Much of the fascination and challenge of studying nonlinear systems arises from the complicated spatial, temporal and even spatial-temporal behavior they exhibit. On the level of mathematics this complicated behavior can occur at all scales, both in the state space and in parameter space. Somewhat paradoxically, 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, most of our understanding of specific systems comes from experimental observation or numerical simulations and thus it is important that these techniques be computationally efficient.
Algebraic Topology is the classical mathematical tool for the global analysis of nonlinear spaces and functions, within which homology is perhaps the most computable subset. In particular, it provides a well understood framework through which the information hidden in large datasets can be reduced to compact algebraic expressions that provide insight into underlying geometric structures and properties.
The material described through these web pages represents our ongoing effort to develop and apply efficient and effective topologically based methods to the analysis of nonlinear systems.