18 problems

Let $V$ be a cluster variety and let $V^\vee$ be its mirror cluster variety. Write $V^\vee(\mathbb{Z}^T)$ for the tropical points of $V^\vee$. Fock–Goncharov's cluster duality conj…

Let $B$ be an exchange matrix. The scattering fan–mutation fan coincidence conjecture. The scattering fan $\operatorname{ScatFan}^T\!(B)$ coincides with the mutation fan…

Let $Q$ be the quiver with two vertices $0,1$, no loops, and $k\in\mathbb{Z}_{\geq 1}$ arrows from $0$ to $1$. For $n\in\mathbb{Z}_{\geq 1}$, define the signed quantum numbers … Us…

Let $b,c>0$ with $bc>4$, and let $r$ lie in the Badlands interval … For positive integers $i,j$, let $\tau(i,j)$ be the corresponding wall-function coefficient. Asymptotic growth c…

Elgin–Reading–Stella's conjectures. For $i,j>0$, the coefficient $\tau(i,j)$ satisfies all of the following: it is a polynomial in $b$, $c$, and $g$; this polynomial has $g$ as a f…

Let $\mathcal{F}$ be the fundamental domain and let $\mathcal{F}+\mathbb{Z}$ denote its integer translates. Consider the scattering of rays associated to orbifold quivers, includin…

Let $Y$ be a Fano manifold with smooth anticanonical divisor $D$. For a suitable submonoid $P\subset \operatorname{Pic}(Y)$, let $(X,W)$ be the intrinsic mirror dual, where … is a…

Multiplicative-basis conjecture. The selected subset of $x$-variables acts as a complete multiplicative basis for the desired $\hat{y}$-variables.

Degenerate scattering diagram conjecture. Tropicalizing only a subset of the functions in $\mathcal{O}(\mathcal{X}^{\vee})$ can produce a degenerate scattering diagram in which cer…

Two-limiting-ray conjecture. The asymptotic chambers of only two limiting rays are relevant for the 8-point MHV amplitude.

Let $(\mathbf{s,w})$ be a seed with non-degenerate potential. Let ${\mathfrak D}_{\mathbf{s,w}}$ and ${\mathfrak D}_{\mathbf{s}}$ be the stability and cluster scattering diagrams,…

Let $(\mathbf{s,w})$ be a seed with non-degenerate potential. The Donaldson–Thomas scattering diagram ${\mathfrak D}^\mathrm{DT}_{\mathbf{s,w}}$ and the initial-data scattering dia…

Let $\\f{D}_{\operatorname{in}}$ be an initial scattering diagram of the form specified in the source, with incoming wall functions determined by Laurent polynomials $p_i(t)$. A sc…

Let $M_\gamma$ be a moduli space of Gieseker semistable sheaves, let $\mathcal T_\gamma$ be the set of oriented weighted trees in the scattering construction, and let…

Let $\mathcal{A}_{\mathbb{T}_{1}^{2}}$ be the cluster variety associated with the once-punctured torus, let $\mathcal{A}_{\mathbb{T}_{1}^{2}}^{\vee}$ denote its Langlands dual, and…

Let $B$ be an exchange matrix and let $k\in\{1,\ldots,n\}$. Write $\mu_k(B)$ for mutation of $B$ in direction $k$, and let $\eta_k^{B^T}:V^*\to V^*$ be the associated piecewise-lin…

Let $B$ be the base of the log Calabi–Yau surface, and let $\beta\in\operatorname{NE}(Y)$. Let $\gamma$ be a broken line in $B$ for the canonical scattering diagram, and choose a n…

Let $\varphi$ be a tropical function on the tropicalization of a cluster variety, or of a fiber of a cluster variety, let $\s{Y}$ denote that variety or fiber, and let $\f{w}$ be a…