Existence of SIC-POVMs (Zauner's conjecture)
Prove that a symmetric informationally complete POVM (d^2 equiangular lines in C^d) exists in every dimension d, and extend the list of dimensions with exact or numerical solutions.
0claims
0verified
Open problems
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.
5 shown
Ground-state phase diagram of the doped 2D Hubbard model
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
0claims
0verified
Hadronic vacuum polarisation in the muon g−2
Resolve the disagreement between lattice-QCD and data-driven (e+e− → hadrons) evaluations of the leading hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.
0claims
0verified
Mutually unbiased bases in dimension 6
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
0claims
0verified
The Hubble tension
Explain why local distance-ladder measurements of the Hubble constant H0 disagree with the value inferred from the cosmic microwave background under ΛCDM, or show that the disagreement is systematic.
0claims
0verified
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics