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 two-dimensional real manifolds with a conformal structure. Several discretizations of Riemann surfaces exist, e.g., involving discretized Cauchy-Riemann 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 unit-diameter 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 three-dimensional Euclidean space to study the ropelength problem for links in a flat three-torus. These of course lift to triply periodic structures in Euclidean space, which can include both compact and noncompact components.

B02: Discrete Multidimensional Integrable Systems
  • Classifying and Structuring Multidimensional Discrete Integrable Systems
  • Yuri Suris, Alexander Bobenko
  • In recent years, there has been great interest among differential geometers for discovered discrete integrable systems, since many discrete systems with important applications turned out to be integrable. However, there is still no exhaustive classification of these systems, in particular concerning their geometric and combinatoric structure. This is the goal of B02.

B03: Numerics of Riemann-Hilbert Problems and Operator Determinants
  • The Search for the Optimal Contour
  • Folkmar Bornemann
  • Riemann-Hilbert Problems (RHP) are another way of expressing equations satisfying a special property and have some advantages over the traditional forms. Take for example an equation describing the motion of a water wave and its current state: Both the traditional form and the RHP form of the equation enables us to calculate the state of the wave at any point in time. But with the RHP form we can accomplish this without knowing or calculating anything about the state of the wave in between.

B08: Curvature Effects in Molecular and Spin Systems
  • Understanding Crystallization
  • Marco Cicalese, Gero Friesecke
  • Many basic phenomena in solid mechanics like dislocations or plastic and elastic deformation are in fact discrete operations: small breakdowns of perfect crystalline order. The goal of this project is thereofore to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization.

B09: Structure Preserving Discretization of Gradient Flows
  • Analyzing Discrete Curves of Steepest Descent in Discrete Energy Landscapes
  • Oliver Junge, Daniel Matthes
  • Many evolution equations from physics, like those for diffusion of dissolved substances or for phase separation in alloys, describe processes in which a system tries to reach the minimum of its energy (or the like) as quickly as possible. The descent towards the minimum is restrained by the system’s inertia. In this project, we discretize these evolution equations by optimizing discrete curves in a discrete energy landscape with respect to a discrete inertia tensor.

C01: Discrete Geometric Structures Motivated by Applications and Architecture
  • Geometry Supporting the Realization of Freeform Architecture
  • Alexander Bobenko, Helmut Pottmann, Johannes Wallner
  • Many of today's most striking buildings are nontraditional freeform shapes. Their fabrication is a big challenge, but also a rich source of research topics in geometry. Project A08 addresses key questions such as: "How can we most efficiently represent and explore the variety of manufacturable designs?" or "Can we do this even under structural constraints such as force equilibrium?" Answers to these questions are expected to support the development of next generation modelling tools which combine shape design with key aspects of function and fabrication.

C02: Digital Representations of Manifold Data
  • Felix Krahmer, Gitta Kutyniok
  • Whenever data is processed using computers, analog-to-digital (A/D) conversion, that is, representing the data using just 0s and 1s, is a crucial ingredient of the procedure. This project will mathematically study approaches to this problem for data with geometric structural constraints.

C03: Shearlet approximation of brittle fracture evolutions
  • Massimo Fornasier, Gitta Kutyniok
  • A brittle material, subjected to a force, first deforms itself elastically, then it breaks without any intermediate phase. A model of brittle fractures was proposed by Francfort and Marigo, where the displacement is typically a smooth function except on a relatively smooth jump set determining the fracture. This approach has the advantage not to require a pre-defined crack path, but has the drawback that any mesh discretization is a geometrical bias. Despite encouraging numerical results, FEM are expected to retain a certain geometrical bias and so far a proof of convergence of the proposed algorithm remains out of reach. Differently from finite elements, shearlets are frames with rotation invariance and optimal nonlinear approximation properties for the class of functions, which are smooth except on smooth lower dimensional sets. In this project we intend to compare anisotropic and adaptive mesh refinements with adaptive frame methods based on shearlets. In particular, by taking advantage of the shearlet property of optimally approximating piecewise smooth functions, we aim at reaching not only a proof of the convergence of the frame adaptive algorithms but also their optimal complexity.

C04: Persistence and Stability of Geometric Complexes
  • Ulrich Bauer, Herbert Edelsbrunner
  • The following more detailed questions are studied within this project: The definition and construction of geometric complexes from data. Topological persistence. The homology of dynamical systems. The convergence of variants of Crofton's formula obtained with persistent homology to compute intrinsic volumes. The approximation of persistent homology through simplification of the representative complexes.

C05: Computational and structural aspects of point set surfaces
  • Implementation of Manifold Structures in Point Clouds
  • Konrad Polthier
  • Point set surfaces have a more than 15 year long history in geometry processing and computer graphics as they naturally arise in 3D-data acquisition processes. A guiding principle of these algorithms is the direct processing of raw scanning data without prior meshing. However, a thorough investigation of a differential geometric representation of point set surfaces and their properties is not available. Inspired by the notion of manifolds, we develop new concepts for meshless charts and atlases and use these to establish sound formulations of discrete differential operators on point set surfaces. On this solid basis of meshless differential operators, we develop novel algorithms for important geometry processing tasks, such as feature recognition, filtering operations, and surface parameterization.

CaP: Communication and Presentation
  • Research on Transporting Maths Content from the SFB to the Public
  • Alexander Bobenko, Jürgen Richter-Gebert, Günter Ziegler
  • Images from geometry can serve as a visual pathway into mathematics. In the SFB they are leading in particular into pioneer research. This is to be taken as an opportunity: We would like to learn more about how mathematics can be communicated by means of visual content. And we would like to communicate new results from the SFB.