When the Clay Mathematics Institute announced the Millennium Prize Problems in 2000, it was not predicting that mathematics would solve seven famous questions within a generation. It was doing something more interesting: drawing a public map of some of the deepest places where mathematical knowledge still stopped. Twenty-six years later, that map has changed surprisingly little. One problem has been solved. Six remain open.
The seven problems carry a total prize fund of US$7 million, with US$1 million allocated to each problem. There is no time limit for solving them. Clay’s current list still identifies the Birch and Swinnerton-Dyer Conjecture, Hodge Conjecture, P vs NP, Riemann Hypothesis, and Yang-Mills & the Mass Gap as unsolved, while the Poincaré Conjecture is listed as solved. Navier–Stokes remains on the open list as well. Clay Mathematics Institute.
The scorecard in 2026
- Poincaré Conjecture — solved. Grigoriy Perelman posted his proof in 2002–2003, building on Richard Hamilton’s Ricci-flow program. Clay awarded the Millennium Prize in 2010; Perelman declined the prize money.
- Riemann Hypothesis — open. The conjecture says that all nontrivial zeros of the Riemann zeta function have real part 1/2. Enormous computational evidence exists, but no proof is known.
- Yang-Mills and the Mass Gap — open. Physicists have compelling theoretical and experimental reasons to expect a mass gap, but a rigorous mathematical construction establishing the required properties is still missing.
- Navier–Stokes Existence and Smoothness — open. The challenge is to prove whether smooth three-dimensional solutions of the equations always remain smooth, or whether a finite-time singularity can form.
- Hodge Conjecture — open. It asks how far topological information about algebraic varieties can be represented by algebraic cycles. Important special cases are known, but the general conjecture remains unresolved.
- Birch and Swinnerton-Dyer Conjecture — open. It connects rational points on elliptic curves with analytic information encoded in their L-functions. Major partial results exist, but the full conjecture remains open.
- P vs NP — open. The central question is whether every problem whose solution can be verified efficiently can also be solved efficiently. Despite decades of work, neither P = NP nor P ≠ NP has been proved.
The striking fact is that these are not seven versions of the same problem. They span topology, number theory, algebraic geometry, fluid mechanics, quantum field theory, and theoretical computer science. What they share is not a common technique but an unusually high level of resistance to proof.
1. Poincaré: the one door that opened
The Poincaré Conjecture is the exceptional case. Poincaré asked in 1904 whether the three-dimensional sphere is characterized by simple connectivity. Perelman’s work, based on Ricci flow and the broader geometrization program, established the conjecture and much more: the proof completed the geometrization picture for three-manifolds. Clay now places the problem in its solved archive.
Its history is instructive. A problem can remain apparently immovable for decades and then yield when a new framework changes what “solving” it means. Poincaré therefore stands not merely as one solved item on a scoreboard, but as a warning against assuming that today’s formulation captures tomorrow’s method.
2. Riemann: overwhelming evidence is not a proof
The Riemann Hypothesis may be the most famous of the six open problems. It sits at the heart of our understanding of the distribution of prime numbers. Clay notes that the first 10 trillion nontrivial zeros have been checked computationally, yet that does not establish the statement for infinitely many zeros.
This distinction captures something fundamental about modern mathematics: computation can provide extraordinary evidence while leaving the logical gap between “all cases tested” and “all cases” completely intact. The unresolved question is not whether the hypothesis looks true. It is whether mathematics can prove that it must be true.
3. Navier–Stokes: when equations meet physical reality
The Navier–Stokes equations have been used for generations to model fluids, yet the rigorous mathematical question is much harder than successful physical prediction. In three dimensions, mathematicians still do not know whether appropriately smooth initial data necessarily lead to globally smooth solutions, or whether singular behavior can develop.
That makes Navier–Stokes a particularly revealing Millennium problem: practical science can work remarkably well while the mathematical foundations of the equations remain incomplete.
4. Hodge and Birch–Swinnerton-Dyer: hidden structure
The Hodge Conjecture and the Birch and Swinnerton-Dyer Conjecture are less familiar to the general public, but they illustrate one of the deepest themes in mathematics: apparently different descriptions of an object may contain the same hidden information.
The Hodge Conjecture concerns which topological features of algebraic varieties can be represented by algebraic cycles. The Birch and Swinnerton-Dyer Conjecture concerns elliptic curves and the relationship between rational points and the behavior of an associated L-function. Both have substantial bodies of partial results, special cases, computational evidence and powerful surrounding theories. Neither has yielded to a complete proof.
5. P vs NP: can finding be as easy as checking?
P vs NP asks whether every problem for which a proposed solution can be checked quickly can also be solved quickly. The problem has become central to theoretical computer science because an answer would clarify the limits of efficient computation. Yet the basic question remains unanswered.
It is tempting to think that ever faster computers or increasingly capable AI systems will eventually settle the question by brute force. But P vs NP is a mathematical statement about computational complexity, not merely about the speed of today’s machines. The unresolved task is to establish a general theorem about what efficient algorithms can or cannot accomplish.
6. Yang–Mills: physics knows the gap; mathematics must prove it
Yang–Mills theory underlies the mathematical description of fundamental interactions. The Millennium problem asks for a rigorous construction of the quantum theory and a proof that it has a positive mass gap. Physical theory and computational evidence strongly support the existence of such a gap, but the mathematical construction and proof demanded by the problem remain open.
This is another example of the frontier between successful physical theory and rigorous mathematical foundation. Nature may be telling us the answer is yes; mathematics still has to prove exactly what that means.
7. What has actually changed by 2026?
The most important update is surprisingly simple: the official scoreboard is still one solved and six open. Clay’s public-lecture program, however, shows that the problems remain active subjects of serious mathematical research. The lecture series running from September 2025 through April 2026 included dedicated talks on all seven problems, and Clay has continued to host research activity around them. A September 2026 workshop on Birch and Swinnerton-Dyer is scheduled as part of the 2026 Clay Research Conference and Workshops.
That activity matters. “Open” does not mean “nothing is happening.” It can mean decades of partial theorems, special cases, computational experiments, new frameworks, unexpected connections and increasingly precise knowledge of where the real obstruction lies. In several cases, the surrounding mathematical landscape has changed dramatically even though the headline conjecture has not.
The deeper lesson
The Millennium Prize Problems were designed to demonstrate that mathematical frontiers remain genuinely open. Twenty-six years later, they have done exactly that. The fact that only one of seven has been resolved is not evidence that mathematics has stalled. It is evidence that the problems were chosen at the edge of what human mathematical methods could see.
And the solved problem offers perhaps the most encouraging lesson. Poincaré was not defeated by a faster version of the old approach. It was solved through a new conceptual framework—Ricci flow and geometrization—that changed the problem’s landscape. The remaining six may require something equally unexpected.
For now, the 2026 status is clear: one million-dollar problem has been solved, six remain unsolved, and the frontier is still very much alive.
Primary reference: Clay Mathematics Institute — The Millennium Prize Problems. See also the official prize rules.