The fields of research I have been working in are ordinary differential equations and dynamical systems, in particular, singular perturbations, invariant manifolds and applications in numerical analysis.
Applications are in the numerical integration of differential equations.
From a numerical point of view singularly perturbed systems are a model class for so-called stiff systems which are difficult to treat. The question is which numerical integration methods are suitable for stiff systems and, in particular, which methods preserve the geometric property of the continuous system, i.e., admit an attractive invariant manifold. These results allow, e.g., to derive global error bounds of stiff integration schemes, to describe the behaviour of stiff integration schemes near a hyperbolic equilibrium and to prove the existence of closed hyperbolic invariant curves for the discrete systems. Another question of interest in this area is how to approximate an invariant manifold numerically in an efficient way. A topic also related to invariant manifolds is the reduction of high dimensional systems of differential equations to lower dimensional ones which still possess the essential properties.