Martin Wechselberger School of Mathematics and Statistics, The University of Sydney Rankine-Hugoniot, Lax and folds: the geometry of advection-reaction-diffusion systems Wednesday 19th May 14:05-14:55pm, New Law School Seminar 030 (Building F10). Hyperbolic balance laws (conservation laws with source terms) have attracted much attention in the biosciences because they play an important role in modeling tactically-driven cell migration. In particular, sharp interfaces, or shocks, in the wave form of cell migration within tissues are observed which motivates the study of advection-reaction-diffusion models where the diffusion is considered a viscous small perturbation. (Sharp interfaces are, of course, well-known in classical physical applications such as in gas dynamics, MHD theory and traffic flows.) In this talk, we will give a twist to the existing hyperbolic PDE theory on viscous balance laws and apply recently developed geometric singular perturbation methods to this class of problems. In particular, I will identify the underlying geometric structures, folds and canards, that lead to the existence of travelling waves with sharp interfaces which provides us with a geometric interpretation of the famous Rankine-Hugoniot jump condition and the Lax entropy condition.