40 problems

In the acyclic case, let the cluster category be the category associated with a quiver without oriented cycles, and let an exceptional object mean an object of this cluster categor…

Positivity conjecture. For any object $M$ and any exceptional object $K$ of ${\mathcal C}$, the integer $r(M,K)$ is nonnegative.

Let $C$ be a cluster, let $(\mathcal{Q}_C,I_C)$ be its associated quiver with relations, and let $T$ be the corresponding tilting object in the cluster category $\mathcal{C}$. Writ…

Derived exchange-pair conjecture. The quotient module categories are equivalent:

Exchange-relation conjecture. The exchange relation in $\operatorname{\mathcal A}(H)$ is

Stackability conjecture. Every triangulation $obreakDelta$ of $obreakmathcal{M}{T}$ is stackable.

Let $obreakmathsf{T}$ be a cluster-tilting object in the cluster category of the type $obreakmathbb{A}n$, and let $obreakmathcal{M}{mathsf{T}}$ be the oriented matroid obtained…

Christ's conjecture. The restriction of $\Phi$ to ${\mathcal H}$ is an equivalence of $k$-linear categories

Christ's conjecture. The Higgs category should be equivalent to the cosingularity category:

Compactness conjecture for X-foliations. If $Q$ is a Dynkin quiver, then every X-foliation is compact. If $Q$ is an Euclidean quiver, then every X-foliation is semi-compact. Moreov…

Acyclicity conjecture. Any $[?]$-foliation is acyclic.

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 $B_{3,9}$ and $B_{4,8}$ be the Grassmannian algebras considered in the source, and let ${\rm CM}(B_{k,n})$ denote their Cohen–Macaulay module categories. Rank-four rigidity con…

Let $\mathcal{A}$ be a cluster algebra with monoidal categorification $\mathscr{C}_\ell$ and additive categorification $(\mathcal{F},\widehat{T})$, let $\mathcal{C}$ be the stable…

Let $\mathcal{C}$ be the stable category of the Frobenius category $\mathcal{F}$, let $T$ be a cluster-tilting object, let $\Sigma$ denote the suspension, and let…

Let $A$ be a finite-type simply-laced generalized Cartan matrix, let $C_{\rm add}(A)$ be the set of cluster-additive functions associated to $A$, and let cluster-hammock functions…

Let $A$ be a finite-type simply-laced generalized Cartan matrix, let $\Lambda$ be the path algebra of the associated Dynkin quiver, and identify $[1,r]\times\mathbb Z$ with the ver…

Let $\xi$ be a height function and let $\ell\geq 2$. Let $\mathscr{C}^{\leq \xi}_\ell$ be the corresponding module subcategory, and let $L(m)\in\mathscr{C}^{\leq \xi}_\ell$ be a re…

Let $\xi$ be an arbitrary height function and let $\ell\geq 2$. Let $Q^{\leq \xi}_\ell$ be the principal quiver, let $A^{\leq \xi}_\ell$ be its Jacobian algebra, and let…

Let $n\in\mathbb{N}_{>0}$, and let $\mathcal{C}(\mathbb{D}_{\overline{n,\infty}})$ be an infinite type $\mathbb{D}$ weak cluster category. A full subcategory is weakly cluster tilt…

Let $n\in\mathbb{N}_{>0}$, and let $\mathcal{C}(\mathbb{D}_{n,\infty})$ be an infinite type $\mathbb{D}$ cluster category whose clusters are weakly cluster tilting. A full subcateg…

Let $B$ be a dimer tree algebra, let $\mathcal{S}$ be its checkerboard polygon with $2N$ vertices, and let $\gamma$ be a 2-diagonal. For the projective modules $P_0(\gamma)$ and…

Let $B$ be a dimer tree algebra, and let $\mathcal{S}$ be its checkerboard polygon. The category $\operatorname{Diag}(\mathcal{S})$ has 2-diagonals as indecomposable objects, with…

Three-quasi-boxes rigidity conjecture. Then $M$ is not rigid.

Real canonical-profile conjecture. Let $M\in {\rm CM}(B_{k,n})$ be rigid indecomposable and assume that $\varphi(M)$ is a real root in $J_{k,n}$. Then $P_M$ is a cyclic permutation…