Minimum opaque barrier for a square
How little total barrier length can intercept every straight line that passes through a unit square?
Algebra and geometry
Within Algebra and geometry · Search all subjects
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?
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?
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?
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?
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?
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?
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?