## Biomathematics, Dynamical Systems, Mathematical Finance and Probability

This key research area comprises a broad spectrum of applied mathematical research with a unifying theme to describe the deterministic or stochastic dynamics of complex systems.

Biomathematics. With the ready availability of large amounts of data (such as DNA sequences, gene expression or population data), biology is increasingly becoming a quantitative science. The interpretation of this data requires complex mathematical models. This is especially true for evolutionary biology and ecology, but also for systems biology. Mathematical methods are used to explain phenomena within single cells and organisms (e.g., reaction-diffusion dynamics), but also at the population or ecosystem level (evolutionary dynamics driven by mutation, selection, and drift, or population dynamics due to birth and death and interactions with the environment). Research topics in our groups concern cellular processes, metabolic networks, the causes of biological diversity, and the analysis of adaptation and speciation. We use ordinary and partial differential equations, stochastic processes, and statistical and computational methods.

Dynamical systems. The dynamics of complex systems is not only of interest in biology, but also in numerous other applied areas such as physics, meteorology or economics. In many cases, the behavior is chaotic (depending sensitively on the initial conditions), making it impossible to calculate the time history explicitly. The mathematical research field of dynamical systems describes such systems and attempts to understand them. In smooth ergodic theory, the behavior of typical time courses (in the sense of an invariant measure) is considered in order to arrive at statistical statements such as mixing rates and the central limit theorem. Topics covered in our groups include iterated mappings, infinite measure-conserving systems, and flow mixing rates.

Probability and mathematical finance. Probabilistic ideas and techniques play a crucial role not only in modern pure and applied mathematics, but also more generally in society at large. Although the mathematical objects arising in this field are typically rough and fractal, the challenge is to be able to nonetheless perform analysis and/or geometry with them. We therefore try to describe how it is possible to "smoothly" map one stochastic process into another (this is the theory of optimal stochastic transport), how to integrate with respect to such objects (stochastic calculus, exemplified by Itô's famous formula), and how to give a geometric description of physical systems at their critical point (random geometry).  There are extensive connections to a number of other parts of pure and applied mathematics such as analysis, geometry and combinatorics.

## Working Groups

Biomathematics
Dynamical Systems
Mathematical Finance and Probability