160 problems

Let $F:\{0,1\}^{\mathbb{Z}^2}\to\{0,1\}^{\mathbb{Z}^2}$ be the global map of Conway's Game of Life, and let…

Let $A$ and $B$ be finite essential adjacency matrices, and let $L_k(A)$ and $L_k(B)$ denote the corresponding Leavitt path algebras over a field $k$. Hazrat's graded classificatio…

Let $f$ be a finite sum of quadratic monomial rotation-symmetric Boolean functions, and let $W_f(n)$ denote the Hamming weight of its induced Boolean function on $n$ variables. Let…

Given two square nonnegative integer matrices $A\in\mathbb{N}^{m\times m}$ and $B\in\mathbb{N}^{n\times n}$, determine whether they are strong shift equivalent: that is, whether th…

The decoration conjecture. If $q<q'$ and $w\succcurlyeq w'$, then $P_q^w$ forces $P_{q'}^{w'}$. Moreover, $P_q^w$ and $P_{q'}^{w'}$ have the same braid type if and only if $q=q'$ a…

Let $G$ be a group and $A$ a finite alphabet. A cellular automaton is a $G$-equivariant continuous map $T\colon A^G\to A^G$. Gottschalk's conjecture. If $T$ is injective, then it i…

Let $\sigma$ be a Pisot irreducible substitution, meaning that the characteristic polynomial of its incidence matrix is the minimal polynomial of a Pisot number. Let…

Let $G$ be a finitely generated group. The domino problem asks whether a given finite set of Wang tiles admits a tiling of $G$. Ballier–Stein conjecture. $G$ has undecidable domino…

Entropy-minimal subshift conjecture. There exists an entropy minimal subshift representing $A$ if and only if, for every arithmetic progression $P$,

Let $\sigma$ be an irreducible Pisot substitution, and let the associated substitution dynamical system and its strong coincidence condition be understood. Strong coincidence conje…

Periodic orbit-closure conjecture. The orbit closure

Nivat's conjecture. Then $c$ is $1$-periodic.

Let $G$ be a finitely generated group. A strongly aperiodic subshift of finite type (SFT) is an SFT with no periodic configurations. The strongly aperiodic SFT conjecture. $G$ admi…

Let $\beta$ be a Pisot unit, meaning an algebraic integer greater than $1$ whose other algebraic conjugates have modulus less than $1$ and whose norm is a unit. Call $\beta$ weakly…

Let $\mathcal A$ be a finite alphabet with at least two elements, and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a two-dimensional configuration. For $k,n\in\mathbb N$, let…

Morita–singularity classification conjecture. The following conditions are equivalent: 1. There is an isomorphism of partially ordered $\mathbb{Z}[x,x^{-1}]$-modules

Let $G$ be a group and let $A$ be a finite set. Let $\tau\colon A^G\to A^G$ be a cellular automaton. It is post-surjective if, for every $x,y\in A^G$ such that $y$ is asymptotic to…

Let $\gamma$ be a periodic trajectory in a cyclic quadrilateral tiling, and let $\Omega_{\gamma}$ be the domain bounded by it. Tree conjecture for cyclic quadrilateral tiling billi…

Let $A$ be a square matrix over a subring $\mathcal R$ of $\mathbb R$. Its nonzero spectrum is the multiset of its nonzero eigenvalues. Boyle–Handelman's strong generalized spectra…

Let $(X,T)$ be the full two-shift, let $\alpha$ be the partition used to define information, and consider the space of one-step Markov measures on $X$. A measure is fully supported…

Non-factorability conjecture. On a large class of groups, not every SI SFT is an SFT factor of a contractible SFT.

Second minimal odd-orbit digraph conjecture. For every integer $k\geq 3$, the digraph of any second minimal $(2k+1)$-orbit has one of $4k-3$ possible types up to inverses.

The linear repetitivity conjecture. Every aperiodic linearly repetitive Delone set is densely repetitive.

Let $\mathcal{T}_{\chi}$ and $\mathcal{T}_{\psi}$ be the tiling spaces associated with substitutions $\chi$ and $\psi$, and let $\tilde{\chi}$ and $\tilde{\psi}$ denote the associa…

Let $\mathcal{M}$ be the class of maps under consideration, let $\operatorname{invlim} f$ denote the inverse limit of $f$, let $\chi_f$ be the substitution associated with $f$, and…