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 allows 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.