31 problems

Let $A$ be a finite dimensional algebra over an algebraically closed field. A brick is an $A$-module whose endomorphism ring is a division ring, and a generic module is an indecomp…

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. 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.

Second brick-Brauer-Thrall conjecture. If $A$ is brick-infinite (equivalently, $\tau$-tilting infinite), there is a dimension vector $\mathbf{d}$ such that…

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…

Let $A$ be an algebra of rank $n$, and let $\mathcal{F}_A$ denote its $\tau$-tilting fan in $\mathbb{R}^n$. Call $A$ brick-infinite if it has infinitely many bricks up to isomorphi…

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

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…

Second Brauer–Thrall conjecture. If $A$ is tau-tilting infinite, then there exists a dimension $d\geq 0$ and infinitely many pairwise non-isomorphic bricks…

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

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

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

Tau-tilting finiteness–brick-discreteness conjecture. The following are equivalent:

Let $\Lambda$ be a finite-dimensional algebra, and let $\operatorname{mod} \Lambda$ denote its category of finite-dimensional right $\Lambda$-modules. A module is a brick if its en…

Let $\Lambda$ be a $\tau$-tilting finite algebra, and let $\mathcal{D} \sqcup \mathcal{U}[1]$ be a semibrick pair, meaning a collection of bricks in the indicated degrees satisfyin…

Exceptional wild Schur algebra conjecture. All wild Schur algebras contained in $(\star)$ are $\tau$-tilting finite.

Cartan monotonicity conjecture. If