12 problems

Let $G=(V,A)$ be a directed acyclic graph, and let $\Sigma$ denote the covariance matrix of its arc-indicator variables. Covariance-matrix sparsity conjecture. The covariance matri…

Let $G=(V,A)$ be a directed acyclic graph, with arcs represented by indicator random variables $A_{ij}$ and covariance matrix $\Sigma$. Two arcs are incident when they share a comm…

Feasible indexing conjecture. There exists an indexing satisfying this successor-set ordering and, whenever two consecutive nodes have equal successor-set order,

Sullivant's saturation conjecture. The vanishing ideal satisfies

Let $G$ be a directed acyclic outerplanar graph, and let $\operatorname{sn}(G)$ denote its stack number. Heath, Pemmaraju and Trenk's conjecture. The stack number of the class of d…

Assume the higher d-separation conjecture: whenever two vertices of a DAG have minimum d-separating-set size at least $2$, the vanishing ideal is not toric. Let $G$ be a chordal di…

Let $G$ be a directed acyclic graph (DAG), and let $i,j$ be vertices of $G$. A d-separating set for $i$ and $j$ is a set of vertices that d-separates them; consider the minimum car…

Let $G$ be a directed acyclic graph (DAG) with toric vanishing ideal $I_G$. The relevant constructions are safe gluing, gluing at sinks, and adding a sink. Recursive classification…

Let $G_1$ and $G_2$ be directed acyclic graphs (DAGs), and let $I_{G_i}$ be their vanishing ideals and $ST_{G_i}$ their shortest trek ideals. Suppose that $I_{G_1}=ST_{G_1}$ and…

Let $\Gamma$ be a finite directed acyclic graph. Its universal dynamics monoid $D(\Gamma)$ is the smallest quotient of $\operatorname{\mathbf{HK}}_\Gamma$ through which all evaluat…

Let $G=(V,A)$ be a directed acyclic graph, and consider two arcs incident on a common node. Incident-arc range conjecture. The covariance between the two arcs takes values in the i…

Let $G=(V,A)$ be a directed acyclic graph, and consider two arcs that are incident on a common node. Let covariance and correlation refer to the corresponding arc-indicator random…