8 problems

Let $j$ denote the involution of the real line induced by Dyer's outer automorphism of the extended modular group $\operatorname{PGL}(2,\mathbb{Z})$. The map $j$ preserves the set…

Let $[x]_q$ denote the $q$-deformation of a real number $x>0$, viewed as a power series in the complex variable $q$, and let $R(x)$ be its radius of convergence. For the golden rat…

Let $c_1,c_2,\ldots,c_k$ be integers, and let $M_q(c_1,c_2,\ldots,c_k)$ be the corresponding matrix over $\mathbb{Z}[q,q^{-1}]$. Its trace is a Laurent polynomial in…

Let $N$ be an integer greater than $2$, and let the size of an irreducible solution of $(E_N)$ mean the size notion defined earlier in the paper. Uniform size-bound conjecture. The…

Let $N\in\mathbb{N}^{*}$ with $N\geq 2$. A solution of $(E_N)$ is called irreducible when it cannot be decomposed as a sum of two nontrivial solutions, as in the preceding discussi…

Let $\Gamma={\rm PSL}_2(\mathbb Z)$ be the modular group, and let $M_h$ be a subgroup obtained from Neumann's construction. A subgroup of $\Gamma$ is maximal nonparabolic if it is…

Let $\Gamma={\rm PSL}_2(\mathbb Z)$ be the modular group, and let $M_h$ denote a subgroup arising from Neumann's construction. A maximal nonparabolic subgroup is a maximal subgroup…

Canonical 3d-dissection conjecture. Every element of $\mathrm{PSL}(2,\mathbb{Z})$ corresponds to a unique $3d$-dissection.