Pattern Selection: Growth, Fronts, and Defects

This research project aims to mathematically describe physical and biological growth processes through direct modeling and analysis of coherent structures. Particular model examples create regular patterns such as stripes or hexagonal lattices of spots. The goal of this research is to find mathematical characterizations of patterns that explain the ubiquity of such regular patterns across nature and, at the same time, to provide recipes for the self-organized manufacturing of microstructure. The project includes research activities that will train graduate and undergraduate students.

The project focuses on perturbation methods that are applicable to systems without a technically viable spatial dynamics interpretation, including nonlocal and multi-dimensional problems. It will explore the interaction of the domain growth and pattern-formation and mechanisms of the speed-selection for traveling fronts. In particular, the research aims to develop techniques for the study of perturbation and bifurcation problems in infinite dimensional dynamical systems in the presence of essential spectrum using rigorous core/far-field decompositions. Beyond local perturbation analysis, these strategies will be implemented in numerical continuation algorithms, with a priori and a posteriori error bounds. Particular goals set forth are wavenumber and pattern predictions in growing domains, characterization of invasion speeds in pattern-forming systems, and depinning asymptotics in heterogeneous media.