12 problems

Geometric second brick–Brauer–Thrall conjecture. Every brick-infinite algebra admits infinitely many bricks without open orbits.

Gluing conjecture. Every geometrically irreducible algebra is a gluing of algebras with at most two simple modules.

Conjecture. Theorem 1 would remain valid without any assumptions on the algebras $A$ and $B$ other than their being isomorphic as graded $S$-modules; equivalently, the conditions i…

Two-simple-module conjecture. Up to a trivial glueing procedure, every geometrically irreducible algebra has, up to isomorphism, at most two simple modules.

Geometrically irreducible minimal algebra conjecture. Every geometrically irreducible minimal algebra has at most two simple modules.

Let $A=KQ/{\mathcal I}$ be a finite-dimensional algebra presented by a quiver with relations. Call $A$ minimal if ${\mathcal I}=0$, or if every generating set of relations satisfie…

Let $A=KQ/{\mathcal I}$ be a finite-dimensional algebra presented by a quiver with relations. An algebra is geometrically irreducible when its module varieties are irreducible. For…

Conjectures 4.6. The following hold: the source excerpt does not include the individual assertions that follow this introductory sentence.

Multiplicity-free conjecture. An algebra is Schur-representation-finite if and only if it is MF.

Multiplicity-free conjecture. The theorem characterizing the MF property for tame algebras should hold without the assumption that $A$ is tame.

Dense orbit conjecture. The following statements are equivalent:

MF–Schur-representation-finiteness conjecture. The MF property should correspond to Schur-representation-finiteness in general.