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5. Geometry, topology, and visualization |
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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.
The new field of discrete differential geometry (DDG) can be said to be a Berlin speciality (Bobenko, Pinkall, Polthier, Sullivan, Suris). The many related strands of research in DDG have been demonstrated at the series of Oberwolfach workshops in the area (2006, 2009, 2012) and in two new books on the subject. 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.
The particular strength of this area in Berlin is reflected by the DFG Research Unit Polyhedral Surfaces (coordinated by Bobenko and Ziegler), by the MATHEON chairs "Mathematical Visualization'' (Sullivan) at TU and "Mathematical Geometry Processing'' (Polthier) at FU, and by the visualization group at ZIB (Deuflhard, Hege). A proposal for a new SFB "Discretization in Geometry and Dynamics'' (Transregio with TU München) is currently under review.
Specific topics of research in geometry 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),
geometric knot theory (Sullivan),
and convergence of discrete curvatures (Polthier).
Algebraic topology in Berlin has been strengthened by recent hiring at FU (Reich). Active areas of research include the Farrell--Jones Conjecture for algebraic K-theory of group rings, with its connections to conjectures of Bass, Baum--Connes, and Kaplansky and also to geometric group theory. Topological restrictions on equivariant maps (with applications related to the square peg problem) have also been studied (Ziegler).
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