Arithmetic geometry
Arithmetic questions studied through geometric objects.
10 problemsAlgebra and geometry
Shapes, spaces, and algebraic structures.
Explore Algebra and geometry
Topology
Spaces and the properties preserved by continuous change.
15 problemsDiscrete geometry
Arrangements, distances, coverings, and extremal geometric configurations.
11 problemsConvex geometry
Convex bodies, their boundaries, volumes, and dual shapes.
4 problemsAlgebras and modules
Algebraic actions, extensions and projective resolutions.
6 problemsAlgebraic geometry
Shapes defined by polynomial equations, their singularities and algebraic cycles.
8 problemsCollections
Browse collectionsDiophantine equations
Equations over integers and rational numbers, from elliptic curves to the limits of decision algorithms.
7 problemsGeometric extremes
Packing, covering, distances and other limits set by geometry.
11 problemsFields and rational points
Field symmetries and rational points on curves and higher-dimensional varieties.
7 problemsA few problems to explore
64 totalBirch 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.
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?