21 problems

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

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

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

Cluster-category correspondence conjecture. There is a one-to-one correspondence between the cluster variables of $\operatorname{\mathcal A}(H)$ and…

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

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 $\mathcal{A}$ 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 $\xi$ 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…

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…

Profile-count conjecture. The number of profiles of rigid indecomposable rank $2$ modules in ${\rm CM}(B_{k,n})$ is $N_{k,n}$.

Let $\mathcal{C}$ be a $2$-Calabi-Yau triangulated category with cluster structure, let $T$ be a cluster tilting object, and let $\phi$ be a cluster map sending cluster tilting obj…

Assume $3\le k\le n/2$. Let $J_{k,n}$ be the associated Kac–Moody algebra, let $\alpha$ be a real root of degree $2$, and let $\varphi$ be the map from indecomposable modules to ro…

Fix integers $(k,n)$ with $k\ge 3$. Let $I$ and $J$ be tightly $3$-interlacing $k$-subsets, meaning that they are $3$-interlacing and $|I\cap J|=k-3$. Let $L(I,J)$ denote the rank…

Let $\Lambda$ be a $\tau_2$-finite algebra of global dimension at most $2$. Let $\mathcal{D}^{\rm b}(\Lambda)$ be the bounded derived category of $\Lambda$-modules, let…

Piecewise hereditary characterization. The orbit category coincides with the cluster category for $\Lambda$ if and only if $\Lambda$ is piecewise hereditary.