Birch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Algebra and geometry
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Hodge conjecture
Some topological features of a complex algebraic shape look as though they should come from smaller algebraic shapes inside it. Must they?
Hadwiger–Nelson problem
How many colors are needed to color every point of the plane so that points exactly one unit apart always have different colors?
Square peg problem
Must every simple closed curve in the plane contain four points that are the corners of a square?
Smooth four-dimensional Poincaré conjecture
A four-dimensional space can have the topology of a sphere. Must every smooth version be smoothly equivalent to the standard sphere?
Andrews–Curtis conjecture
A set of generators and relations may describe the trivial group in a complicated way. Can a small set of permitted moves always simplify it?
Schanuel’s conjecture
The conjecture predicts how many algebraically independent numbers must appear when independent complex numbers are paired with their exponentials.
Bombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Nonnegative inverse eigenvalue problem
Which lists of numbers can be the eigenvalues of a square matrix whose entries are all nonnegative?
Falconer distance conjecture
A sufficiently large fractal set is predicted to determine a set of pairwise distances with positive ordinary length.
Sphere packing in five dimensions
How densely can identical nonoverlapping spheres fill five-dimensional Euclidean space?
Erdős–Szekeres convex polygon conjecture
How many points in the plane guarantee a convex polygon with a specified number of vertices? The proposed exact threshold remains unproved.
Artin’s primitive root conjecture
For a fixed eligible integer, how often do its powers generate every nonzero residue modulo a prime?
Leopoldt’s conjecture
Do the independent units of a number field remain independent when measured using p-adic arithmetic?
Kakeya conjecture
A set can contain a unit line segment pointing in every direction and still have zero volume. Must it nevertheless have full fractal dimension?
VP versus VNP
Can every polynomial described as a sum of efficiently computable terms itself be computed by a small arithmetic circuit?
Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Symmetric Mahler conjecture
Do cubes and their dual shapes minimize a symmetric body’s volume multiplied by its polar’s volume?
Kissing number in five dimensions
How many equal five-dimensional balls can touch one central ball without overlapping each other?
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?
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?
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?