UCT problem for nuclear C*-algebras
Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?
Mathematics
Zeeman collapsibility conjecture
Does every contractible two-dimensional polyhedron become collapsible after taking its product with one interval?
Bochner–Riesz conjecture
How much smoothing of a spherical frequency cutoff guarantees bounded Fourier reconstruction in higher dimensions?
Tuza’s triangle packing conjecture
Can all triangles in any finite simple graph be destroyed by removing at most twice its maximum number of edge-disjoint triangles?
Ryser’s hypergraph conjecture
Does every finite multipartite uniform hypergraph have a vertex cover at most r−1 times its largest matching?
List edge-coloring conjecture
Do separate allowed-color lists on edges always need no more colors than ordinary edge coloring?
Elliott–Halberstam conjecture
How far can averaged control of prime-counting errors in arithmetic progressions extend as the modulus grows?
Farrell–Jones conjecture
Can algebraic invariants of every group be assembled from its virtually cyclic subgroups?
Residual finiteness of hyperbolic groups
Can finite groups detect every nontrivial element of every hyperbolic group?
Cannon’s conjecture
Does a hyperbolic group whose boundary is a two-sphere act geometrically on hyperbolic three-space?
Quillen’s conjecture
Can the topology of a finite group’s elementary abelian 2-subgroups detect its normal 2-subgroup structure?
Serre’s uniformity conjecture
Do non-CM elliptic curves over the rationals always have full mod-p Galois image for primes p>37?
Lehmer’s nonvanishing conjecture
Is every coefficient of Ramanujan’s discriminant modular form nonzero?
Bott–Grove–Halperin conjecture
Does nonnegative sectional curvature force finite total rational homotopy in a closed simply connected manifold?
Purely cosmetic surgery conjecture
Can two distinct surgery slopes on the same nontrivial knot produce the same oriented three-manifold?
Volume conjecture
Does exponential growth of colored Jones evaluations recover the volume of every hyperbolic knot complement?
Normality of π
Do all finite strings of decimal digits occur in π with the frequencies expected from uniformly chosen digits?
Graham’s pebbling conjecture
Moving a pebble across an edge costs two pebbles. Is the worst-case supply needed on a product graph at most the product of the supplies needed on its two factors?
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?
Lovász Hamiltonian path conjecture
Does complete symmetry among a connected graph’s vertices guarantee a route visiting every vertex exactly once?
Chvátal’s toughness conjecture
Is there one fixed level of resistance to vertex removal that guarantees a round trip visiting every vertex exactly once?