10 problems

Let $\Lambda$ be the graded commutative algebra formed from the antisymmetric subspace of connected tri-/univalent graphs with three labeled univalent vertices, and let $\Lambda_0$…

Let a graph have all but finitely many vertices lying on finitely many infinite paths. Fold each of these paths so that it produces countably infinitely many edges between two dist…

Fau's graph-algebra conjecture. $C^*(E)$ is nowhere scattered if and only if $C^*(E)$ is $\mathcal{Z}$-stable, if and only if $E$ has Condition (K) and distinct detours.

Graph-algebra characterization conjecture. The following conditions are equivalent:

Let $E$ and $F$ be graphs and let $\mathsf{k}$ be a field. Consider the Leavitt path algebras $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$, and the graph $C^*$-algebras $C^*(E)$ and…

Let $S_n$ be the star graph on $n$ vertices, with central vertex labeled by $1$, and let … For a vertex subset $S$, write $Y_S$ for the associated cluster-algebra element and…

Gauge-coaction normality conjecture. Every gauge coaction on a representation of $\Lambda$ is normal. Consequently, the other results in the relevant section should require only th…

Let $E$ and $F$ be finite graphs, and let $R$ be a field. Write $\mathcal{G}_E$ and $\mathcal{G}_F$ for their graph groupoids, and let $H_0^{\operatorname{gr}}$ denote graded group…

Six-vertex infinite-dimensionality conjecture. The algebra $\mathcal E_G$ is infinite-dimensional if and only if $G$ contains a subgraph isomorphic to one of the graphs shown in Fi…

Let $\mathcal Z$ be the Jiang--Su algebra and let … For a graph $G$, let $B(G)$ denote the associated CCR algebra. Non-realizability claim. The algebra $A_{\aleph_0\aleph_1}$ is no…