Tate conjecture over finite fields
Do all Galois-invariant cohomology classes of the relevant degree come from algebraic cycles over a finite field?
Mathematics
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?
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?
Baum–Connes conjecture
Can the proper-action geometry of a countable discrete group recover the K-theory of its reduced operator algebra?
Grothendieck’s section conjecture
Does the arithmetic fundamental group of a curve identify exactly its rational points?
Planar first-passage percolation fluctuations
How widely does the fastest travel time vary across a square grid with independent random edge times?
Relativistic Vlasov–Maxwell global regularity
Can smooth three-dimensional collisionless plasma equations develop a singularity from regular initial data?
Total coloring conjecture
Can every finite simple graph color its vertices and edges together using at most two colors beyond its maximum degree?
Cartan–Hadamard conjecture
Must a region in a nonpositively curved space have at least the boundary area of a Euclidean ball with the same volume?
Abundance conjecture
Does numerical nonnegativity of a minimal model’s canonical divisor guarantee a globally defined geometric map?
Bloch’s constant
What is the largest disk radius guaranteed on a one-to-one image patch of every normalized holomorphic map?
Lefschetz standard conjecture
Can a basic reversible cohomology operation on a complex algebraic variety always be reversed using algebraic cycles?
Green–Griffiths–Lang conjecture
Must all entire complex curves in a general-type projective variety lie in one proper algebraic subset?
Kannan–Lovász–Simonovits conjecture
Can local changes control the fluctuations of functions under log-concave probability laws with a constant independent of dimension?