Berlin researchers in geometry have been very active in investigating the interplay of the two fields of differential geometry (studying smooth curves and surfaces like the solutions to many variational problems) and discrete geometry (studying polyhedral surfaces, like the typical representations used in computers). In particular, problems in mathematical visualization and geometry processing require novel discretization techniques in geometry.
Berlin Research Groups
The field of discrete differential geometry (DDG) can be said to be a Berlin speciality (Bobenko, Pinkall, Polthier, Sullivan, Suris). Among other things, it has been noted repeatedly that with proper choices of discretization for many smooth problems the discrete surfaces that arise are not merely approximations but share special structural properties of the smooth limits. For instance, for surfaces defined by integrable equations, the analogous discrete integrable systems shed great light on the essence of integrability. One can follow Whitney to develop discrete exterior calculus by requiring exact sequences at the discrete level. The related theory of discrete harmonic maps and discrete minimal surfaces can also be explained in terms of finite element methods and, therefore, directly links with numerical applications. Various techniques related to circle packings and circle patterns seem to give good notions of discrete conformal and holomorphic maps.
Such geometric research, focusing on curves and surfaces in low-dimensional space, has many practical applications in addition to its theoretical interest. Typical systems for computer-aided design (CAD) rely heavily on spline surfaces for geometric modeling and design. In contrast, geometry acquisition systems like 3D scanners provide point sets and simplicial meshes not yet suitable for CAD systems. Here the labor-intensive reverse engineering process can be simplified by switching from spline representations to novel methods from discrete differential geometry. Similarly the requirements of computer graphics -- both for real-time interactive games and for high-quality rendering of films -- provide a rich source of problems in geometry processing: how to efficiently manipulate digital representations of geometric structures.
Much of the Berlin research in DDG research in Berlin is conducted within the SFB/Transregio "Discretization in Geometry and Dynamics'' (coordinated by Bobenko). Generally, the term "discretization" refers to any procedure that turns a differential equation into difference equations involving only finitely many variables, whose solutions approximate those of the differential equation. The SFB/Transregio brings together scientists from the fields of geometry and dynamics, to join forces in tackling the numerous problems raised by the challenge of discretizing their respective disciplines..
The particular strength of this area in Berlin is also reflected by the MATHEON chairs "Mathematical Visualization'' (Sullivan) at TU and "Mathematical Geometry Processing'' (Polthier) at FU, and by the Visual Data Analysis department at ZIB.
Algebraic and geometric topology in Berlin is represented at FU by the Topology group (Reich, Vogt), whose active areas of research include the Farrell-Jones Conjecture for algebraic K-theory of group rings, with its connections to conjectures of Novikov, Bass, Baum-Connes, and Kaplansky and also to geometric group theory and moduli spaces of curves. Configuration spaces and equivariant topology and their application to problems from combinatorics and discrete geometry are also studied intensively in Ziegler's discrete geometry group.
Specific topics of research include:
Discrete integrable systems, quad nets and isothermic surfaces (Bobenko, Suris),
Conformal surface theory and constrained Willmore surfaces (Pinkall),
Special parametrizations of discrete surfaces (Polthier),
Global geometry of constant-mean curvature surfaces (Sullivan),
Polyhedral geometry and combinatorics (Ziegler),
Instrinsic Delaunay triangulations (Bobenko),
Hirota--Kimura discretization (Suris),
Configuration spaces and equivariant topology (Ziegler),
Algebraic K-theory of group rings (Reich),
Rigidity conjectures in geometric topology (Reich),
Geometric knot theory (Sullivan), and
Convergence of discrete curvatures (Polthier).
This two-semester sequence gives a graduate-level introduction to classical geometries, discrete differential geometry, and mathematical visualization. While noneuclidean and projective geometry are often taught at the undergraduate level at other universities, it is rare to find them at this advanced level. Discrete differential geometry is an area of special research interest in Berlin.
Outline of the contents:
◦ Projective geometry
◦ Spherical and hyperbolic geometry
◦ Möbius geometry
◦ Lie sphere geometry
◦ Basic structures in discrete geometry
◦ Polytope theory
◦ Combinatorial geometry
◦ Geometric combinatorics
Discrete differential geometry and visualization
◦ Discrete surfaces and polyhedral complexes
◦ Discretizations of mean and Gauss curvature
◦ Delaunay triangulations
◦ Subdivision surfaces
◦ Discrete Hodge theory
◦ Geometry processing
◦ Review of point-set topology
◦ Fundamental group, covering spaces, van Kampen theorem
◦ Homology and cohomology
◦ Cell complexes and manifolds
Spectrum of Advanced Courses
Convex Geometry; Computational Topology; Homotopy Theory; Stable Homotopy Theory; Equivariant Topology; Geometric Knot Theory; Mathematical Visualization; Topological Combinatorics; Discrete Riemann Surfaces; Tropical Geometry; Lattice Polytopes