KASS seminar

Abstract:

This talk is based on joint work in progress with Fredrik Viklund at KTH.

In the classical Hele-Shaw flow a domain in the complex plane grows according to Darcy's Law, thus modelling the propagation of a viscous fluid trapping in a thin layer. We introduce a competitive version of the flow where several domains on a compact Riemann surface similarly strive to expand but at the same time hinder each other. We conjecture that, with some easily understood exceptions, this flow will exist for all time and converge to an equilibrium configuration. Interestingly, such an equilibrium gives rise to a very special kind of quadratic differential on the Riemann surface, which is best understood in terms of its associated half-translation surface. In this talk I will try to explain all this, and time permitting maybe also say something about a connection to wall-crossing, and/or a related discrete model called competitive erosion.