Prof. Dr. Christian Seis, Angewandte Mathematik Münster: Institut für Analysis und Numerik

The Research Training Group is dedicated to educating mathematicians in the field of complex random systems. It provides a strong platform for the development of both industrial and academic careers for its graduate students. The central theme is a mathematically rigorous understanding of how probabilistic systems, modelled on a microscopic level, behave effectively at a macroscopic scale. A quintessential example for this RTG lies in statistical mechanics, where systems comprising an astronomical number of particles, upwards of 10^{23}, can be accurately described by a handful of observables including temperature and entropy. Other examples come from stochastic homogenisation in material sciences, from the behaviour of training algorithms in machine learning, and from geometric discrete structures build from point processes or random graphs. The challenge to understand these phenomena with mathematical rigour has been and continues to be a source of exciting research in probability theory. Within this RTG we strive for macroscopic representations of such complex random systems. It is the main research focus of this RTG to advance (tools for) both qualitative and quantitative analyses of random complex systems using macroscopic/effective variables and to unveil deeper insights into the nature of these intricate mathematical constructs. We will employ a blend of tools from discrete to continuous probability including point processes, large deviations, stochastic analysis and stochastic approximation arguments. Importantly, the techniques that we will use and the underlying mathematical ideas are universal across projects coming from completely different origin. This particular facet stands as a cornerstone within the RTG, holding significant importance for the participating students. For our students to be able to exploit the synergies between the different projects, they will pass through a structured and rich qualification programme with several specialised courses, regular colloquia and seminars, working groups, and yearly retreats. Moreover, the PhD students will benefit from the lively mathematical community in Münster with a mentoring programme and several interaction and networking activities with other mathematicians and the local industry.

In this project, the evolution of toroidal bubble vortices is to be investigated. Bubble vortices are special vortices that occur in two-phase fluids. A typical and fascinating example is an air bubble ring in water created by dolphins or beluga whales. The underlying mathematical model is given by the two-phase Euler equations with surface tension. One major goal is a thorough mathematical construction of steady rings that move without changing their shape, and of perturbations of these. Such traveling waves are known for the classical Euler equations, but their existence is unknown for surface tension dependent models. Of particular interest is the role of the surface tension for the shape of the ring, which will be investigated. A second objective of this project is to understand how the effect of surface tension can be exploited to rigorously justify certain nonlinear motion laws of one or more interacting bubble rings. The understanding of such motion laws for the classical Euler equations is poor, and it is expected that the regularising effect of surface tension helps to mathematically tame the problem.