5 problems

Bijectivity conjecture. The map $\pi\colon\operatorname{silt}\nolimits^{\mathcal F}{\mathcal T}\to{\it d}\strut\kern-.2em\operatorname{-ctilt}\nolimits{\mathcal C}$ is bijective fo…

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 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 $\Lambda$ 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…