Pattern formation occurs in a wide variety of natural contexts, from animal coat markings to convection cells in the Sun. Experimentally, patterns have been studied in many different systems, including Rayleigh-Bénard convection, solidification, chemical reactions and Faraday waves. Despite the physical differences between these systems, the patterns that appear display common features, indicating some kind of universal underlying structure. There are currently many different mathematical approaches to the study of patterns, including numerical simulation, amplitude equations and equivariant bifurcation theory, and this programme will bring together mathematicians and experimentalists for an interchange and fusion of ideas. This synthesis of techniques will be needed to tackle current problems in the theoretical foundation of the subject, and to explain recent experimental results. A short instructional course aimed at younger researchers will introduce the central topics and current approaches.
In small domains, simple regular patterns are observed, and these can be understood through the application of methods from symmetric bifurcation theory. But as the domain size increases, the range of validity of this approach shrinks to zero and so other methods are required. Similar difficulties are encountered in large networks of coupled oscillators. Patterns in one space dimension can, generically, be described by the Ginzburg-Landau equation, but this has not been rigorously generalised to two dimensions, because of the problem of orientational degeneracy. There are therefore many fascinating aspects of two-dimensional patterns that are currently not well understood. Specifically, the programme will address the following topics.
- Recent experimental and numerical results on pattern formation in large domains.
- Bifurcation theory and symmetry in extended systems, including the validity of amplitude equations and phase equations in two dimensions.
- Networks of coupled oscillators and synchronisation.
- Complex spatial structures such as giant spirals, quasipatterns, spiral defect chaos, localised states, defects and domain boundaries.
- Systems where, even in one dimension, the Ginzburg-Landau equation does not apply, for example because of the presence of a large-scale neutral mode.
- Applications, including convection, Faraday waves and neuroscience.