For a problem to be treated as solved, a proposed proof must survive expert scrutiny and Clay’s own prize procedures. A public claim, a preprint or an announcement that a proof may exist is not automatically the same as a completed Clay award. That distinction matters in 2026 because Navier–Stokes has generated a major announcement without yet receiving the same settled status that Clay gives Poincaré.
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| Problem | Clay page label | What it asks, in broad terms | Evidence or status note |
|---|---|---|---|
| Poincaré Conjecture | Solved | Whether a closed, simply connected three-dimensional manifold must be the three-dimensional sphere. | Perelman posted the proof in 2002–2003; Clay announced the prize award on March 18, 2010. |
| Navier–Stokes Equation | Active | Whether the equations governing fluid flow always have globally existing, smooth solutions in the required setting. | Clay said on September 11, 2026, that the problem had “apparently been settled,” while evaluation continued. |
| Birch and Swinnerton-Dyer Conjecture | Unsolved | How information about elliptic curves modulo primes relates to the rank of their rational points. | No Clay-recognized solution is listed. |
| Hodge Conjecture | Unsolved | How far the topology of algebraic varieties can be represented by algebraic cycles. | Some special cases are known; the general problem remains open, including the noted four-dimensional case. |
| P vs NP | Unsolved | Whether every problem whose answer can be checked efficiently can also be solved efficiently. | No Clay-recognized solution is listed. |
| Riemann Hypothesis | Unsolved | Whether every nontrivial zero of the Riemann zeta function has real part one-half. | No Clay-recognized solution is listed. |
| Yang–Mills and the Mass Gap | Unsolved | To give a rigorous mathematical construction of quantum Yang–Mills theory with a positive mass gap. | No Clay-recognized solution is listed. |
Thus, using Clay’s labels, one problem is solved, one is active, and five are unsolved. “Active” is not a synonym for “solved” or “unsolved”; it is especially important here because the label has not yet been updated to reflect the September announcement.
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The solved problem: Poincaré Conjecture
What the conjecture says
Henri Poincaré formulated the conjecture in 1904. In intuitive terms, it asks whether the three-dimensional analogue of a sphere is uniquely characterized by simple connectivity. A loop drawn on an ordinary spherical surface can be continuously shrunk to a point. A loop that winds around the central hole of a doughnut cannot be shrunk away without leaving the surface. The conjecture asks whether the corresponding no-hole condition, applied to a closed three-dimensional manifold, forces that manifold to be the three-dimensional sphere.
The phrase “three-dimensional sphere” needs care. It means a three-dimensional manifold, often denoted S³, rather than the familiar two-dimensional spherical surface S². S³ can be represented as a set of points in four-dimensional space, but the conjecture concerns the intrinsic three-dimensional manifold.
How Perelman resolved it
Grigoriy Perelman announced the solution in preprints posted to arXiv in 2002 and 2003. His work developed ideas from Richard Hamilton’s Ricci-flow program. Ricci flow deforms a manifold’s metric in a way that can reveal its geometric structure; controlling singularities in that flow was a central challenge. Perelman’s results established the Poincaré Conjecture as a special consequence of the broader Thurston geometrization conjecture.
The order of events is often misstated. The proof announcement belongs to 2002–2003. Clay’s formal prize-recognition milestone came later, on March 18, 2010, when it announced Perelman as the recipient of the Millennium Prize for resolving Poincaré. The 2010 date is therefore not the date when the proof first appeared.
Why Clay calls it solved
After mathematicians examined the preprints and subsequent expositions, the result was accepted as a valid proof. Clay’s current Millennium Problems page places Poincaré in its solved section, and the 2010 announcement records the award. Those are two related but distinct facts: mathematical resolution was established through verification of the proof, while the institute’s prize announcement was an administrative recognition of that achievement.
Navier–Stokes: an apparent resolution still under review
What the problem asks
The Navier–Stokes equations model the motion of fluids such as water and air. The Millennium problem is not simply to find useful numerical solutions for particular flows. It asks for a rigorous answer about existence and smoothness: starting from suitable smooth initial conditions in three dimensions, do solutions remain smooth for all time, or can a singularity develop in finite time? A complete solution must prove the required global regularity or demonstrate a mathematically valid breakdown.
What Clay announced in September 2026
On September 11, 2026, Clay published a statement saying: “Today, CMI shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled.” The statement also says, “The process is deliberately unhurried, but we will provide updates.”
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Those words establish an important qualification. Clay was acknowledging an apparent settlement and describing an ongoing evaluation and credit process; it was not announcing that a Millennium Prize had already been awarded. As currently listed by Clay, Navier–Stokes remains under “Active problems.” The most accurate description is therefore “apparently settled, pending evaluation,” not “officially solved.”
How to interpret the apparent settlement
- A claim or proof announcement is evidence that a proposed solution exists, not by itself a completed prize decision.
- Clay’s statement does not identify the claimant or provide technical details of the proposed argument.
- The institute has not, in the cited announcement, declared that its review is complete or that the $1 million prize has been awarded.
- Future updates could change the status label once the proof, attribution and formal requirements have been evaluated.
Keeping these points separate prevents a common error: treating the September announcement as equivalent to Perelman’s completed Poincaré recognition.
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Birch and Swinnerton-Dyer Conjecture
This conjecture concerns elliptic curves, equations whose rational solutions have a rich arithmetic structure. It predicts a connection between the curve’s rank—the number of independent rational points—and data obtained by examining the curve modulo different primes. Establishing the conjecture in full would link local information at many primes to a global description of rational points. Clay continues to list it as unsolved.
Hodge Conjecture
The Hodge Conjecture asks which topological features of complex algebraic varieties arise from algebraic subvarieties, or algebraic cycles. Hodge theory supplies classes that look topological or analytic; the conjecture predicts that certain classes with the right properties should come from genuinely algebraic cycles. Clay notes that important special cases are known, but the general statement remains open, including the highlighted four-dimensional case.
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P vs NP asks whether every problem for which a proposed answer can be checked efficiently can also be solved efficiently. “Efficiently” is formalized using polynomial-time algorithms. A proof that P equals NP would show that verification and discovery are equivalent in this broad sense; a proof that they differ would establish a fundamental limit on efficient computation. Neither direction has been proved, so Clay lists the problem as unsolved.
Riemann Hypothesis
The Riemann Hypothesis concerns the zeros of the Riemann zeta function, a function deeply connected with the distribution of prime numbers. It predicts that every nontrivial zero lies on the vertical line with real part one-half in the complex plane. Extensive computations and partial theorems do not replace a proof of the full statement. Clay’s current page continues to classify it as unsolved.
Yang–Mills and the Mass Gap
Yang–Mills theory is central to modern quantum field theory. The Millennium problem requires a rigorous mathematical construction of the theory and a proof of a positive mass gap: roughly, that the theory’s excitations have a minimum nonzero mass rather than allowing arbitrarily low-energy massive states. Physical evidence and successful calculations are not by themselves the mathematical proof Clay requests, and the problem remains unsolved on its list.
How the prize status should be reported
There are several dates and labels that should not be collapsed into one headline:
- May 24, 2000: Clay announced the seven problems and the $7 million fund, with $1 million allocated to each.
- 2002–2003: Perelman posted the preprints that announced the Poincaré solution.
- March 18, 2010: Clay announced Perelman’s Millennium Prize award for Poincaré.
- September 11, 2026: Clay said Navier–Stokes had “apparently been settled” and that evaluation would proceed deliberately.
Consequently, the direct answer depends on the meaning of “solved.” Under Clay’s formal current label, Poincaré is the sole solved problem. If recent developments are included, Navier–Stokes must be mentioned as an apparent resolution undergoing review, not as a completed Clay prize award. The other five—Birch and Swinnerton-Dyer, Hodge, P vs NP, Riemann, and Yang–Mills and the Mass Gap—remain under Clay’s unsolved heading.
Further reading on Poincaré
Clay’s Poincaré materials include The Poincaré Conjecture, edited by James Carlson, as a more detailed mathematical resource. It is best treated as background reading on the theorem and its proof history, separate from the institute’s short status label.
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