Active Projects of the SFB Transregio 109

B02: Discrete Multidimensional Integrable Systems
  • Classifying and Structuring Multidimensional Discrete Integrable Systems
  • Alexander Bobenko, Yuri Suris
  • 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.

B09: Structure Preserving Discretization of Gradient Flows
  • Analyzing Discrete Curves of Steepest Descent in Discrete Energy Landscapes
  • Daniel Matthes, Oliver Junge
  • 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 as well as to desingularization of space-discretizations of fast-slow partial differential equations.

B11: Geometric rigidity in spin systems
  • Marco Cicalese, Barbara Zwicknagl
  • The project B11 aims at developing a deeper understanding of typical low-energy states and to derive coarse-grained continuum theories of variational lattice spin theories. One of the main difficulties in this respect is the lack of understanding of elementary low-energy singularities such as vortices or line defects in spin systems. Our goal is to address these issues for simple but relevant lattice models. For that, we will combine methods from different areas: direct methods in the calculus of variations, abstract methods of Gamma-convergence and scaling laws analysis via domain branching constructions.

B12: Coarse cohomological models of dynamical systems
  • Ulrich Bauer, Oliver Junge
  • Many processes (aka dynamical systems) in nature or technology exhibit complicated behavior. In particular, it is often inherently impossible to reliably predict future states of the process over long time spans. Reliable predictions, however, can be made on statistical or otherwise coarsened properties of the system. In this project, we construct coarse models of the behavior of a system which, in contrast to existing approaches, incorporate information about cycling motion, generalizing the classic notion of periodic or quasiperiodic dynamics.