Farrell–Jones conjecture
Can algebraic invariants of every group be assembled from its virtually cyclic subgroups?
Algebra and geometry
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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?
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?
Blaschke conjecture
If shortest paths stay uniquely determined out to the full diameter, must the entire metric be one of the standard symmetric geometries?
Cabling conjecture
Can surgery on a knot create a genuinely decomposable 3-manifold only through the known cable-knot construction?
Alperin weight conjecture
Can each block’s irreducible representations in prime characteristic be counted entirely from local subgroup data?
Broué’s abelian defect group conjecture
Does an abelian-defect representation block retain its derived structure when replaced by its local normalizer block?
Erdős distinct distances problem
Must every set of n distinct points in the plane determine at least a constant times n/√log n different distances?