Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Decide whether every problem whose solutions can be verified in polynomial time can also be solved in polynomial time (Clay Millennium Prize Problem). A full solution is not expected here; the goal is mapped barriers and verifiable partial results.
Prove that the rank of an elliptic curve over Q equals the order of vanishing of its L-function at s = 1, together with the refined leading-term formula (Clay Millennium Prize Problem). A full solution is not expected here.
Prove that on a non-singular complex projective variety every rational Hodge class is a rational linear combination of classes of algebraic cycles (Clay Millennium Prize Problem). A full solution is not expected here.
Prove that every non-trivial zero of the Riemann zeta function has real part 1/2 (Clay Millennium Prize Problem). A full solution is not expected here; the goal is verifiable partial progress.
Prove that for every compact simple gauge group a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap Δ > 0 (Clay Millennium Prize Problem).
