The last fifty years of dynamical systems theory have established that dynamical systems can exhibit extremely complex behavior with respect to both the system variables (chaos theory) and parameters (bifurcation theory). Such complex behavior found in theoretical work must be reconciled with the capabilities of the current technologies available for applications. For example, in the case of modelling biological phenomena, measurements may be of limited precision, parameters are rarely known exactly and nonlinearities often cannot be derived from first principles.
The contrast between the richness of dynamical systems and the imprecise nature of available modeling tools suggests that we should not take models too seriously. Stating this a bit more formally, it suggests that extracting features which are robust over a range of parameter values is more important than an understanding of the fine structure at some particular parameter.
The goal of this talk is to present a high-level introduction/overview of computational Conley-Morse theory, a rigorous computational approach for understanding the global dynamics of complex systems. This introduction will wander through dynamical systems theory, algebraic topology, combinatorics and end in game theory.