22 problems

Let $A$ be a finite-dimensional algebra over a field $K$. Write $K_0(\operatorname{\mathsf{proj}} A)$ for the Grothendieck group of finitely generated projective $A$-modules, and l…

Let $A$ be a finite-dimensional algebra. Say that $A$ is brick-infinite if it admits an infinite family of pairwise non-isomorphic bricks, and brick-continuous if it admits an infi…

Norine–Thomas's conjecture. Every minimal brick $G$ has at least

Second brick-Brauer-Thrall conjecture. An algebra $A$ admits infinitely many non-isomorphic bricks if and only if $\operatorname{mod} A$ contains infinitely many non-isomorphic bri…

Let $A$ be a finite-dimensional algebra over a field $K$. A brick is a module $M\in\operatorname{\mathsf{mod}} A$ whose endomorphism algebra $\operatorname{\mathsf{End}}_A(M)$ is a…

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

Stable-discreteness conjecture. An algebra $A$ is stably-discrete if and only if it is brick-discrete.

Second brick-Brauer-Thrall orbit characterization. Every $X\in\operatorname{brick}(A)$ has open orbit if and only if $A$ is brick-finite.

Kothari et al.'s conjecture. Every essentially 4-edge-connected cubic near-bipartite brick $G$ other than $K_4$ has at least

Let $A$ be an algebra. Call $A$ brick-infinite if it has infinitely many bricks up to isomorphism, and call a set of pairwise Hom-orthogonal bricks in $\operatorname{mod} A$ a semi…

Let $A$ be an algebra, and let $\theta\in K_0(\operatorname{proj} A)$. A module $M\in\operatorname{mod} A$ is $\theta$-stable if $\theta(\underline{\dim}M)=0$ and…

Let $A$ be an algebra, and let $\theta\in K_0(\operatorname{proj} A)$. A module $M\in\operatorname{mod} A$ is $\theta$-stable if $\theta(\underline{\dim}M)=0$ and…

Let $A$ be an algebra, and call a set of pairwise Hom-orthogonal bricks in $\operatorname{mod} A$ a semibrick. Call $A$ brick-infinite if it has infinitely many bricks up to isomor…

Let $A$ be a finite-dimensional algebra over a field $k$. Write $\operatorname{brick}(A,d)$ for the set of isomorphism classes of bricks of dimension $d$, and call $A$ brick-finite…

Faithful-component form of the second brick–Brauer–Thrall conjecture. There is always a faithful brick component containing a one-parameter family of bricks.

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

Lovász's conjecture. Every minimal brick has two adjacent vertices of degree three.

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

Brick-infinite equivalence conjecture. The following are equivalent:

Brick-discrete–brick-finite conjecture. Brick-discrete algebras are the same as brick-finite algebras.

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…

Linear degree-three-vertex conjecture. There exists $\alpha > 0$ such that every minimal brick $G$ has at least