Mathematics Problem Archive

(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 $g^n\in H$....

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...

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

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?...

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....

(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 $z\mapsto z^2+c$, $c\in\mathbb{C}$....

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

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....

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{...

Give a (polynomial time) construction of the tight frame partition satisfying the properties required in the Kadison-Singer problem (or the related We...

A spanning tree of a graph G is called $\epsilon$-thin if it contains at most an $\epsilon$ fraction of the edges of each cut. Is there a function $f:...

Let $M$ be a matroid of rank n whose ground set S can be partitioned into n disjoint bases $B_1,\dots,B_n$. Is it true that $B_1,\dots,B_n$ always hav...

Erdős: Suppose I have a sequence of positive integers whose reciprocals sum to infinity. Must that sequence contain arbitrarily long arithmetic progre...

How quickly do the gaps between successive primes grow? Is it slower than n^(ľ) for every ľ > 0? See this....

Erdős: Is there a prime between n^(2)and (n+1)^(2 )for every n > 0?...

Is the least quadratic residue modulo p at most p^(ľ)^( )for any ľ > 0?...

For positive integers $N,q,a$, let $Q(N;q,a)$ be the number of perfect squares among $a,a+q,\ldots,a+(N-1)q$, and let $Q(N)=\max_{q,a\geq1}Q(N;q,a)$. ...

For $\phi(n)=\sup_{\deg p=n}\int_{|z|<1}|p'|/(1+|p|^2)\,dm$, let $\alpha=\limsup\log\phi(n)/\log n$. For a regular compact set $E$, define $\beta_E$ f...

Does every three-dimensional convex polytope have a non-self-intersecting edge unfolding? Does a minimum spanning tree of the dual edge graph, with a ...

On a centrally symmetric convex body, must every pair of points at maximum intrinsic surface distance be antipodal? Resolve this even for rectangular ...

Determine the minimum number of $d$-simplices needed to triangulate the $d$-dimensional cube, and its asymptotic growth with $d$....

Does the hyperbolic plane admit a smooth isometric immersion into $\mathbb R^4$? More generally, determine the least Euclidean dimension for such an i...

Are the Hermite constants associated with densest lattice sphere packings always rational? Determine their arithmetic nature in dimensions where the e...

What is the largest possible ratio between the sum of edge lengths of a tetrahedron that can pass through or fit inside another tetrahedron and the su...

Does there exist a nondegenerate triangle whose side lengths, three medians, three altitudes, and area are all rational?...

If two Birkhoff billiard maps $T_1$ and $T_2$ satisfy $T_1=ST_2S^{-1}$ for a homeomorphism $S$, must their tables be similar? Relatedly, can one hear ...

(a) Is the set of $n$-periodic orbits of a smooth strictly convex Birkhoff billiard nowhere dense for every $n$? (b) Does every polygonal Birkhoff bil...

In the space of area-preserving $C^1$ diffeomorphisms of a compact manifold, is it generic that the dynamics is either hyperbolic or has zero Lyapunov...

For the infinite-dimensional Calogero–Moser system, in which particles on the real line interact through the inverse-square potential, does the dynami...

Let $p$ be a prime for which $$f(u_1,u_2)=1+u_1u_2+u_1^2u_2+u_1^3u_2+u_1^4+u_2^2+u_1^4u_2^2$$ is irreducible, and consider the algebraic $\mathbb{Z}^2...

Choose a random subset $Q$ of the primes by independently retaining each prime with probability $1/2$. Is it almost surely true that $$\limsup_{n\to\i...

Given $\varepsilon>0$, does there exist a polynomial $f(x)=\prod_{i=1}^d(x-\alpha_i)\in\mathbb{Z}[x]$ whose logarithmic Mahler measure $$m(f)=\sum_{i:...

Let $\log_p:\mathbb{C}_p^*\to\mathbb{C}_p$ be the branch of the $p$-adic logarithm with $\log_p(p)=0$, and let $T_\lambda:x\mapsto\lambda x$ on $\math...

Let $M$ be a compact Riemannian manifold of constant negative curvature and $(g_t)_{t\in\mathbb{R}}$ its geodesic flow. Does there exist a probability...

For sofic shifts $S,T$ with $h(S)\ge h(T)$, give necessary and sufficient conditions for a factor map from $S$ onto $T$. The most fundamental unequal-...

Characterize the functions that occur as beta functions of mixing Markov shifts....

For a $\mathbb Z^2$ shift of finite type $\alpha$, characterize the possible sets $E_1(\alpha)$ of expansive directions, especially under the assumpti...

For multiplicatively independent integers $p,q>1$, can there exist a nonatomic Borel probability measure on the circle, other than Haar measure, that ...

For the symbolic system $X$ constructed in Section 14 of the source from a commuting pair of full-shift endomorphisms, is Haar measure its only invari...