Geometry/topology Seminar
Tuesday, April 3, 2012, 4:30pm, 119 Physics
Sanjeevi Krishnan (University of Pennsylvania)
Directed Poincare Duality
Abstract:- 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. [video]
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