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OP-0057Condensed matterOpen

Does the two-dimensional Hubbard model superconduct?

The simplest model believed to capture high-temperature superconductivity, and its phase diagram is still contested.

Community consensus· 1987

OP-0055Quantum informationSolved

Tsirelson's problem and the Connes embedding conjecture

Resolved in 2020 by a complexity-theoretic argument that nobody in operator algebras saw coming.

Community consensus· 1976

OP-0051Quantum informationOpen

The quantum PCP conjecture

Is estimating the ground state energy of a local Hamiltonian to constant relative precision still QMA-hard?

Community consensus· 2002

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-0039Computational geometryOpen

The Erdős unit distance problem

How often can the same distance occur among n points in the plane? The gap between the bounds is enormous.

The Open Problems Project (computational geometry)· 1946

OP-0037Algorithms and data structuresOpen

A strongly polynomial algorithm for linear programming

Smale's ninth problem: is there an LP algorithm whose arithmetic operation count does not depend on the bit size of the input?

Smale's problems for the 21st century· 1998

OP-0036CryptographyOpen

Do one-way functions exist?

The minimal assumption of modern cryptography, and it has never been proved from anything weaker than itself.

Community consensus· 1976

OP-0033Computational complexityOpen

A superlinear circuit lower bound for an explicit function

After fifty years, the best known bound for any function in NP is about 3.1n gates.

Community consensus· 1949

OP-0029Computational complexityOpen

Does randomness help? BPP versus P

Widely believed to be equal, and provably tied to circuit lower bounds nobody can prove.

Community consensus· 1977

OP-0028Algorithms and data structuresOpen

Is the exponent of matrix multiplication equal to 2?

Every improvement since 1969 has shaved digits off 2.37. Nobody can rule out 2 + o(1), and nobody can achieve it.

Community consensus· 1969

OP-0027Computational complexityOpen

The Unique Games Conjecture

Half of it was proved in 2018. Whether the full statement holds decides the approximability of dozens of problems at once.

Community consensus· 2002

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-0024AlgebraOpen

The Jacobian conjecture

If a polynomial map has constant non-zero Jacobian determinant, must it be invertible?

Smale's problems for the 21st century· 1939

OP-0022ProbabilityOpen

Conformal invariance of critical percolation on the square lattice

Smirnov proved it for the triangular lattice in 2001. The square lattice remains untouched.

Community consensus· 1992

OP-0020Geometry and topologyOpen

The smooth four-dimensional Poincaré conjecture

Is every smooth 4-manifold homeomorphic to S^4 also diffeomorphic to it?

Kirby's problem list in low-dimensional topology· 1961

OP-0019Number theoryOpen

The Littlewood conjecture

Can two real numbers be simultaneously badly approximable in the multiplicative sense?

Community consensus· 1930

OP-0018AnalysisDisputed

The invariant subspace problem for separable Hilbert spaces

Does every bounded operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Community consensus· 1935

OP-0017AnalysisPartially solved

The Kakeya set conjecture in dimensions at least 4

The three-dimensional case fell in 2025. Every dimension above it is still open.

Community consensus· 1971

OP-0012Graph theoryOpen

Hadwiger's conjecture

Is every graph with no K t minor properly colourable with t − 1 colours?

Open Problem Garden· 1943

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-0008Number theoryDisputed

The abc conjecture

A claimed proof exists and is not accepted by the wider community — the conjecture's status is genuinely contested.

Community consensus· 1985

OP-0007Number theoryOpen

The binary Goldbach conjecture

Is every even integer greater than 2 a sum of two primes?

Community consensus· 1742

OP-0006Number theoryOpen

The twin prime conjecture

Are there infinitely many primes p such that p + 2 is also prime?

Community consensus· 1849

OP-0005Number theoryOpen

The Collatz conjecture

Does the 3n + 1 iteration reach 1 from every positive integer?

Community consensus· 1937

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