50 problems

Extension conjecture. Theorem 1 holds in the same form for $\sigma>r$; in particular, for any fixed $\sigma\geq 1$ and $r\geq 1$ with $\sigma>r$, $X_{\sigma,n}/n^{\sigma/2}$ conver…

Let $\mathcal{T}$ be the phase space of the ultra-discrete $(g+1)$-periodic Toda lattice, let $\mathcal{C}$ be the moduli space of compact tropical curves $\Gamma_C$, and let … be…

Let $S$ be a finite set, let $d>1$, and let $X=S^{\mathbb{Z}^d}$. A $d$-dimensional cellular automaton is a continuous map $T:X\to X$ commuting with the shift action of…

Low-growth existence conjecture. There exists $N>1$ and a surjective cellular automaton $f$ on $N$ symbols such that $\nu(f,S_N)<N$. The surrounding discussion poses related quanti…

Let $\Sigma_N$ be the space of doubly infinite sequences over an alphabet of $N$ symbols, and let $f:\Sigma_N\to\Sigma_N$ be a surjective one-dimensional cellular automaton. Let…

Let $m=(m^{(a)}_j)$ be a content such that … where … Let $\Omega(m)$ be the Bethe-ansatz multiplicity, let $W$ be the Weyl group of $A_n$, and let $\lambda(m)$ be the associated do…

Let $p\in P(m)$ be a state with soliton content $m$, and let $(T^{(r)}_l)^t(p)\neq0$ for every $t\geq0$. Denote by ${\mathcal P}^{(r)}_l$ the least common multiple defined earlier…

Even-index period formula conjecture. The period function satisfies

Odd-index period formula conjecture. The period function satisfies

Twin-period recurrence conjecture. Assume that $p$ and $p+3$ are twin periods. Then the next twin pair is $p'$ and $p'+3$, where

Surjectivity characterization conjecture. $F$ is surjective if and only if either

Let $F$ be a surjective binary cellular automaton with radius $r$, and let $N^1_1(F,r,k)$ denote the corresponding symbol-frequency quantity; write $\mathrm{id}$ for the identity c…

Unique-active-transition idempotency conjecture. The automaton $\tau$ is not idempotent if and only if there exists $t\in S$ such that

Let $F_2$ be the free group on generators $a$ and $b$, and let $S\subset F_2$ be a shape. Say that $S$ has finite filling when the solitaire filling process generated by $S$ can fi…

Non-commutativity and linguization of Oshida-Rule. It is not possible to prove from the supplied formulation that this identity holds using matrices $A$; equivalently, it is not po…

Let $m$ be a positive integer, let $e$ denote the distinguished initial vector, and let $h$, $h'$, $f_1$, and $H_{245}=L\circ S$ be the maps defined above, with…

Let $G$ be a finitely generated group and let $A$ be a finite alphabet. Let $M\subset G$ be finite, let $S=A^{A^M}$, and let $s\in S^G$. Suppose that the NUCA map…

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 $G$ be a group, and let $\mathbb{Z}*G$ denote its free product with the infinite cyclic group. A group is dual surjunctive if every post-surjective cellular automaton over it i…

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…

Asymptotic label conjecture. There exists an index $j\notin I$ such that

A doubly periodic configuration in the Game of Life is a configuration with two independent spatial periods, and a Garden of Eden is a configuration with no preimage. Undecidabilit…

For the Game of Life, consider finite-support Gardens of Eden and rectangular orphans, with the encodings specified in the paper and polynomial-time many-one reductions. Co-NP-comp…

Let $\Sigma$ be a homogeneous $B_3$-space covering a finite homogeneous $PSL(2,\mathbb{Z})$-space $\overline{\Sigma}$. Suppose that the cycles of $\overline{\Sigma}$ are simply int…

Let … be the spatial period parameter and $r\geq 1$ the neighborhood radius, and consider the limit as $n\to\infty$. The asymptotic cycle-length conjecture. If … n^{-}Cnto 1 … is e…