41 problems

Cartan monotonicity conjecture. If

Cerulli–Labardini–Schröer conjecture. Every component cluster has cardinality at most $n$, and the $E_A$-rigid component clusters are precisely the component clusters of cardinalit…

Let $A$ be a graded skew-gentle algebra, and let the three conditions in Theorem 1.3 be the conditions concerning $\tau$-tilting finiteness, silting-discreteness, and the correspon…

Let $\Lambda$ be a finite-dimensional algebra. A module $M\in\operatorname{mod}\Lambda$ is a brick if $\operatorname{End}_{\Lambda}(M)$ is a division algebra. The algebra $\Lambda$…

Let $\Lambda$ be a finite-dimensional algebra. It is $\tau$-tilting finite if it has finitely many isomorphism classes of basic $2$-term silting complexes in…

Let $\Lambda$ be a finite-dimensional algebra. It is $\tau$-tilting infinite if it has infinitely many isomorphism classes of basic $2$-term silting complexes. A module $M$ is…

Let $\Lambda$ be a finite-dimensional algebra. A semibrick in $\operatorname{mod} \Lambda$ is a set of bricks whose distinct members have no nonzero morphisms between them. Enomoto…

Let $A$ be a finite-dimensional symmetric algebra. The algebra $A$ is $\tau$-tilting finite if it has only finitely many isomorphism classes of support $\tau$-tilting modules, and…

E-finiteness conjecture. If $A$ is $E$-finite, then $A$ is brick-finite.

Let $\Lambda$ be a finite-dimensional algebra. The group $\operatorname{\mathrm{int}\text{-}\mathrm{heart}}(\operatorname{tors}\Lambda)$ has heart-controlled generators, meaning th…

Let $\Lambda$ be a finite dimensional algebra. Let $\mathsf{W}\subseteq\operatorname{mod}\Lambda$ be a $\tau$-perpendicular subcategory of $\operatorname{mod}\Lambda$, and let…

Let $\Lambda$ be a finite-dimensional algebra and let $g$ be a g-vector. Write $[g]_{\operatorname{TF}}$ for the $\operatorname{TF}$-equivalence class of $g$, let…

Let $\Lambda$ be a finite-dimensional algebra, let $K_0(\operatorname{proj}\Lambda)_{\mathbb{R}}$ be its real Grothendieck group, and let $g$ be a g-vector. Write…

Tameness conjecture for g-tame incidence algebras. If $\operatorname{Inc}_\K(P)$ is $\mathbf{g}$-tame, then $\operatorname{Inc}_\K(P)$ is tame.

Wildness and g-tameness conjecture. The incidence algebra $\operatorname{Inc}_\K(P)$ is wild if and only if it is not $\mathbf{g}$-tame.

The p-hyperfocal subgroup conjecture. The group algebra $kG$ is $\tau$-tilting finite if and only if either $R$ is cyclic, or $p=2$ and $R$ is isomorphic to a dihedral, semidihedra…

Let $\Lambda$ be the finite-dimensional algebra under consideration, and let … be a generically $\tau$-reduced component, with $|\Lambda|$ denoting the number of isomorphism classe…

Stable Brauer–Thrall II' conjecture. There exist $\bd\in K_0(A)^+$ and $\theta\in\K_0(A)^*$ such that

The $\tau$-reduced Brauer–Thrall II' conjecture. There exist $\bd\in\K_0(A)^+$ and a generically indecomposable $\cZ\in\Irr^\tau(A,\bd)$ such that

Demonet's conjecture. $A$ is $\tau$-tilting finite if and only if

Buan–Marsh conjecture. If two complete $\tau$-exceptional sequences over $\Lambda$ differ in only one place, then they do not exist; equivalently, the result proved for $\tau$-tilt…

Tau-rigidity conjecture. If $M$ is an indecomposable syzygy, then $M\oplusc3M$ is $c4$-rigid.

Syzygy $c4$-rigidity and reachability conjecture. Indecomposable syzygies over dimer tree algebras are $c4$-rigid and reachable.

Tau-rigidity conjecture. Every indecomposable Cohen–Macaulay module over $A$ and $AG$ is $c4$-rigid.

Uniqueness conjecture. The statement of Theorem $$ holds for all finite-dimensional algebras.