Bifurcations and patterns on spheres

Bifurcation from a state of spherical symmetry is a subject of considerable mathematical complexity that goes back to Alan Turing's 1952 paper on morphogenesis. There are applications in biology, such as tumour growth, and in physics, for example convection patterns in the Earth's mantle. More recently there has been interest in patterns which form in neural field models on the surface of a sphere. Neural fields model large scale electrical brain activity and the cortical white matter system is topologically close to a sphere. This research project will look at some aspects of bifurcation theory and pattern formation in a spherical geometry, such as: Localised patterns on spheres. Localised patterns can appear in planar pattern formation problems via a 'snaking' bifurcation diagram, but it is not known how this extends to the spherical case. Recognition of patterns on spheres. Given a non-spherical, bifurcated
shape, how can the determination of the symmetry group of this pattern be automated? Neural field models on spherical domains. These are integro-differential equations used in neuroscience.Here, a spherical domain is a better representation of the brain than the more commonly used linear or planar geometry. The project will involve a combination of different methods, including equivariant bifurcation theory, stability analysis and asymptotic methods, and numerical computations.