Mathematics Problem Archive

(Thurston) Is every closed hyperbolic 3-manifold finitely covered by one that fibers over the circle?...

If $A$ and $B$ are countably infinite groups, does $A\times B$ have fixed price $1$?...

The Bieberbach conjecture Is it true that $|a_n|\leq n$ for $f$ in $S$ with equality only for $f(z)\equiv f_\theta(z)$? The result is known to be true...

Given a polygonal room with perfectly reflecting sides and a point light source, characterize when every point of the room is illuminated. In particul...

Does there exist a family of bounded-degree expander graphs, with spectral gap bounded away from zero and diameter tending to infinity, whose coarse R...

Let $S\subset\mathbb R$ be a unital subring and let $A$ be a square matrix over $S$ whose nonzero spectrum satisfies the spectral-conjecture condition...

If a quadratic polynomial $f_c$ is infinitely renormalizable of bounded type, must $J(f_c)$ be locally connected? In particular, is the Julia set of t...

Can a nowhere-dense Julia set of a rational map have positive Lebesgue measure?...

Consider minimal immersions $S^n\to S^N(1)$ with total area at most a fixed constant $A$, identifying immersions which differ by a motion of the ambie...

Let $B$ be a set of $n+2$ hyperplanes in general position in complex projective space $\mathbb{P}^n(\mathbb{C})$. Prove that there is no nondegenerate...

Let $\Delta$ be a ball in $\mathbb{C}^n$, $n\geq2$, and let $F$ be a complete Hermitian manifold. Does every holomorphic map from $\Delta\setminus\{0\...

Are there any complete surfaces of negative curvature in Euclidean 3-space whose principal curvatures are bounded away from zero?...

Does every unimodular random countable locally finite rooted graph arise as a local weak limit of finite graphs?...

Let $S$ and $S'$ be random walks on $\mathbb{R}$ with the same i.i.d. step-length distribution, starting at $0$ and $x$. Suppose the step lengths are ...

f Γ is an irreducible arithmetic group of rank ≥ 2, then every homomorphism Γ→ ModS has finite image. For many arithmetic groups Γ the conjecture can ...

[Ehrenpreis Conjecture] Given two closed Riemann surfaces, there are finite un- branched covers with homeomorphic total spaces which are arbitrarily c...

(Orbit closures for moduli spaces). Determine the closures of the orbits for the GL+(2, R)-action on H(α) andQ(β). Are these closures always complex-a...

Let M = H3/Γ be a finite volume hyperbolic 3-manifold, does Γ contain a surface subgroup. This was answered in [ 8] for non-compact but finite volume ...

Forn> 3, does Aut(Fn) have property (T)? The corresponding question for mapping class groups is also open. If Aut( Fn) were to have Property (T), then...

Classify all finite loop spaces. This is the long term project of Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the Lie groups are...

Does the possible Kervaire-invariant-one element $\theta_6\in\pi_{126}^{S}$ exist; equivalently, is $h_6^2$ a permanent cycle in the mod-2 Adams spect...

We now know that Morava E-theory admits an action of the stabillizer group S. This is the famous Hopkins-Miller result, which one day I hope will see ...

Improve on Benson-Carlson-Rickard. Recall their theorem: if G is a finite p-group and k is an algebraically closed field, then thick subcategories in ...

Classify the localizing subcategories of the stable k[G]-module category. These should be in 1-1 correspondence with arbitrary subsets of Proj H^*(G,k...

Currently we know how to do equivariant stable homotopy theory only when the structure group G is compact Lie. But I bet we can do it when the group G...