With easy access to huge data sets (such as DNA sequences, gene expressions, or population data), biology increasingly becomes a quantitative science. Interpretation of these data requires complex mathematical models. This applies, in particular, to evolutionary biology and ecology. Mathematical methods are used to explain how emergent phenomena on the level of populations and ecosystems can be explained from fundamental processes of population genetics (such as mutation, selection, recombination) and population dynamics (birth- and death processes, interactions among species and with the environment). Research topics in our groups concern the causes of biological diversity, the analysis of adaptation processes, and speciation theory. We use ordinary and partial differential equations, stochastic processes, and statistical methods. Another focus is on approaches from game theory.

The dynamics of complex systems is of interest not only in biology, but also in many other applied fields, such as physics, meteorology, or economy. In many cases, the behavior is chaotic (sensitive to initial conditions), such that the explicit time-evolution is impossible to compute. The area of Dynamical Systems describes such systems and tries to understand them. In smooth ergodic theory, one considers the behavior of typical (in terms of an invariant measure) evolutions in order to make statistical statements, such as rates of mixing and the central limit theorem. Topics studied in our research groups include iterated maps, infinite measure preserving systems, and mixing rates for flows.

## Working Groups

PhD students and Postdocs are listed on the webpages of the two working groups.