Mathematics Problem Archive

(Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?...

(Eilenberg-Ganea) Is there a group G of cohomological dimension 2 and geometric dimension 3?...

Does every Artin group have a finite $K(G,1)$?...

(Andrews-Curtis) If K and L are simple homotopy equivalent finite 2-complexes, can one transform K to L by a sequence of elementary collapses and expa...

(Belegredek) Is there a 3-complex X (not necessarily aspherical) which is not homotopy equivalent to a 2-complex but H³(X; {G}) = 0 for all local coef...

Study the quasi-isometry group QI(Rn). How big is it?...

(Bridson) Is G×Z quasi-isometric to G for G =Thompson’s group? Any f.g. group?...

(Lubotzky) Does Out(Fn) have the congruence subgroup property?...

(Yves de Cornulier) Let G be residually torsion-free nilpotent. Is G Haagerup (= a-(T)-menable)?...

(Henry Glover) Does for every finite graph G the following 1−2−∞- conjecture hold: G is planar, a double cover of G is planar, or no finite cover is p...

Let $L$ be a flag triangulation of $S^{2k-1}$, let $f_i$ be the number of $i$-simplices of $L$, and define $$\chi=1-\sum_{i=0}^{2k-1}(-1)^i\frac{f_i}{...

Do there exist groups G with balanced presentation (same number of generators and relations), with H1(G) = 0 and with unsolvable word problem?...

Is there a sequence of (perfect, of course) groups with balanced presentations among which one cannot recognize trivial groups?...

Let $G$ be a closed transitive subgroup of the automorphism group of a rooted tree. Must every level stabilizer of $G$ contain an element acting witho...

Prove the conjectured local-to-global principle for linear ordinary differential equations known as the Grothendieck–Katz $p$-curvature conjecture....

Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?...

Study geometric properties of Markov spectrum....

Determine when the minimizer $y^*$ in Theorem 2.1 of the source is smooth....

With the terminology of Problem 2.20, denote by $\mathcal{F}(f)$ the set of points where the sequence $\{f_n(z)\}$ is not normal. Fatou asks if there ...

Let $B$ be the Bergman space of square integrable functions in $\mathbb{D}$, that is, those functions $f(z) = \sum^\infty_{n=0} a_nz^n$ for which $\su...

Let $\Omega$ be a simply-connected domain in $\mathbb{C}$ with at least two boundary points, and let the function $\phi$ map $\Omega$ analytically and...

For $-\infty<p<+\infty$ let \[B(p):=\sup\{\beta_f(p):f\text{ conformal map of }\mathbb{D}\text{ into }\mathbb{D}\}\] where \[\beta_f(p)=\limsup_{r\to1...

In their famous Acta paper , Hardy and Littlewood introduced the celebrated Hardy-Littlewood maximal function in connection with complex function theo...

For an iteration without memory that uses $n$ evaluations of a function or its derivatives per step, is its convergence order always at most $2^{n-1}$...

Given a complex polynomial $f$ of degree $d\geq2$ and $z\in\mathbb{C}$, must there be a critical point $c$ of $f$ such that $|f(z)-f(c)|\leq |f'(z)|\,...

Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? ...

How many mutually unbiased bases are there in 6 dimensions?...

What is the value of the (real) Grothendieck constant?...

3n+1 ("Collatz" or "Ulam") problem: Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three a...

Are the positive integer powers of 3/2 mod 1 uniformly distributed in the unit interval? One would think so, but apparently this is a hard question. S...

Let $M(n)$ be the number of monotone Boolean functions of $n$ variables, equivalently the number of antichains of subsets of an $n$-element set. Deter...

Let $f$ be entire and suppose every zero of every derivative $f^{(n)}$, $n\ge0$, lies in the closed lower half-plane. Must $f$ lie in the compact-open...

For a regular connected compact plane set $E$, let $\beta_E=\limsup_{\varepsilon\to0}\log l(\varepsilon)/(-\log\varepsilon)$, where $l(\varepsilon)$ i...

For real matrices $A\in\operatorname{Mat}_{n\times n}$, $B\in\operatorname{Mat}_{n\times p}$, and $C\in\operatorname{Mat}_{m\times n}$ with $n=mp$, de...

Determine the least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors....

Given a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to ...

Can the $3\times6$ rectangle with a $2\times2$ corner removed tile a rectangle using an odd number of congruent copies?...

Can sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a no...

Find the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine...

Are there quasiperiodic global minimals for metrics on the torus that are close to a flat three-dimensional torus? Here a geodesic is a global minimal...

Characterize mixing shifts of finite type up to topological orbit equivalence....

If a nonnegative integer matrix $A$ has a unique, simple, nonzero eigenvalue $n$, prove that $A$ is strong shift equivalent over $\mathbb Z_+$ to the ...

Classify shifts of finite type up to topological conjugacy; in particular, give a decision procedure determining whether two nonnegative integer matri...

Given a mixing shift of finite type $S_A$, determine the range of the dimension representation $\operatorname{Aut}(S_A)\to\operatorname{Aut}(G_A)$....

Let $A,B$ be irreducible integer matrices of the same spectral radius. Suppose $\operatorname{tr}(A^n)>0$ implies $\operatorname{tr}(B^n)>0$ for every...

Prove that two mixing Markov shifts admit a magic-word isomorphism exactly when they have the same beta function, the same ratio group $\Delta$, and t...

Suppose a $\mathbb Z^2$ action $\alpha$ has $\alpha^{\boldsymbol n}$ an SFT for some $\boldsymbol n$. Can $\alpha$ have infinitely many expansive comp...

If $S$ and $T$ are mixing shifts of finite type, prove that $S^i$ and $T^j$ can commute for all sufficiently large integers $i,j$....

Determine all shift-invariant Borel probability measures and all subsystems of the symbolic system $X$ constructed in Section 14 of the source from th...

For $d>1$, if a continuous factor map sends a $\mathbb Z^d$ SFT $X$ onto a sofic shift $Y$, must $X$ contain a sofic subshift $W$ with $h(W)=h(Y)$ and...