Research in this area focuses on the analysis of non-linear partial differential equations (PDEs) using a variety of techniques such as calculus of variations, dynamical systems techniques, energy and ladder methods, and geometry. The methods are used to analyse PDEs in various contexts such as non-linear elasto-statics, reaction-diffusion systems, dispersive wave equations, Navier-Stokes equations, delay equations and both dissipative and Hamiltonian equations. 

Ergodic theory deals with the probabilistic aspects of deterministic dynamical systems. For chaotic systems, one cannot predict particular trajectories over long time periods, but the average behaviour of typical trajectories is predictable. In this sense, laws of probability theory have counterparts in dynamical systems. Examples are the central limit theorem, mixing rates, return statistics and approximation by Brownian motion. Systems studied by our group include billiards, Lorentz gases, low-dimensional maps, systems with intermittency and piecewise isometries. Other interests are  thermodynamic formalism, stochastic tests for chaos and applications to signal processing. 

Our main focus in this area is a geometric approach to Hamiltonian systems and their manifestation in areas such as mechanics and fluid dynamics. Topics range from fundamental aspects of symplectic geometry to their applications, including bifurcation theory, structure preserving discretisations and data assimilation. Specific problems studied are, for example: N-body problems, Josephson junctions, weather prediction, oceanographic waves and satellite dynamics and control. For the applications, we collaborate with the Surrey Space Centre and we are a partner in the National Centre for Earth Observation (NCEO), leading the theme on data assimilation.

The focus of our statistical research is in Bayesian statistics, design of experiments and process control. Particular emphasis is given to medical and industrial applications of theoretical results. An important topic in the area of Bayesian statistics is degradation modelling and model choice diagnostics. Our work in experimental design concentrates on connectivity and robust design choice when observation loss is likely. 

We develop and analyse mathematical models and numerical simulations for many applications including biological systems and life sciences. Examples are population dynamics and epidemiology, industrial eco-systems, localised pattern formation, systems biology of TB, physiological and biological fluid flows, non-linear electronic systems, disease and pharmacological modelling,  reaction- diffusion and predator-prey systems, modelling in tumour growth, the carbon cycle, weather prediction and climate modelling, and surface chemistry.