In Geometric Modeling, mathematical tools for the explicit description of geometric objects are developed and analyzed. Here the focus is on complex structures, as they arise in various applications: One may think of a car body, a piece of cloth, or a dinosaur in an animated film. The surfaces considered in Differential Geometry and Geometric Modeling typically have a fairly complicated structure. For processing, it is necessary to approximate them in a function space of reduced complexity, say a spline space. For that reason, the development of tools for efficient approximation of geometric objects is an important task, giving rise to interesting mathematical questions in the field of multivariate approximation theory.

Classical Differential Geometry and Geometric Analysis studies surfaces minimizing geometric functionals for which examples arise in the sciences and engineering. In the simplest case, say for a biological cell, a bounding surface encloses a given volume in such a way that the area is minimized. Other interfaces minimize functionals involving curvatures, for instance the Willmore functional. Critical points are characterized by Euler equations which are non-linear partial differential equations. The mathematical goal is to establish new solutions or properties of solutions, not only in Euclidean three-space but also in other Riemannian ambient spaces, using methods of analysis and Riemannian Geometry.