8 problems

Classification conjecture. The dissections obtained by gluing towers are exactly the dissections into triangles and quadrilaterals that admit a unitary frieze.

Let a positive integral frieze be a frieze of Dynkin type $E_8$ whose entries are positive integers. The $E_8$ frieze count conjecture. The number of positive integral friezes of t…

Let a positive integral frieze be a frieze of Dynkin type $E_7$ whose entries are positive integers. The $E_7$ frieze count conjecture. The number of positive integral friezes of t…

A frieze of Dynkin type is a mesh frieze on an associated Auslander–Reiten quiver. Frieze-count conjecture for type $E_6$. The number of friezes in type $E_6$ is $868$. More genera…

Let $\mathcal{C}$ be a cluster category of type $E_7$, and let a mesh frieze be a frieze on its Auslander–Reiten quiver. A frieze is non-unitary if it has no entry equal to $1$. No…

Let $\zeta$ be a root of unity, and let $\mathbb{Z}[\zeta]$ be the ring generated by $\zeta$ over $\mathbb{Z}$. A non-zero $\mathbb{Z}[\zeta]$ frieze of Dynkin type is a frieze who…

Unitary-frieze bound conjecture. The value of a frieze at any node of a Dynkin diagram is less than the maximal value of that node over the set of unitary friezes.

Frieze-count conjecture. The numbers of friezes of types $E_6$, $E_7$, $E_8$ and $F_4$ are $868$, $4400$, $26592$ and $112$, respectively.