David Witt Nyström: Competitive Hele-Shaw flows and quadratic differentials

Abstract:

In the classical Hele-Shaw flow a domain in the complex plane grows according to the gradient of its Green's function, thus modelling the propagation of a viscous fluid trapped in a thin layer. We introduce a competitive version of the flow where several domains in the complex plane (or more generally in a Riemann surface of finite type) similarly strive to expand but at the same time hinder each other. Interestingly, stationary flows correspond to a special class of quadratic differentials whose associated half-translation surfaces have a simple description. We also introduce a discrete model, closely related to Propp's competitive erosion model, which conjecturally allows us to simulate the flow. My talk will focus on the geometric aspects of all this. This talk is based on joint work with Fredrik Viklund.