25 problems

Reduced skein–cluster conjecture. Under these assumptions,

Localized skein–cluster conjecture. Under these assumptions,

Let $Br_\Delta$ be the braid group associated with a triangulation $\Delta$, and let … be its abelianization. Abelianization non-injectivity conjecture. The canonical homomorphism…

Let $\Sigma$ be a marked surface, let ${\mathcal A}_\Sigma$ be its noncommutative cluster algebra, and let $I_{p,1}(\Sigma)$ denote the set of the relevant once-punctured indices.…

Let $\Sigma$ be a marked surface with $I_{p,0}(\Sigma)\cup I_{p,1}(\Sigma)=\varnothing$, let $\Delta$ be a triangulation, let $T_\Delta$ be the associated algebraic object, and let…

Let $\Sigma$ be a connected oriented marked surface other than a sphere with $4$ punctures or a projective plane with $2$ punctures, and let ${\bf F}:{\bf TSurf}^t\to{\bf Grp}$ be…

Let $\Sigma$ be an oriented marked surface and let $\Delta$ be a triangulation. Let $Br_\Delta^+$ be the submonoid of $Br_\Delta$ generated by the elements $T_{\gamma,\Delta}$ for…

Let $\Sigma\in {\bf Surf}$ be a marked surface and let $\Delta$ be a triangulation of $\Sigma$. Torsion-freeness conjecture. The group $Br_\Delta$ is torsion-free. The examples imm…

Let $f:\Sigma\to\Sigma'$ be a morphism as in Lemma, and let $f_*:Br_\Delta^f\to Br_{f(\Delta)}$ be the induced homomorphism. Relative braid homomorphism injectivity conjecture. In…

Let $f_\Gamma:\Sigma\to\underline \Sigma=\Sigma/\Gamma$ be the quotient map in the assumptions of Proposition. Write $Br_\Sigma^\Gamma$ for the subgroup of $Br_\Sigma$ fixed by…

Let $(S,M,D)$ be a $d$-angulation of a marked surface, let $(Q,W)$ be its associated quiver with superpotential, and let $\Gamma$ be the associated $d$-dimensional Ginzburg algebra…

Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a marked surface with punctures, and let $\mathcal{B}^{\circ}$ denote the set of bangle functions associated with it. Musiker–Schiffler–Will…

Let $\Sigma$ be a connected, triangulable unpunctured marked surface. Write $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{\mathbb{Z}_q}[\partial^{-1}]$ for the boundary-localization of th…

Let $(S,M)$ be a marked surface and let $\mathcal{A}^{\mathrm{bd}}$ be the cluster algebra from $(S,M)$ with boundary frozen variables. For a triangulation $T$ and an arc $\gamma$,…

Let $(S,M,A)$ be a graded marked surface with a full arc system $A$. Say that it has enough marked intervals when … and, for each $k$, the inclusion $\iota_k:(D^2_k,M_k)\to(S,M)$ i…

Let $(S,M)$ be a graded marked surface with an arc system $A$ such that the inclusions of disks $(D_i,\Lambda_{m_i})\hookrightarrow (S,A)$ in an arc decomposition … determine monom…

Let $(S,M)$ be a marked surface and let $\mathcal{S}$ be an $\mathcal X$-seed pattern from $(S,M)$. For every triangulation $T$ of $(S,M)$ and every $\gamma\in T$, let…

Let $(\Sigma,M)$ be a marked surface, and let $\sigma$ and $\tau$ be tagged triangulations related by a flip at the tagged arc $i$, so that $f_i(\sigma)=\tau$. For a tagged arc $i$…

Let $(S,M)$ be a graded marked surface with a full arc system $A$, and suppose that $(S,M)$ has enough marked intervals, meaning that for each disk $(D_k^2,M_k)$ in the decompositi…

Let $\Sigma$ be a marked surface, let $\Delta$ be a triangulation, and let $\mathcal{A}_\Sigma$ be the associated algebra. The natural homomorphism … comes from the inclusion of…

For each $n\ge 2$, let $\mathcal{U}_n$ be the subalgebra generated by all $x_{ij}$, $i\ne j$, and $x_{i,i^\pm}^{-1}$, and let $\mathcal{A}_\Delta$ be the algebra associated with a…

Let $n'\ge n$ and let ${\bf j}:[n]\hookrightarrow[n']$ be an injective map. The assignments $x_{ij}\mapsto x_{{\bf j}(i),{\bf j}(j)}$ define a homomorphism…

Color-independence conjecture. Changing all colors of marked points on any subset of boundary components does not change the cluster type of $\mathcal{A}(S_{g,\sigma})$.

Uniqueness conjecture. The quiver $Q(\tau)$ admits only one non-degenerate potential up to weak right equivalence.

Let $S$ be a marked surface with empty boundary, and let $\tau$ be an ideal triangulation of $S$. Labardini-Fragoso associates to $\tau$ a quiver with potential $(Q(\tau),W(\tau))$…