Research Interests

My main research interest is in partial differential equations (PDEs) displaying complex, even chaotic, spatiotemporal behaviour.  Such PDEs arise, for instance, in some models of surface growth, fluid instabilities and pattern formation.  I study these systems from a variety of perspectives, ranging from rigorous analysis, through asymptotics and numerical bifurcation analyses, to extensive numerical simulation and computation of statistics, in order to gain as complete an understanding as possible.  I am currently particularly interested in a model for short-wave pattern formation with Galilean invariance, which we have recently found to display unexpected - and still poorly understood - dynamics such as spatiotemporal chaos with strong scale separation, coexistence of chaotic and stable domains, and coarsening to chaos-stabilized fronts.

Detailed features of the solutions of these nonlinear PDEs are typically analytically inaccessible, but one can still obtain rigorous analytical estimates on global, long-time or averaged properties of the solutions.  It is especially helpful when numerical simulations can motivate the analysis, and vice versa.

I also work on rigorous estimates is in fluid dynamics, with a focus on deriving variational upper bounds on heat transport in turbulent convection; in this area, most of my attention has concentrated on whether the conductive properties of the boundaries significantly affect these bounds.

A topic I have recently become interested is the study in compartmental and network-based models in mathematical epidemiology.  A particular application is to modelling HIV disease transmission and treatment protocols, especially regarding the effectiveness of the expansion of highly active antiretroviral treatment in combating the HIV epidemic.