Unsolved Problems
Showing 1-11 of 11 problems
Category
Problem Set
Status
The Poincaré Conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....
Smooth 4-Dimensional Poincaré Conjecture
Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?...
Arithmetic Langlands, Topology, and Geometry
Explore homotopy theory's role in Langlands programs....
Borel Conjecture
Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...
Volume Conjecture
Do quantum invariants of knots relate asymptotically to hyperbolic volume?...
Novikov Conjecture
Are certain combinations of Pontryagin classes homotopy invariant?...
Baum-Connes Conjecture
Is the assembly map in K-theory an isomorphism for all locally compact groups?...
Borel Conjecture
Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...
Hilbert-Smith Conjecture
If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?...
Novikov Conjecture
Are certain polynomials in Pontryagin classes homotopy invariants?...
Volume Conjecture
Do quantum invariants of knots determine their hyperbolic volume?...