Blaschke conjecture
If shortest paths stay uniquely determined out to the full diameter, must the entire metric be one of the standard symmetric geometries?
Mathematics
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?
Landau’s constant
How large a round disk must every normalized holomorphic map of the unit disk cover somewhere in its image?
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?