B01: Complexification of Discrete Time (completed)
- How Complex Time Helps to Make Algorithms and Simulations More Stable
Time may not always be considered to be a linear flow. If a time dependent problem is regular enough one may replace a one-dimensional (real) time variable by a two-dimensional (complex) time variable. By this it suddenly becomes possible to bypass critical singular or nearly-singular situations and to broaden the practical applicability of simulations. The research goal of B01 is to exploit these detours in time for problems in dynamic geometry and elementary particle mechanics.
- Classifying and Structuring Multidimensional Discrete Integrable Systems
- Principal Investigators: 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 (continued as B03: Numerics of Riemann-Hilbert Problems and Operator Determinants)
- The Search for the Optimal Contour
- Principal Investigators: Folkmar Bornemann
- Investigators: Georg Wechslberger
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.
- Impact of Discretization for Integrable Hamiltonian and Nonholonomic Dynamics
- Principal Investigators: Yuri Suris
Continuous time dynamical systems have to be discretized in order to find solutions numerically using a computer. This generally leads to a perturbation of the original dynamical behaviour. In particular, the long-term evolution might change drastically, e.g., regular behavior may turn into chaotic behavior. This raises the question of how to control the corresponding discrepancies. The research goal of B4 is to study this question for integrable and nonholonomic systems, both from quantitative and qualitative perspectives, exploiting the detour of embedding discrete dynamics into the dynamics of a non-autonomously perturbed continuous time system.
B06: Potential Energy Surfaces (completed)
- Discretizing the Forces for Molecular Quantum Dynamics
Potential energy surfaces are at the origin of the analytic description of molecular quantum dynamics and they give insight in it. B06 explores the accessibility of these high-dimensional structures for simulations of chemical processes which are fundamental for our understanding of basic principles in nature.
B07: Lagrangian Multiform Structure and Multisymplectic Discrete Systems (continued as B02: Discrete Multidimensional Integrable Systems)
- Investigating integrability of variational systems
- Principal Investigators: Yuri Suris, Matteo Petrera
- Investigators: Raphael Boll
Integrability is a remarkable property shared by some dynamical systems. Its understanding allows to study more complicated non-integrable systems by means of perturbative techniques. Integrability is, on the one hand, a beautiful mathematical laboratory, on the other hand, a rigid constraint. Such rigidity is reflected in the existence of some characterizing geometric features of the space where integrable systems live. The project B07 is devoted to the study of discrete integrable systems of variational origin, namely coming from variational principles. The systems under investigation offer a sort of magnifying lens on the building blocks of integrability. Standard integrability features, as for instance existence of conserved quantities, turn out to be consequences of some more fundamental structures.
B08: Crystallization, Defects, and Lattice Elasticity (continued as B08: Curvature Effects in Molecular and Spin Systems)
- Understanding Crystallization
- Principal Investigators: Marco Cicalese, Gero Friesecke
- Investigators: Rufat Badal, Lucia De Luca, Dominik Jüstel, Franscesco Solombrino
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: Discrete Gradient Flow in Mass Transportation Metrics (continued as B09: Structure Preserving Discretization of Gradient Flows)
- Analyzing Discrete Curves of Steepest Descent in Discrete Energy Landscapes
- Principal Investigators: Oliver Junge, Daniel Matthes
- Investigators: Daniel Karrasch, Horst Osberger, Simon Plazotta, 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.