Mathematics Problem Archive

Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a de...

Conjecture For every rational $\epsilon > 0$ and every rational $\Delta$, there is no polynomial-time algorithm for the following problem. Given is a...

Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?...

Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smooth...

Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalen...

Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?...

Problem Let $\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal? 1. The funcoid corresponding t...

Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum ...

Graham: For every non-equilateral triangle T, show that it is possible to color the plane with three colors so that there is no monochromatic (congrue...

Straus: Is every polygonal room in the plane illuminable from some point? See this....

/Clark : What is the (homogeneous adjacency) spectrum of the Fano plane?...

Do there exist seven planar points in general position—no three collinear and no four concyclic—such that every pairwise distance is an integer?...