A topological max-flow-min-cut theorem

This note surveys a novel algebraic-topological version of the max-flow-min-cut (MFMC) theorem for directed networks with capacity constraints. Novel features include the encoding of capacity constraints as a sheaf of semimodules over the network and a realization of flow and cut values as a directed homology taking values in the sheaf. We survey the theorem and give applications to (1) multicommodity flows, (2) multi-source/multi-target flows, and (3) boolean-lattice-valued flows.