12 problems

Hazrat's conjecture. The following are equivalent:

Equivariant K-theory classification conjecture. There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

ER Conjecture 5.1. For all $\mathsf{xyz}$ other than $\mathsf{010}$ and $\mathsf{101}$, the equivalence relation $\overline{\underline{\mathsf{x\smash{\mathsf{y}}z}}}$ is generated…

ER geometric classification conjecture. For every $\mathsf{xyz}$ other than $\mathsf{x10}$, the equivalence relation on finite-vertex graphs induced by $\mathsf{xyz}$-isomorphism o…

Generating-moves conjecture. In all six cases, the equivalence relation is generated by the moves that leave the graphs invariant in the relevant sense. Explicitly,

Let $\Gamma$ be a strongly connected graph. Its Perron–Frobenius eigenspace is the eigenspace corresponding to the Perron–Frobenius eigenvalue of its vertex matrix. Quantum vertex-…

Graph $C^*$-algebras are $C^*$-algebras associated to directed graphs, and filtered $K$-theory is the collection of ordered $K_0$-groups and $K_1$-groups of their subquotients toge…

Let $E$ be a directed graph, let $C^*(E)$ be its graph $C^*$-algebra, and let $\mathcal B=\{B_n\}_{n\in\mathbb Z}$ be the associated Fell bundle, with unit fibre $B_e$. A projectio…

Algebraic-to-analytic K-theory conjecture. This map is an isomorphism for every finitely separated graph $(E,C)$. This conjecture asserts that the algebraic and $C^*$-algebraic pro…

Let $E$ be a graph and let $C^*(E)$ be its graph $C^*$-algebra. Suppose that $C^*(E)$ has finitely many ideals, and write…

Let $FK(-)$ denote ideal-related $K$-theory, and consider graph $C^*$-algebras with finitely many ideals. ERR conjecture. The functor $FK(-)$ should classify graph $C^*$-algebras w…

Let $G$ be a finite graph with no sinks or sources such that $C^*(G)$ is simple, and let $E_1$ and $E_2$ be one-sink extensions of $G$. Write $[E_i]$ for the class of the extension…