My research focuses on robust and adaptive methods for the solution of problems in computer graphics and geometric modeling. Robustness means that I am interested in using computers to prove properties of curves and surfaces. This usually takes the form of solving equations in several variables. The main tools for achieving robustness are interval computation methods using interval arithmetic and affine arithmetic. Interval methods provide guaranteed numerical results that are not affected by rounding errors in floating-point computations. More importantly, interval methods allow us to analyse the global behaviour of functions over whole regions of the space without sampling it. Adaptiveness means that I want to concentrate the computational effort near interesting regions of the space, such as near a solution curve or in regions where the surface curvature is high. Global analysis with interval methods leads naturally to adaptive methods.
I am also interested in using computers to prove properties of non-linear dynamical systems. Again, interval methods are the main tools. One result of this research is a computer-aided proof that the Jouanolou foliation of low degree admits no nontrivial minimal sets. Recently, I have been working on adaptive algorithms for generating guaranteed images of Julia sets and fractal basins for Newton's method.
I am also interested in programming languages and I am one of the designers of the Lua language.