Algebraic geometry
Polynomial equations define spaces whose shapes, singularities and algebraic cycles pose different open problems.
8 problemsTopology
Spaces and the properties preserved by continuous change.
Collections
Browse collectionsGroups and operators
Structure in groups, representations and infinite-dimensional spaces.
7 problemsKnots and manifolds
Smooth structures, curved spaces and the manifolds produced by knot surgery.
8 problemsA few problems to explore
15 totalSmooth 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?
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?
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?
UCT problem for nuclear C*-algebras
Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?