The purpose of this proposal is to being together robust control and algebraic topology. It is argued that such robust control issues as structured singular values, multivariable phase margin, Kharitonov's theorem, etc. receive their natural and unifying formulation in the context of algebraic topology, as formulated by Poincare, Cartan, Eilenberg, and many others. The central mathematical issue is whether the mapping of the high dimensional manifold of structured uncertainties into the Nyquist template commutes with the boundary. While in the simple Kharitonov case it does, in the over-whelming majority of situations the Nyquist mapping does not commute with the boundary. However, the simplicial approximation theorem provides us with an approximate Nyquist map that does commute with the boundary. Fast implementation of the simplicial approximation theorem opens the road to a variety of fast, "simplicial" algorithms. Finally, performing some algebra on the simplicial approximation yield the "topography" of the stability boundary, that can be very complicated.
