Zariski cancellation for affine three-space
If adding one affine coordinate makes a complex variety look like four-dimensional affine space, was it already affine three-space?
Mathematics
Borel conjecture
For certain closed manifolds, does matching their homotopy structure force them to be the same topological shape?
Novikov conjecture
Do a manifold’s higher signatures stay unchanged when its homotopy structure is preserved?
Brocard–Ramanujan problem
Can a factorial plus one be a perfect square beyond the three known examples?
Singmaster’s conjecture
Is there a fixed limit on how often any number greater than one can appear in Pascal’s triangle?
Erdős–Ulam problem
Can points be dense throughout the plane while every distance between two of them is rational?
Moser’s convex worm problem
What is the smallest-area convex region that can contain every planar curve of length one after repositioning it?
Minimum opaque barrier for a square
How little total barrier length can intercept every straight line that passes through a unit square?
Seymour’s second-neighborhood conjecture
Must a directed graph without two-way edges have a vertex reaching at least as many new vertices in two steps as in one?
Borsuk’s conjecture in dimension four
Can every bounded four-dimensional set be split into five pieces of strictly smaller diameter?
Kaplansky’s zero-divisor conjecture
Can two nonzero finite algebraic combinations of elements of a torsion-free group multiply to zero?
Conway’s thrackle conjecture
Can a planar thrackle have more edges than vertices?
Gyárfás–Sumner conjecture
Does excluding any fixed induced tree control a graph’s coloring needs once its largest clique is bounded?
Barnette’s Hamiltonian conjecture
Does every cubic, bipartite, planar, 3-connected graph contain a cycle visiting every vertex exactly once?
Sidorenko’s conjecture
Do random graphs minimize the density of every fixed bipartite pattern among graphs of the same edge density?
Invariant subspace problem
Must every bounded linear operator on complex Hilbert space preserve a nontrivial closed subspace?
Vaught’s conjecture
Can a complete countable theory have an intermediate number of countable models?
Inverse Galois problem
Can every finite group occur as the Galois group of a finite extension of the rational numbers?
Uniform boundedness of rational torsion
Can rational torsion be bounded by abelian-variety dimension and number-field degree?
Furstenberg’s ×2, ×3 conjecture
Must an atomless circle distribution preserved by doubling and tripling be uniform?
Banach–Mazur rotations problem
Does exact unit-sphere symmetry force a separable real Banach space to be Hilbert?
Birkhoff billiard conjecture
Are ellipses the only smooth strictly convex planar billiards that are integrable near the boundary?
Auslander–Reiten conjecture
Do vanishing higher extension groups force a module to be projective?
Finitistic dimension conjecture
For each finite-dimensional algebra, are finite projective dimensions of finitely generated modules bounded?