Projects
Active Projects of the SFB Transregio 109
A01: Discrete Riemann Surfaces
 Investigating the Facets of Discrete Complex Analysis
 Alexander Bobenko, Ulrike Bücking, Boris Springborn

Riemann surfaces arise in complex analysis as the natural domain of holomorphic functions. They are oriented twodimensional real manifolds with a conformal structure. Several discretizations of Riemann surfaces exist, e.g., involving discretized CauchyRiemann equations, patterns of circles, or discrete conformal equivalence of triangle meshes. Project A01 aims at developing a comprehensive theory including discrete versions of theorems such as uniformization, convergence issues and connections to mathematical physics.
A02: Discrete Parametrized Surfaces
 Developing a Theory of Discrete Surfaces with Constant Mean Curvature
 Tim Hoffmann, Alexander Bobenko

In recent years, an exhaustive theory has been developed to understand and construct discrete minimal surfaces. We aim to produce something similar for the construction and classification of discrete surfaces with constant mean curvature (cmc). In particular, for discrete minimal surfaces the study of Koebe polyhedra that serve as Gauß maps has been fruitful  we are interested in analoga for discrete surfaces with constant mean curvature.
A03: Geometric Constraints for Polytopes
 Exploring the Subtle Interplay of Geometry and Combinatorics
 Günter Ziegler, Raman Sanyal, Carsten Lange

Polytopes are solid bodies bounded by flat facets. Alternatively, they can be described as the convex hull of their vertices. Thus a polytope can be presented by information on two aspects, a geometric one: "What are the coordinates of the vertices" and a combinatorial one: "Which vertex is incident to which face". There are many interrelations between these two levels: Combinatorial requirements enforce restrictions on the geometry, and vice versa. A03 studies aspects of this interplay.
A05: Conformal Deformations of Discrete Surfaces
 The Construction of a Discrete Differential Geometry Version of Certain Conformal Deformations
 Ulrich Pinkall

Two geometries can be considered equivalent if there exists an angle preserving transformation between them; this is a so called conformal transformation. In the smooth case, conformal equivalences are quite well understood. However, mimicking their construction in the discrete case brought up not only interesting properties and algorithms, but also interesting problems  first and foremost the question of how to construct conformal deformations with certain prescribed properties.
A11: Secondary fans of Riemann surfaces
 Study the combinatorics and geometry of polyhedral fans whose cones correspond to ideal tesselations
 Michael Joswig, Boris Springborn

A famous construction of Gelfand, Kapranov, and Zelevinsky associates to each finite point configuration in $\mathbb{R}^n$ its secondary fan, which stratifies the space of height functions by the combinatorial types of coherent subdivisions. A completely analogous construction associates to each punctured Riemann surface a polyhedral fan, whose cones correspond to the ideal tessellations of the surface that occur as horocyclic Delaunay tessellations in the sense of Penner's convex hull construction. We suggest to call this fan the secondary fan of the punctured Riemann surface. The purpose of this project is to study these secondary fans of Riemann surfaces and explore how their geometric and combinatorial structure can be used to answer questions about Riemann surfaces, algebraic curves, and moduli spaces.
A12: Ropelength for periodic links
 Explore the idea of ropelength in periodic entanglement
 John M. Sullivan, Myfanwy E. Evans

Ropelength is a mathematical model of tying a knot or link tight in real rope: we minimize the length of a curve while keeping a unitdiameter tube around the curve embedded. We have previously developed a theory of ropelength criticality; this allows explicit descriptions of critical configurations of links like the Borromean rings and the clasp. These configurations, whose exact geometry is quite intricate, are conjectured to be minimizers, but only known to be critical. This project will move beyond threedimensional Euclidean space to study the ropelength problem for links in a flat threetorus. These of course lift to triply periodic structures in Euclidean space, which can include both compact and noncompact components.