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Clay Mathematics Institute (CMI) lists the Poincaré Conjecture as solved. Grigoriy Perelman announced its proof in preprints posted in 2002 and 2003, and CMI awarded him the Millennium Prize for it in 2010. Navier–Stokes has a newer, less settled status: on September 11, 2026, CMI said the problem had “apparently been settled,” while describing its evaluation and credit process as ongoing. Its main problem list still labels Navier–Stokes active.
What counts as “solved” here?
The clearest answer is the one CMI gives on its Millennium Problems page: the Poincaré Conjecture appears in the solved section. That is also the problem for which CMI announced a prize award, naming Perelman as the recipient on March 18, 2010.
Navier–Stokes needs a separate qualification. CMI’s September 11, 2026 statement described the problem as “apparently” settled, but said its evaluation and credit process would be deliberately unhurried. At the same time, CMI’s main list continued to place Navier–Stokes under “Active problems.” Those statements are not interchangeable: an apparent resolution is not the same as a completed CMI review or a recorded prize award.
So, if “solved” means the problem CMI formally lists as solved and for which it has announced a Millennium Prize award, the answer is one: Poincaré. If the question includes a recent claim of an apparent resolution still under evaluation, Navier–Stokes must also be mentioned, with that caveat intact.
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This distinction is why a simple count can be misleading. A problem’s label on CMI’s list, a public announcement that a proof exists, and CMI’s decision to recognize a prize recipient are separate status markers. The Poincaré case has all the clarity of a solved label and a formal award announcement. Navier–Stokes is the current exception: CMI has reported an apparent settlement but has not, in the cited statement, said its review is finished.
All seven Millennium Prize Problems and their current labels
CMI announced the seven problems at a Paris meeting on May 24, 2000. Its current page states a total prize fund of $7 million, with $1 million allocated to the solution of each problem. The figure describes the allocation; it does not establish that every prize has been paid.
| Problem | CMI page label | What it asks, in brief |
|---|---|---|
| Poincaré Conjecture | Solved | Whether simple connectivity characterizes the three-dimensional sphere. |
| Navier–Stokes Equation | Active | Questions about existence and smoothness of solutions to equations governing fluid flow; CMI said in September 2026 that it had apparently been settled, with evaluation ongoing. |
| Birch and Swinnerton-Dyer Conjecture | Unsolved | Connects information about points on elliptic curves modulo primes with the rank of rational points. |
| Hodge Conjecture | Unsolved | Asks how much of the topology of algebraic varieties can be described using algebraic cycles. |
| P vs NP | Unsolved | Asks whether every problem whose answer can be checked efficiently can also be solved efficiently. |
| Riemann Hypothesis | Unsolved | Predicts that every nontrivial zero of the zeta function has real part one-half. |
| Yang–Mills & the Mass Gap | Unsolved | Requires a mathematical proof of a mass gap for quantum Yang–Mills theory. |
The labels above reflect CMI’s page, including its distinction between “Active” and “Unsolved.” The latter five problems remain under CMI’s unsolved heading; the September 2026 development makes Navier–Stokes the status exception to explain rather than a settled prize entry.
In particular, “active” should not be silently converted into “solved” just because CMI has acknowledged a settlement announcement. Conversely, “unsolved” does not mean there is no ongoing mathematics or no known special case. These labels describe CMI’s stated status for the prize problems as a whole, not the absence of every partial result.
Why the Poincaré Conjecture is the established solution
Henri Poincaré formulated the conjecture in 1904. In accessible terms, it asks whether a particular property—simple connectivity—identifies the three-dimensional sphere. Simple connectivity means that every loop on the space can be continuously contracted to a point without tearing the space. A loop drawn on a spherical surface can be shrunk this way; a loop encircling the hole of a doughnut cannot.
The conjecture extends that kind of distinction to the three-dimensional sphere, a three-dimensional manifold. It is not asking whether an ordinary two-dimensional spherical surface has the familiar shape of a ball. That difference matters because the result concerns the structure of a space, not a visual resemblance to a round object.
Perelman announced the solution in preprints posted to arXiv in 2002 and 2003. The proof built on Richard Hamilton’s theory of Ricci flow, and Perelman’s work also proved Thurston’s geometrization conjecture, of which Poincaré is a special case. These are connected mathematical results, but the Millennium Prize award at issue was for resolving the Poincaré Conjecture.
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For a reader who wants a next step after the intuitive explanation, CMI’s Poincaré page lists The Poincaré Conjecture, edited by James Carlson, among its further resources. It is an optional mathematical reading recommendation, not a condition for understanding the status answer.
What changed for Navier–Stokes in September 2026?
On September 11, 2026, CMI published a statement saying it shared the mathematical community’s excitement as it contemplated the announcement that Navier–Stokes had “apparently been settled.” It added: “The process is deliberately unhurried, but we will provide updates.” CMI’s wording deliberately stops short of declaring that its evaluation is complete.
As of the cited current problem list, Navier–Stokes remained under CMI’s “Active problems” heading. The most accurate way to describe the situation is therefore to report both pieces of information together: CMI’s list labels it active, and CMI’s September 11 statement says it has apparently been settled while review and credit are still being evaluated. It is not accurate to present that statement as an awarded Millennium Prize.
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Navier–Stokes concerns equations governing fluid flow. The prize problem asks about existence and smoothness of solutions, so a claim of resolution must be considered in relation to those mathematical questions, not merely to an application or a numerical result. The September statement does not provide enough detail to explain the claimed proof’s argument; the responsible status summary is the one CMI itself gave, including the word “apparently.”
What the other five unsolved problems concern
Birch and Swinnerton-Dyer Conjecture
This conjecture links two kinds of information about elliptic curves: data about points when the curve is considered modulo primes, and the rank of its rational points. In broad terms, it asks for a relationship between local information across primes and the structure of rational solutions. CMI lists the conjecture as unsolved; that label does not mean there are no known results or partial progress, only that the stated prize problem has not been marked solved on the page.
Hodge Conjecture
The Hodge Conjecture concerns algebraic varieties and asks how much of their topology can be represented through algebraic cycles. CMI notes that certain special cases are known, while the four-dimensional case remains unknown. That qualification is useful: “unsolved” does not mean every instance is equally open, and established special cases do not amount to a solution of the full conjecture.
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P vs NP
P vs NP asks whether every problem for which a proposed answer can be checked efficiently can also be solved efficiently. The distinction is between verifying a candidate answer and finding one. If those capabilities were shown to be equivalent in the sense asked by the problem, it would resolve a foundational question about efficient computation. CMI continues to list it as unsolved; the familiar phrasing is a plain-language orientation, not a claim that every computational task fits a single practical runtime rule.
Riemann Hypothesis
The Riemann Hypothesis predicts that all nontrivial zeros of the zeta function have real part one-half. The prediction is a precise statement about where those zeros lie, rather than a claim that the zeta function has no other zeros. CMI lists the hypothesis as unsolved.
Yang–Mills & the Mass Gap
This problem calls for a rigorous mathematical proof of a mass gap in quantum Yang–Mills theory. Its framing matters: the prize problem is not simply to use the theory or produce a physical description, but to establish the required result mathematically. CMI’s page places it among the unsolved problems.
How to read the prize count and status claims
There are three different kinds of statement in accounts of the Millennium Problems, and they answer different questions:
- A page label reports how CMI currently categorizes a problem. For Poincaré, that label is solved; for Navier–Stokes, the main list still says active.
- A proof or settlement announcement reports that a result has been presented or described as a resolution. The September 2026 Navier–Stokes statement uses the qualifier “apparently” and leaves evaluation ongoing.
- A prize announcement records CMI’s formal award decision. CMI announced Perelman’s Poincaré prize on March 18, 2010.
Keeping these categories apart prevents two common errors: dating Perelman’s proof to the later prize announcement, and treating the Navier–Stokes announcement as if CMI had already completed its review and awarded the prize. CMI’s stated allocation is $1 million per problem, but the total allocation of $7 million should not be paraphrased as seven completed or paid awards.
That also answers a frequent follow-up: has CMI paid seven million dollars in prizes? Its page gives a $7 million prize fund and allocates $1 million to each of the seven problems. The 2010 press release records the award to Perelman for Poincaré. Those figures and that award record do not establish that all seven allocations have been paid.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Frequently asked questions
How many Millennium Prize Problems have been solved?
CMI’s current page marks one as solved: the Poincaré Conjecture. Navier–Stokes has a September 2026 announcement describing it as “apparently” settled, but CMI says its evaluation and credit process is ongoing and its main list still labels it active.
Has the Navier–Stokes problem officially been solved?
CMI has not described a completed review or prize award in its September 11, 2026 statement. It called the problem “apparently” settled and said it would provide updates. Report that qualification rather than treating the announcement as a final CMI decision.
Did Perelman solve the Poincaré Conjecture in 2010?
No: CMI says Perelman’s solution appeared in preprints posted in 2002 and 2003. March 18, 2010 is the date CMI announced the Millennium Prize award for the solution.
Are the other five problems completely without known results?
“Unsolved” is the status of the full prize problem, not a claim that there are no partial results. For example, CMI notes known special cases of the Hodge Conjecture, while saying the four-dimensional case remains unknown.
How much is each Millennium Prize worth?
CMI says $1 million is allocated to each of the seven problems, for a stated total prize fund of $7 million. The allocation should not be confused with a record that every award has been made.
Frequently Asked Questions
How many Millennium Prize Problems have been solved?
CMI lists one as solved: the Poincaré Conjecture. Navier–Stokes has been described by CMI as “apparently” settled, but evaluation is ongoing and the main list still labels it active.
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CMI’s September 11, 2026 statement says it has apparently been settled and that the evaluation and credit process is ongoing. It is not yet a recorded CMI prize award.
When did Perelman solve the Poincaré Conjecture?
Perelman announced the proof in preprints posted in 2002 and 2003. CMI announced the prize award on March 18, 2010.
Have all seven $1 million prizes been paid?
CMI states that $1 million is allocated to each problem, for a $7 million total prize fund. That allocation does not establish that every prize has been paid; CMI’s 2010 announcement records the Poincaré award to Perelman.
The Bottom Line
Bottom line: Poincaré is the one Millennium Prize Problem CMI lists as solved and for which it announced an award. Navier–Stokes warrants a prominent update, not a second definitive solved count: CMI said on September 11, 2026 that it had “apparently” been settled, while evaluation remained ongoing and the status page still listed it as active. The other five problems remain under CMI’s unsolved heading.
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