A discrete Schwarz minimal surface      

DFG Research Unit
Polyhedral Surfaces

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Project Heads

Alexander I. Bobenko (Coordinator)
Michael Joswig
Ulrich Pinkall
Konrad Polthier
John M. Sullivan
GŁnter M. Ziegler (Deputy coordinator)

Overview

The theory of polyhedral surfaces is a very concrete special case (and testing ground) of an active mathematical terrain where differential geometry (providing the classical theory for smooth surfaces) and discrete geometry (concerned with polytopes, simplicial complexes, etc.) meet and interact.

The Schwarz P minimal surface:
smooth Pinkall, Polthier, Rossman Goodman-Strauss, Sullivan Bobenko, Hoffmann, Springborn

The goal of the research group is to combine discrete and differential geometry in order to attack fundamental problems in the area of polyhedral surfaces. Areas to which we devote considerable joint effort include discrete surfaces of constant mean curvature (including minimal surfaces), discrete notions of curvature, cubical complexes (including quad-meshes and quad-surfaces), and the existence and rigidity of special kinds of polyhedral surfaces. While all of these problems have interest from the pure mathematics standpoint we adopt here, many are also motivated by questions from such diverse settings as computational geometry, mesh generation, and mathematical physics.

There are seven tightly related projects:

Project B1: Discrete differential geometry of surfaces: special classes and deformations
continues the previous project Discrete surfaces from circles and spheres.

Project B2: Geometry of discrete integrability
is carried over from the first period.

Project JZ: Non-positive curvature and cubical surfaces
continues the previous project Branched coverings and combinatorial holonomy.

Project P: Spectral curves of polygons and triangulated tori
is carried over from the first period.

Project Po: Discrete implicit surfaces
This is a new project.

Project S: Restricting valence for polyhedral surfaces and manifolds
continues the previous project Discrete minimal surfaces in the cubic lattice.

Project Z: Realization spaces of polyhedral surfaces
is carried over from the first period.

If you go to the projects page, you will get an impression of how these different projects are related.

Moreover there are several related application-projects within the DFG-Research Center Matheon, in particular Projects F1, F4 and F5.

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Emanuel Huhnen-Venedey . 02.01.2012.