Sidorenko’s conjecture
Do random graphs minimize the density of every fixed bipartite pattern among graphs of the same edge density?
Mathematics
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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?
Tate conjecture over finite fields
Do all Galois-invariant cohomology classes of the relevant degree come from algebraic cycles over a finite field?
Dynamical uniform boundedness conjecture
Can finite arithmetic orbits be bounded by map degree and number-field degree alone?
Meyniel’s conjecture
Can O(√n) pursuers catch a visible evader on every connected n-vertex graph?
Hypergraph dualization
Can a classical polynomial-time algorithm recognize all minimal hitting sets?
Carmichael’s totient conjecture
Can an Euler totient value come from exactly one positive integer?
Generalized Ramanujan conjecture
Are the local components of every cuspidal automorphic representation of GLn tempered?
Erdős–Turán additive basis conjecture
Must an additive basis of nonnegative integers have arbitrarily many representations of some sums?
Ryser’s Latin square conjecture
Does every odd-order Latin square have a transversal?
Erdős–Lovász Tihany conjecture
Can a graph be split into two parts retaining prescribed coloring needs?
Planar self-avoiding walk scaling limit
Do critical self-avoiding walks on a fixed square lattice converge to SLE₈/₃ as the grid becomes finer?
Amenability of Thompson’s group F
Can Thompson’s group F admit an average that is unchanged by every group translation?
Hartshorne’s complete intersection conjecture
Must a smooth complex projective variety of sufficiently small codimension be defined by exactly that many polynomial equations?
Nagata conjecture
How low can a curve’s degree be when it must pass repeatedly through many very general points?
Zariski multiplicity conjecture
Does a complex hypersurface singularity’s local topological shape determine its multiplicity?