The Department of Applied Mathematics weekly seminar is given by scholars and researchers working in applied mathematics, broadly interpreted.
Title: Universal spreading into unstable states
Abstract: The emergence of complex spatial structures in physical systems often occurs after a simpler background state becomes unstable. Localized fluctuations then grow and spread into the unstable state, forming an invasion front which propagates with a fixed speed and selects a new stable state in its wake. The mathematical study of these invasion processes has historically been limited to systems with restrictive monotonicity properties (in PDE terms, a comparison principle). Such systems, however, inherently cannot describe the formation of complex spatiotemporal patterns, which is of particular interest both in nature and in manufacturing applications. On the other hand, formal calculations in the physics literature have long outlined a universal approach for predicting invasion speeds and associated selected states, valid for systems which do not obey comparison principles and instead exhibit complex spatiotemporal dynamics. This prediction scheme is often referred to as the marginal stability conjecture. In this talk, I will discuss the first proof of the marginal stability conjecture and explore how this can be used to make concrete predictions for physical systems.