Active Projects of the SFB Transregio 109

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. Here, we investigate and classify multidimensional discrete integrable systems.

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.

B10: Geometric desingularization of non-hyperbolic iterated maps
  • A coherent theory for geometrically resolving singularities in time-discrete dynamical systems
  • Christian Kühn, Yuri Suris
  • Singularities are ubiquitous in dynamical systems. They often mark boundaries between different dynamical regimes and also serve as organizing centers for the geometry of phase space and parameter space. In this project, we aim to extend geometric desingularization methods developed in the context of continuous-time systems to various classes of discrete-time maps.