7 problems

Musiker–Schiffler–Williams' punctured-surface basis conjecture. A similar basis result should hold for cluster algebras associated with punctured surfaces.

Let $\Sigma$ be a sphere with $n\geq 3$ punctures, and let $\rho\colon\pi_1\Sigma\to\operatorname{SL}_2\operatorname{\mathbb{C}}$ be a representation. Write $[\rho]$ for its conjug…

Let $T_1,\dots,T_s$ be tensor diagrams such that each $[T_i]$ is a cluster variable of $\mathscr{A}(\mathcal{M})$. Let $varphi_i$ be a tagging of $T_i$, with invariant…

Let $P$ be the weight-vector set used for the cluster algebra $\mathscr{A}(\mathcal{M})$, let $\mathbb{M}_\circ$ be the set of punctures, and let $pi_p$ denote projection to the we…

Let $\Sigma$ be a sphere with at most three punctures, equipped with a complete Riemannian metric of finite area, and let ${\rm scg}(\Sigma)$ be the infimum of the lengths of its c…

Let a BPS graph have branch points $\{b\}$ and faces $\{F\}$. Define the proposed quiver superpotential by … where $C_b$ and $C_F$ are the cycles in the quiver path algebra obtaine…

Arrangement (a) conjecture. The Hopf differential $\Phi^h$ of Corollary $$ has divisor data described by arrangement (a).