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?...

(Canary) Let $G$ be word-hyperbolic and $H$ a finitely presented subgroup of $G$. Suppose that for every $g\in G$ there is $n>0$ such that $g^n\in H$....

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

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

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

Compute the asymptotic dimension of CAT(0) groups, Out(Fn), mapping class groups, nonuniform lattices, Thompson’s group. Is there a group of finite ty...

(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) Classify the mapping tori of automorphisms $\mathbb{Z}^n\to\mathbb{Z}^n$ up to quasi-isometry....

(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...

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

Does every countable group have fixed price; that is, do all its free probability-measure-preserving actions have the same cost?...

Given a scheme of generic fat points $X \subset \mathbb{P}^{n_1-1}\times\ldots\times \mathbb{P}^{n_t-1}$, what is the multi-graded Hilbert function $\...

For a non-hyperelliptic algebraic curve, is its Clifford index determined by the extent to which its canonical embedding has linear syzygies?...

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....

Can the Lavrentiev phenomenon occur for elastostatics under growth conditions ensuring that all finite-energy deformations are continuous?...

Prove global existence and uniqueness for suitable initial-boundary-value problems in dynamic nonlinear elasticity....

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 ...

(Fatou's conjecture) Show that the subset $U$ of functions $g$ in $R_d$, such that all the critical points of $g$ are in the basins of attraction of p...

Let $B$ denote the boundary of the Mandelbrot set (or, equivalently, the topological bifurcation set of the family $z\mapsto z^2+c$, $c\in\mathbb{C}$....

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...

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...

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 nonconvex subsets of $\mathbb{R}$ and $\mathbb{R}^2$, is a set spectral if and only if it tiles by translations?...

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)|\,...

Characterize the domains for which there exists a nonzero function whose integral vanishes over every congruent copy of the domain....

Establish global regularity, or exhibit breakdown, for solutions of the Vlasov–Maxwell equations from appropriate smooth initial data....

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

Develop a global-in-time theory of the Cauchy problem for the one-dimensional Euler–Fourier system for initial data constrained only by finite energy ...

Develop a compensated-compactness calculus for symmetric matrices when compensated compactness yields only inequalities. As a first application, prove...

For compressible Navier–Stokes equations with constant viscosities near vacuum, does the unphysical one-dimensional consequence identified by Hoff and...

Does the compressible Euler system in odd spatial dimension admit a nontrivial smooth eternal solution having finite nonzero mass and energy?...

Prove existence of a regular reflection for compressible flow against a wedge without assuming that the flow is irrotational....

Let ${G=(V,E,W)}$ be a graph and ${k}$ a positive integer, is the following true? $\rho_G(k) \leq \mathrm{polylog}(k) \sqrt{\lambda_k}. \ \ \ \ \ (2)$...

Construct deterministic matrices $A\in\mathbb{C}^{M\times N}$ (or $A\in\mathbb{R}^{M\times N}$ ) satisfying the $(s,\frac13)$ -RIP for $s\approx\frac{...

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