I work on nonlinear partial differential equations in mathematical models from the applied sciences. In particular I am interested in models that exhibit sharp fronts or free boundaries. Such models appear in a wide variety of applications, notably in combustion models for flame fronts, the lubrication approximation for thin viscous flows, Darcy flows and other processes in porous media, and harmonic map flows in the context of nematic liquid crystals. The central theme is the global behaviour of solutions, the possible formation of singularities and blow up phenomena. The interaction between rigorous methods and formal methods is essential for progress in this area.