13 problems

Connectedness problem. For every $w\in W$, the cluster tilting graph of $\operatorname{Sub}\nolimits\Lambda/I_w$ is connected.

The maximal-rigid-to-cluster-tilting conjecture. Any maximal rigid object whose quiver has no loops or 2-cycles is a cluster tilting object.

Let $A$ be a finite-dimensional algebra, and let $d$ be an integer for which $A$ admits a $d$-cluster tilting module. Darpö–Iyama finiteness conjecture. For a fixed finite-dimensio…

Let $A$ be a local commutative algebra with an $n$-cluster tilting module. Rigidity conjecture. Then $n=1$ and there is an integer $m\geq 1$ such that … The conjecture is one of th…

Let $A$ be the local finite-dimensional algebra described in the paper, and let $M$ be its 2-precluster tilting module. Marczinzik's conjecture. The module $M$ is 2-cluster tilting…

Let $\mathcal{M}$ be a small $d$-abelian category. Write $\operatorname{Ind}(\mathcal{M})$ for its ind-completion, and let $\operatorname{Mod}\mathcal{M}$ be the category of functo…

Assume that $k$ is of characteristic $0$. Let ${\mathcal C}$ be a 2-Calabi–Yau algebraic triangulated category with a cluster-tilting object $T$. For a quiver with potential…

Let $k$ be a field of characteristic zero. Let ${\mathcal C}$ be a 2-Calabi–Yau triangulated category with cluster-tilting objects, and let…

Let $\mathfrak{Per}\,\Gamma$ be the perfect category associated with a 4d $\mathcal{N}=2$ theory, let $\mathcal{K}(\phi)$ denote the full additive subcategory of $\phi$-calibrated…

Mutation compatibility conjecture. For $2$-Calabi–Yau-tilted algebras which are Jacobian algebras, the mutations of $2$-Calabi–Yau-tilted algebras coincide with those of Jacobian a…

Maximal-rigid-to-cluster-tilting conjecture. Any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object.

Let $\Lambda$ be a $2$-representation-finite algebra, and let $\Phi$ be its Coxeter transformation. A root is $\Phi$-positive if … for every $m\in\mathbb{Z}$. The cluster-root conj…

The complements conjecture. Each liftable almost complete $m$-cluster tilting object has exactly $m+1$ complements in generalized $m$-cluster categories arising from $m$-rigid good…