12 problems

Let $u$ be the parameter of the quantum transfer matrix, let $r\geq 1$, and let $a\in\{1,2,\dots,r\}$ and $m\in\mathbb{Z}_{\geq 1}$. Denote by $\widetilde{T}^{(a)}_{m}(v)$ the fuse…

Colored-mutation root-count conjecture. The number of $\Gamma$-mutations in one period is the total number of roots in the root systems associated to the $\Gamma$-components,…

Mizuno's conjecture. The following identity holds:

Zamolodchikov–Ravanini–Tateo–Valleriani conjecture. The solution is periodic with

Let $ell greater than or equal to 2$, let $\gamma=\gamma(Xr,ell)$ be the mutation loop, and let $H{\gamma}$ be its associated group. For $\sigma eed in H{\gamma}$, let $\lambda eed in Q$…

Let $X_r$ be a finite type Dynkin diagram and let $\ell$ be a positive integer with $ell greater than or equal to 2$. Let $J(\eta)$ be the Jacobian matrix at the unique…

Linear recurrence conjecture. The iterates satisfy a constant-coefficient linear recurrence relation of order $4(N-1)(N-2)$. For odd $N$, it has the form

Tateo's dilogarithm identities.

The periodicity conjecture. All solutions to this system are periodic in $t$ of period dividing $2(h+h')$. This conjecture predicts a uniform periodicity phenomenon for algebraic…

Let $X_r$ and $X'_{r'}$ be simply laced Dynkin diagrams of finite type, with index sets $I$ and $I'$, ranks $r$ and $r'$, and Coxeter numbers $h$ and $h'$. Let $Y_{ii'}(u)$ be a po…

Let $X_r$ and $X'_{r'}$ be simply laced Dynkin diagrams of finite type, with index sets $I$ and $I'$, Coxeter numbers $h$ and $h'$, and ranks $r$ and $r'$. A family of positive rea…

Zamolodchikov periodicity conjecture. Every solution of this system is periodic in $t$, with period dividing $2(h+h')$. The periodicity conjecture is a central structural statement…