### Abstract

Much of the fascination and challenge of studying infinite dimensional nonlinear dynamical systems arises from the complicated spatial and/or temporal behavior that they exhibit. On the mathematical side, this complicated behavior can occur on all scales both in phase space and parameter space. Somewhat paradoxically, from a scientific perspective this points to the need for a coherent set of mathematical techniques that is capable of extracting coarse but robust information about the structure of these systems. Furthermore, since most of our understanding of specific systems comes from experimental observation or numerical simulations, it is important that these techniques can be applied in a computationally efficient manner. Finally, it is important that the experimentally observed patterns be quantified in an efficient manner both for purposes of parameter identification and for modeling purposes. In this talk it will be argued that computational homology has an important role to play in these endeavors.