Baum–Connes conjecture
Can the proper-action geometry of a countable discrete group recover the K-theory of its reduced operator algebra?
Mathematics
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Grothendieck’s section conjecture
Does the arithmetic fundamental group of a curve identify exactly its rational points?
Bag containment of conjunctive queries
Can an algorithm decide whether one database query always produces each answer no more often than another?
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?
Positivity problem for linear recurrences
Can an algorithm decide whether an integer sequence defined by a linear recurrence never becomes negative?
Deterministic NC perfect matching
Can a deterministic parallel algorithm find a perfect matching in every general graph with polylogarithmic time and polynomially many processors?
Kannan–Lovász–Simonovits conjecture
Can local changes control the fluctuations of functions under log-concave probability laws with a constant independent of dimension?
UCT problem for nuclear C*-algebras
Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?
Zeeman collapsibility conjecture
Does every contractible two-dimensional polyhedron become collapsible after taking its product with one interval?
Bochner–Riesz conjecture
How much smoothing of a spherical frequency cutoff guarantees bounded Fourier reconstruction in higher dimensions?
Tuza’s triangle packing conjecture
Can all triangles in any finite simple graph be destroyed by removing at most twice its maximum number of edge-disjoint triangles?
Ryser’s hypergraph conjecture
Does every finite multipartite uniform hypergraph have a vertex cover at most r−1 times its largest matching?
List edge-coloring conjecture
Do separate allowed-color lists on edges always need no more colors than ordinary edge coloring?
Elliott–Halberstam conjecture
How far can averaged control of prime-counting errors in arithmetic progressions extend as the modulus grows?
Farrell–Jones conjecture
Can algebraic invariants of every group be assembled from its virtually cyclic subgroups?
Residual finiteness of hyperbolic groups
Can finite groups detect every nontrivial element of every hyperbolic group?
Cannon’s conjecture
Does a hyperbolic group whose boundary is a two-sphere act geometrically on hyperbolic three-space?