19 problems

Let $Q$ be a quiver of rank at least $4$, and let $[A_2]$ denote the finite-type $A_2$ quiver. Say that $[A_2]$ embeds into the mutation class of $Q$ when it occurs there under the…

Let $Q$ be a quiver, and let $\operatorname{uRed}(Q)$ denote its unrestricted red size, namely the maximal number of red vertices obtained after an arbitrary mutation sequence. Two…

Let $Q$ be a connected quiver on $n$ vertices that does not admit a reddening sequence. The red size $\operatorname{Red}(Q)$ is the maximal number of red vertices obtained after mu…

Let $G$ and $G'$ be connected simple plabic graphs whose quivers $Q_G$ and $Q_{G'}$ are mutation equivalent. Let $L^{\operatorname{plab}}_G$ and $L^{\operatorname{plab}}_{G'}$ be t…

Let $Q$ be a connected quiver, let $V(Q)$ be its vertex set, and suppose that $Q$ does not admit a reddening sequence. A general maximal green sequence is a mutation sequence resul…

Let $Q$ be a quiver and let $\mu_k(Q)$ denote the quiver obtained by mutation at vertex $k$. The unrestricted red size $\operatorname{uRed}(Q)$ is the minimum red size over general…

Let $(Q,W)$ and $(Q',W')$ be quivers with potential related by mutation at a vertex $k$, and let $d,d'$ and stability weights $\theta,\theta'$ be related by the mutation formulas i…

Let a mutation class be originating from an unpunctured surface or orbifold if it arises from an unpunctured surface or orbifold in the sense of the paper. A realization by reflect…

Let $Q$ be a framed quiver and consider the periodic mutation sequence … Assume that performing the mutations $v_1,\ldots,v_p$ on the base quiver returns it to the original quiver.…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. Polynomiality conjecture for deformed $F$-polynomials. The deformed $F$-polynomial $S_n$ is a polynomial…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. For each mutation index $i$, let $r_i$ denote the associated term and let $M$ be the set of relevant mon…

Let $\Gamma_n$ be the $n$-cycle quiver, and let $\Phi^1$ denote the corresponding root set, with $|\Phi^1|=\binom{n+1}{2}-1$. A maximal green sequence for $\Gamma_n$ has a well-def…

Let $Q$ be a quiver in the class $\mathcal{P}$, and let $\mathcal{A}$ and $\mathcal{U}$ be the cluster algebra and upper cluster algebra associated with the cluster algebra defined…

Let $Q$ be a quiver in the class $\mathcal{P}'$, let $(\mathbf{x},\mathbf{y},Q)$ be a seed with quiver $Q$, and let $\mathcal{A}$ and $\mathcal{U}$ be the associated cluster algebr…

Let $Q$ and $Q'$ be quivers such that $\Psi(Q)$ and $\Psi(Q')$ are finite and non-empty, and let $Q\oplus Q'$ denote their direct sum. Direct-sum finiteness conjecture. Then…

Let $Q$ be a minimal mutation-infinite quiver. Let $\widehat{\Psi}(Q)$ denote the subgraph associated with quivers admitting maximal green sequences, as in Theorem. Minimal mutatio…

Let $Q$ be a recurrent quiver and let $\mu_-\mu_+$ be its associated mutation map. Let the associated upper cluster algebra be the intersection of the Laurent polynomial rings of i…

Let $Q$ be a recurrent quiver, meaning that its underlying graph is bipartite and mutating at its black vertices followed by its white vertices returns the same quiver. A positive…

Let $Q$ be a recurrent quiver: its underlying graph is bipartite, and mutating at its black vertices followed by mutating at its white vertices returns the same quiver. A positive…