7 problems

Representation-dimension conjecture. The representation dimension of $A$ is at most three.

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

Let $(Q,I)$ be a bound quiver, and let $A=kQ/I$ be its bound quiver algebra. For a dimension vector $\mathbf{d}$, let $\operatorname{rep}_Q(I,\mathbf{d})$ denote the variety of rep…

Skowroński's conjecture. The period of $A$ is equal to four.

Openness conjecture. The property of being tame is open in the variety of finite-dimensional associative $K$-algebras.

Carroll–Chindris–Weyman conjecture. Let $\Lambda$ be an algebra. Then the following conditions are equivalent:

Let $A$ be a tame algebra, and let $X$ be a nonprojective module in $\operatorname{ind} A$. For an almost split sequence … with indecomposable middle terms $Y_i$, write $s(X)$ for…