a discrete minimal surface
My research lies in the field of discrete differential geometry and its applications in visualization as well as its connections to the theory of integrable systems. The main subject of discrete differential geometry is investigating discrete analogues of objects of classical differential geometry. Choosing "good" discretizations can preserve the integrable structure that lies behind many of the well known types of special curves, surfaces, parametrizations etc. This in turn leads to very stable and "well behaving" discretizations that preserve the integrability. These discrete integrable equations are of interest in mathematical physics.
The discrete nature of the objects makes them ideal for mathematical visualization and computer graphics as well and indeed mathematical visualization has always been a key ingredient in research in discrete differential geometry. While the visualization helps finding structures, the discretization of objects and notions of classical differential geometry is useful in computer graphics and visualization in itself: The applications here range from providing and investigating notions of curvature for polyhedral surfaces
A discrete version of the smoke
over variational methods to smoothen noisy mesh data to discrete versions of physical behaviours.
They can be used for modeling or simulation of deformations ( e.g. elastic rods) or time evolutions (the above shown evolution of vertex filaments in a perfect fluid).
I am co-developer and member of the steering committee of the software package jReality (www.jreality.de), a Java library for doing interactive mathematical and scientific visualizations and experiments, that scales from web applications to fully immersive CAVE-like 3D environments. It currently has stereo projection installations at Technische Universität Berlin and City College New York. A third installation at Technical University Munich is in the planing stage.
A surface in a 3d scene
These environments allow for highly interactive experiments and applications.