My research focuses on robust and adaptive methods for solving 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 analyze the global behavior of functions over entire regions of the space without sampling it. Adaptivity 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. Another result is an adaptive algorithm for generating guaranteed images of Julia sets.
I am also interested in programming languages and I am one of the designers of
the
Lua language.