Stochastics: evolving networks and nonlinear dynamic processes

The research group on probability theory focuses on networks for modelling complex systems with many interdependencies. Interactions in complex systems are often represented by networks. One problem inherent to this approach is that in real-world problems, the structure of those networks changes with time due to interdependencies. The processes themselves influence the topology of the network, as connections disappear and new ones appear. At the same time, the structure of the network also has a considerable impact on the processes. Within the scope of multiple DFG-funded projects, Professor Anita Winter’s research group has developed analytical and probability theory methods that facilitate the mathematically rigorous description of both effects. In the project ‘Evolving Pathogen Phylogenies: a Two-Level Branching Approach’, funded under the DFG Priority Programme ‘Probabilistic Structures in Evolution’, the researchers developed a model for describing virus populations influenced by cell division processes, in particular, their long-term behaviour. 

Additional projects have been launched in the same field. They are taking place within the scope of the DFG Priority Programme ‘Random Geometric Systems’. One of its members, Dr Anton Klimovsky, is an early-career researcher from our faculty. He is contributing an independent project to the programme.

The research group on applied stochastics has secured funding for various projects within the scope of their membership of the CRC ’statistical Modelling of Nonlinear Dynamic Processes’. Their work also focuses on similar mathematical structures that may be applied to problems in financial mathematics as well as hearing acoustics. One of the group’s recent projects has produced new results on variance reduction in Markov chains. Variance reduction methods are important tools for reducing complexity in simulation-based, numerical algorithms, such as various Monte Carlo methods. They are also widely used in Bayesian statistics and machine learning.

Issues in machine learning are also a key topic of other research groups that focus on numerical methods, such as Professor Martin Hutzenthaler’s group (stochastics) and Professor Johannes Kraus’s group (numerics).