9 problems

Bellon–Viallet conjecture. The entropy $epsilon$ is the logarithm of an algebraic integer for every birational self-map of a projective space.

Let $T_{r,s}$ be the trapezoidal poset with parameters $r$ and $s$, and let birational rowmotion act on its labelings. Grinberg–Roby's periodicity conjecture. Birational rowmotion…

Let $P_J$ denote the Painlevé equations indexed by $J\in\{V_{deg},IV,III(D_6),III(D_7),III(D_8),II(JM),II(FN),I\}$, and let $\chi_J$ be their character varieties. Suppose that each…

Let $F:\mathbb{C}^4\rightarrow\mathbb{C}^4$ be the birational map … let $\phi(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$, and for generic $c\in\mathbb{C}$ let $X_c\subset\mathbb{P}^4$ be the…

Birational OY-invariant conjecture. The function $\EuScript{Y}^{\mathrm{B}}\colon\mathcal{A}^{\mathrm{B}}(\mathsf{A}^n)\to\mathbb{R}_{>0}$ is invariant under both…

Birational type B homomesy conjecture. The statistics $h^{\mathrm{B}}_i\colon \mathcal{A}^{\mathrm{B}}_{\kappa}(\mathsf{B}^n)\to \mathbb{R}_{>0}$, for $1\leq i\leq 2n-1$, and…

Let $P=[a]\times[b]$, and let $n$ denote the order parameter for the product of the two chains. Write $\pi_{\mathcal{B}}$ for the birational rowmotion operation on $P$. The order c…

Auriac–Maillard–Viallet conjecture. For every $q\geq 3$,

Rigidity conjecture. Then the family $(\varphi_\alpha\Phi_n)_{\alpha\in U}$ is holomorphically trivial.