The max-flow min-cut theorem, traditionally applied to problems of maximizing the flow of commodities along a network (e.g. oils in pipelines) and minimizing the costs of disrupting networks (e.g. damn construction), has found recent applications in information processing. In this talk, I will recast and generalize max-flow min-cut as a form of twisted Poincare Duality for spacetimes and more singular "directed spaces." Flows correspond to the top-dimensional homology, taking local coefficients and values in a sheaves of semigroups, on directed spaces. Cuts correspond to certain distinguished sections of a dualizing sheaf. Thus max-flow min-cut dualities extend to higher dimensional analogues of flows, higher dimensional analogues of directed graphs (e.g. dynamical systems), and constraints more complicated than upper bounds. I will describe the formal result, including a construction of directed sheaf homology, and some real-world applications.