My field of work is chance. If we want to calculate what the daily high temperature might be in Berlin in a week’s time, it makes sense to include random variables in our mathematical model. The action of a random physical force can be modeled as white noise, which represents the idea of random influences with no prevailing direction. We refer to such models as random dynamical systems.
In simplified terms, a random dynamical system can be imagined as a set of particles that move under the influence of a random force field. In mathematics, we refer to qualitative changes in the long-term behavior of these particles that are determined by a continuous transition of parameter values as bifurcation - the focus of my research.
You can think of bifurcations as the railroad switches of chance. Depending on which track you take, you will observe a different long-term system behavior. The choice of track corresponds to the choice of system parameters. For certain parameters, one point in the system may attract all particles; for others, there may be multiple attractors. Analyzing bifurcations is important because they give us insights into the fundamental structure of a dynamic system.
One area where bifurcations can be observed in the real world is in ecology. When two species interact in the same ecosystem, one species may displace the other. Factors that are somewhat random, like the onset of the seasons, play a role in this. If winter sets in earlier or if the temperature behavior changes within the seasons, the population development of a species may suddenly find itself at a bifurcation switch – and develop in one direction or the other due to a new balance of forces.
Maximilian Engel and I are interested in bifurcations in random dynamical systems with absorption. The idea of absorption helps us deal with the fact that particles within a system can randomly move very far from their starting points over time. We limit our observations to particles that stay within a defined space, and those that leave it are absorbed. In this way, bifurcations in random dynamical systems can be meaningfully described and used in mathematical models. In the example of population change, the concept of absorption can help obtain biologically meaningful solutions through modeling and depict the extinction of a population.
As I see it, one of the goals of mathematics is to develop fundamental tools that can be used in a wide variety of fields. However, the path to application is often very long. In our project, we first need to lay the theoretical foundations. This means we must first develop a theory for bifurcations in random dynamical systems with absorption.
Ideally, the resulting mathematics will be so robust that it can help solve problems that were previously unsolvable and understand processes that are not yet understood. For me, this is one of the cultural achievements of mathematics.