16 problems

Let a finite type simply laced Dynkin diagram be given, and consider its associated system of algebraic relations, called a Y-system. Zamolodchikov's periodicity conjecture. Soluti…

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

Refinement conjecture. For every $s\in H$, there exists $C$ such that $q^{C}f_{A_{\pm}(z)}^{(s)}(q)$ is modular.

Finite-type Y-system conjecture. The matrix $K$ is symmetric and positive definite, and for some $C$ the Nahm sum $f_{K,0,C}(q)$ is modular.

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…

Let $Q$ be a bipartite recurrent quiver admitting a weakly subadditive labeling, and let its associated $Y$-system be the corresponding dynamical system. Arnold–Liouville integrabi…

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.

Dilogarithm conjecture. For this real positive solution, the dilogarithm identity stated immediately after the candidate holds.

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…