Kannan–Lovász–Simonovits conjecture
Can local changes control the fluctuations of functions under log-concave probability laws with a constant independent of dimension?
Analysis and dynamics
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UCT problem for nuclear C*-algebras
Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?
Bochner–Riesz conjecture
How much smoothing of a spherical frequency cutoff guarantees bounded Fourier reconstruction in higher dimensions?
Normality of π
Do all finite strings of decimal digits occur in π with the frequencies expected from uniformly chosen digits?
Pólya’s eigenvalue conjecture
Does the area of a planar region impose a sharp lower bound on every Dirichlet vibration frequency, rather than only on high frequencies in the limit?
Quantum unique ergodicity conjecture
Must every sequence of increasingly high-frequency waves spread uniformly in phase space on a connected compact manifold of negative curvature and dimension at least two?
Gersho’s conjecture in three dimensions
For many representative points in a uniform three-dimensional region, is a body-centered cubic arrangement asymptotically best at minimizing squared distance to the nearest representative?
Kelvin problem
What periodic partition of three-dimensional space into equal-volume cells uses the least shared surface area?
Mazur’s separable quotient problem
Can every infinite-dimensional Banach space be collapsed to an infinite-dimensional space described by a countable dense set?
Landau’s constant
How large a round disk must every normalized holomorphic map of the unit disk cover somewhere in its image?