# Mathematics

The Stochastics research centre works on statistics and probability theory. Research activity in probability theory focuses on the analysis of complex interactive stochastic systems occurring in mathematical physics and mathematical biology.

One focus of research is on systems and questions motivated by mathematical biology and in particular the theory of evolution and cell biology. It has long been known in population genetics that the development of species of different phenotypes is heavily influenced by stochastic fluctuations. For example, a given species is characterised by certain (biological) attributes. The individuals wander randomly in a given geographical structure. Each individual has a random lifetime after which it disappears from the population, possibly leaving behind a random number of children. The number of children depends on available local resources and the current size of the local population competing for those resources. In such systems, the main focus of interest is on which conditions allow individuals of different phenotypes to also coexist over the long term. This is the theme of a proposal with which the research group is participating in the DFG SFB/Transregio 12 “Symmetries and Universality in Mesoscopic Systems”, a joint project between mathematicians and physicists of the Universities of Bochum, Duisburg-Essen, Cologne and the LMU Munich. The research group is also describing the genealogies of the considered systems which code the interactions between the individuals. For this purpose, various topological aspects in spaces of trees are being investigated. The working group plays a leading role in the development of tools for stochastic analysis to study tree-valued Markov chains. These permit in particular analysis of limiting processes with an infinite number of nodes using martingale problems and Dirichlet forms.

In statistics, the main research topic relates to non-parametric statistics and corresponding applications. The focus here is on an approximation concept to evaluate the degree of approximation of stochastic models to given data. It can also be applied to reconstruct phylogenetic trees from DNA data. This is the topic of a new DFG Priority Programme on “Probabilistic aspects of evolution”. The group plans to contribute a proposal to this programme. The research is being conducted in close international cooperation (France, India, Israel, Canada and Singapore).