Interfacial Pattern formation: A Progress Report

Interfacial patterns such as those which arise during dendritic solidification, viscous fingering and electrochemical deposition have long been of scientific and technological interest. Until recently, however, even the simplest steady-state structures have eluded theoretical treatment. In this talk, I review the progress made over the past several years towards remedying this situation. Specifically, I explain how the paradigm of microscopic solvability can be used to explain unique steady-state patterns and why secondary instabilities such as tip-splitting or sidebranching are inevitable. I focus on the Saffman-Taylor finger in a channel to illustrate the general framework