Quillen’s conjecture
Can the topology of a finite group’s elementary abelian 2-subgroups detect its normal 2-subgroup structure?
Mathematics
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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?
Shannon capacity of the seven-cycle
Seven symbols sit around a circle, and neighboring symbols can be confused. How many messages can long strings of these symbols carry with absolutely no decoding mistakes?
Pure subtype system type safety
A type system can let programs and their types share the same language. Can the remaining proof show that the proposed pure-subtype rules stay valid as a program evaluates?
Rabin–Mostowski index problem
A finite automaton can check an infinite branching tree using a hierarchy of recurring priorities. Can an algorithm tell whether the same trees can be recognized with a smaller priority hierarchy?
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?
Blaschke conjecture
If shortest paths stay uniquely determined out to the full diameter, must the entire metric be one of the standard symmetric geometries?
Mazur’s separable quotient problem
Can every infinite-dimensional Banach space be collapsed to an infinite-dimensional space described by a countable dense set?
Cherlin–Zilber algebraicity conjecture
Must every infinite simple group with finite logical dimension come from algebraic geometry?
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?
Pillai conjecture
For each fixed positive gap, are there only finitely many distinct pairs of integer perfect powers separated by that gap?