This talk will present both computational and analytical results about inner and
outer bounds for the region of entropic vectors and associated rate regions in
network coding and multilevel diversity coding systems.  The computational part
of the talk points out three computations: non-Shannon outer bounds, inner
bounds based on representable matroids, and inner and outer bounds for capacity
regions, have a common underlying three step polyhedral computation structure. 
Multiple possible computational techniques for each step, each having widely
varying problem dependent complexity, are reviewed and their efficacy is
compared.  The analytical part of the talk elucidates the information geometric
structure of the Shannon and non-Shannon exposed faces of entropy.