Minicourse: Bifurcations in Reaction-Diffusion Equations

The term ``reaction-diffusion equation'' refers to a (typically nonlinear) parabolic PDE for a function u(x,t) of position and time of the form \partial_t u = \nabla^2 u + F(u). Systems of reaction-diffusion equations are defined analogously. The name comes from the fact concentration of chemicals undergoing reactions obey such equations if the concentrations are not uniform in space. These equations arise in numerous other contexts as well, including the propagation of nerve impulses and cardiac electrophysiology.

The minicourse will begin with a brief treatment of the underlying theory for such equations, and then basic bifurcation theory will be covered. Following this, the class will jointly read Turing's classic paper on the subject. Modern and classical applications of the subject will be introduced, and students will be asked to do a miniproject on one of them.