- Classifying and Structuring Multidimensional Discrete Integrable Systems
- Principal Investigators: Yuri Suris, Alexander Bobenko
- Investigators: Raphael Boll, Matteo Petrera, Mats Vermeeren
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.
- The Search for the Optimal Contour
- Principal Investigators: 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 (continued as B08: Wigner crystallization)
- Understanding Crystallization
- Principal Investigators: Marco Cicalese, Gero Friesecke
- Investigators: Yuen Au Yeung, Annika Bach, Rufat Badal, Lucia De Luca, Dominik Jüstel, Gianluca Orlando, Franscesco Solombrino, Arseniy Tsipenyuk
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.
- Analyzing Discrete Curves of Steepest Descent in Discrete Energy Landscapes
- Principal Investigators: Oliver Junge, Daniel Matthes
- Investigators: Daniel Karrasch, Horst Osberger, Simon Plazotta, Benjamin Söllner, Jonathan Zinsl
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.
- A coherent theory for geometrically resolving singularities in time-discrete dynamical systems
- Principal Investigators: Christian Kühn, Yuri Suris
- Investigators: Maximilian Engel
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.