Biomathematics, dynamical systems, mathematical finance and probability

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.

  Biomathematics
  Dynamical Systems

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 It\^o'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. auch in zahlreichen anderen angewandten Bereichen wie der Physik, Meteorologie oder Wirtschaft. In vielen Fällen ist das Verhalten chaotisch (und hängt empfindlich von den Anfangsbedingungen ab), sodass es unmöglich ist, den zeitlichen Verlauf explizit zu berechnen. Das mathematische Forschungsgebiet der dynamischen Systeme beschreibt solche Systeme und versucht, sie zu verstehen. In der glatten Ergodentheorie wird das Verhalten typischer Zeitverläufe (im Sinne eines invarianten Maßes) betrachtet, um zu statistischen Aussagen wie Mischungsraten und Grenzwertsätzen zu gelangen. Zu den Themen, die in unseren Gruppen behandelt werden, gehören iterierte Abbildungen, unendliche maßerhaltende Systeme und Mischungsraten.


Wahrscheinlichkeitsrechnung und Finanzmathematik. Probabilistische Ideen und Techniken spielen nicht nur in der modernen reinen und angewandten Mathematik eine entscheidende Rolle, sondern auch in der Gesellschaft im Allgemeinen. Obwohl die mathematischen Objekte in diesem Bereich oft grob und fraktal sind, besteht die Herausforderung darin, dennoch eine Analyse mit ihnen durchführen zu können. Wir versuchen daher zu beschreiben, wie es möglich ist, einen stochastischen Prozess "reibungslos" auf einen anderen abzubilden (Theorie des optimalen stochastischen Transports), wie man mit solchen Objekten integrieren kann (stochastische Kalkulation, z. B. die berühmte Formel von It\^o) und wie man eine geometrische Beschreibung physikalischer Systeme an ihrem kritischen Punkt (Zufallsgeometrie) geben kann. Es bestehen weitreichende Verbindungen zu einer Reihe anderer Bereiche der reinen und angewandten Mathematik wie Analysis, Geometrie und Kombinatorik.