Heterogeneity

I will discuss recent work with biologists related to understanding particular structural characteristics in the whiskers of mice, and in synthetic quorum-sensing mechanisms of bacteria. These scientific problems are typically modelled using reaction-diffusion systems, and one is often interested in emergent spatial and spatiotemporal patterns from instabilities of a homogeneous equilibrium. I will use these scientific questions to motivate fundamentally mathematical questions regarding instabilities and the emergence of patterns in complex domains. First I will discuss the well-known effects of how manifold structure impacts the modes which may become unstable in reaction diffusion systems, and hence how the kinds of patterns we may observe on manifolds can change due to geometry directly. More strikingly, I will discuss recent work where coupling between two different simple planar geometries leads to highly non-intuitive results regarding the role of geometry. Finally I will discuss results on instabilities on a large class of time-evolving manifolds, and show that one can derive a meaningful notion of instability of the homogeneous state even in this explicitly time-dependent setting. The technical Theorems in this last part may also have application far beyond the realm of developmental biology, as they generalize notions of instability to a large class of non-autonomous systems.