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OP-0050Mathematical physicsOpen

Yang–Mills existence and the mass gap

Construct quantum Yang–Mills theory on R^4 rigorously, and prove it has a positive mass gap.

Millennium Prize Problems· 2000$1,000,000 prize

OP-0026Computational complexityOpen

P versus NP

Can every problem whose solution is quickly checkable also be quickly solved?

Millennium Prize Problems· 1971$1,000,000 prize

OP-0011CombinatoricsPartially solved

The Erdős–Rado sunflower conjecture

How large must a family of k-sets be before it must contain three sets meeting pairwise in the same core?

Erdős Problems (Thomas Bloom)· Source reports OPEN· 1960$1,000 prize

OP-0010CombinatoricsOpen

Erdős's conjecture on arithmetic progressions

If the reciprocals of a set of integers diverge, must the set contain arbitrarily long arithmetic progressions?

Erdős Problems (Thomas Bloom)· Source reports OPEN· 1936$5,000 prize

OP-0004Number theoryOpen

The Birch and Swinnerton-Dyer conjecture

Does the rank of an elliptic curve equal the order of vanishing of its L-function at s = 1?

Millennium Prize Problems· 1965$1,000,000 prize

OP-0003Geometry and topologyOpen

The Hodge conjecture

Is every Hodge class on a projective complex manifold a rational combination of algebraic cycle classes?

Millennium Prize Problems· 1950$1,000,000 prize

OP-0002Partial differential equationsOpen

Global regularity for the three-dimensional Navier–Stokes equations

Do smooth, finite-energy initial velocity fields always produce solutions that stay smooth forever?

Millennium Prize Problems· 2000$1,000,000 prize

OP-0001Number theoryOpen

The Riemann hypothesis

Do all non-trivial zeros of the Riemann zeta function lie on the critical line?

Millennium Prize Problems· 1859$1,000,000 prize