Unsolved Problems
Showing 1-17 of 17 problems
Category
Problem Set
Status
Smooth 4-Dimensional Poincaré Conjecture
Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?...
The Volume Conjecture
For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement....
Physical Consequences of Perelman's Proof
Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....
Arithmetic Langlands, Topology, and Geometry
Explore homotopy theory's role in Langlands programs....
Unknotting Problem
Can unknots be recognized in polynomial time?...
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?...
Berge Conjecture
Are Berge knots the only knots in S³ admitting lens space surgeries?...
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?...
Unknotting Problem
Can unknots be recognized in polynomial time?...
Volume Conjecture
Do quantum invariants of knots determine their hyperbolic volume?...
Whitehead Conjecture
Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...
Zeeman Conjecture
Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...