In this talk we will describe a new family of combinatorial invariants of simplicial complexes and their relationship to the dynamics of complex decision networks. As a first approximation, one can model a decision network as a Boolean network, and view it as a dynamical system on a finite phase space, which carries the structure of a directed graph. To each point in the phase space one can associate a simplicial complex which carries information about the local dynamics of the network.
The invariants can be thought of as a combinatorial analog of the homotopy theory of topological spaces and are conjecturally related to ``forcing structures'' in the network. We shall also describe potential connections between our combinatorial homotopy invariants and matroid theory.
The talk is accessible to graduate students.