The investigator is developing methods for studying the structure of isolated invariant sets that are applicable to a wide variety of problems, e.g., travelling waves for reaction-diffusion systems, dynamic phase transition, magnetohydrodynamic shock waves, delay equations, to name a few. More specifically, it is concerned with deriving algebraic invariants that describe invariant sets, determining how they can change through bifurcations, and determining which geometric topological structures are imposed on the invariant sets by the algebraic invariants. The guiding philosophy behind these methods comes from the fact that many differential equations that arise as models of biological, chemical, or physical systems share the following three characteristics: (1) they are derived from idealized descriptions of complicated phenomena, (2) they are parameterized systems and exact values of the parameters are not known, (3) they are nonlinear. Thus, from the point of view of applications, describing the global and robust structure of the solutions to these differential equations is of great importance. At the same time, it is well known that nonlinear systems undergo tremendously complicated bifurcations (e.g., the appearance of chaotic dynamics). Therefore, any general theory must rely on tools that are indifferent to the fine structure of the invariant sets, e.g. the Conley index theory.
