# Specialization "Biomathematics"

## Teacher training programme

Biomathematics is not contained explicity in the curriculum for the teacher training programme. However, simple models, for instance from population dynamics, ecology or epidemiology, are perfectly suited for illustrating interesting mathematical applications, for example recurrence relations or differential equations. This area is also wonderful for learning mathematical modelling. In principle bachelor and master's theses can be assigned, but you should get in touch with your prospective supervisor in time about the necessary prior knowledge.

## Bachelor programme

There are no compulsory courses on biomathematics in the bachelor programme but there is an elective module, which is typically offered every second summer semester:

- In the elective module biomathematics and game theory selected concepts and models from biomathematics (in particular from evolution, genetics, ecology, and epidemiology) and game theory (for example prisoner's dilemma, zero-sum games, Nash equilibrium, evolutionary game theory) are presented and illustrated with examples.

For students choosing this elective module there is an abundance of interesting topics for possible bachelor theses; for other students the possibilities are restricted but do exist, in particular with a good knowledge of differential equations and probability theory. Students intending to choose the specialization biomathematics in the master programme should choose the elective module biomathematics and game theory even though it is not a requirement for the master programme. Further particularly recommendable elective modules in view of the master programme in biomathematics are applied statistics or mathematical modelling.

## Master programme

In the master programme "biomathematics" in one of 7 areas of specialization. The basic courses in the specialization biomathematics consists of the following 5 compulsory modules:

- The theory of discrete-time and continuous-time Markov chains is the centre of the module stochastic processes, in particular the classification of states and the long time behaviour. Furthermore special Markov chains, for example random walks and branching processes, and various applications are covered. This course should be taken in the first semester of the master programme since only prior knowledge on probability is necessary but not measure theory. Moreover knowledge about stochastic processes is needed in mathematical population genetics and in advanced courses.
- The module mathematical population genetics deals with the basic models describing the evolution of the genetic composition of a population under processes like selection, mutation, recombination, or random drift. These models, which describe the change of gene frequencies in the course of many generations, are formulated with the aid of differential equations, difference equations, or Markov chains.
- The model mathematical ecology is devoted to models of population growth, models of the interaction of different species (for instance predator-prey or host-parasite interaction, competition for resources) but also models of pattern formation (for example Turing mechanism, morphogenesis, chemotaxis). Likewise the analysis of (systems of) differential equations plays a central role.
- In the module game theory the students are introduced to the mathematical foundations of game theory. It does not deal with board games or gambling but with the modelling of situations of decision-making involving several participants that follow different strategies. The most important applications arise in economy and biology (evolutionary game theory). Among other subjects, the prisoner's dilemma, zero-sum games, Nash equilibria and replicator equations are dealt with.
- In the module Seminar: Biomathematics you have to complete two seminars, for example Seminar (Biomathematics) or Seminar (Mathematical population genetics), and an introductory seminar on one of the compulsory modules. Further seminars and introductory seminars can be taken as advanced courses.

The module stochastic processes is offered every winter semester, whereas the modules mathematical population genetics, mathematical ecology, and game theory are offered in a three semester cycle (ecology in WS2015, game theory in SS2016, population genetics in WS2016 etc.).

The offer of advanced courses for the master programme is closely linked to the research interests of the faculty members working in this area. Usually advanced courses are offered following one of the compulsory basic courses but also other topics from biomathematics, such as systems biology or pattern formation, are taught.

The master's thesis should be connected to one of the advanced courses but other topics can be assigned as master's theses as well. Students should get in touch with possible supervisors as soon as possible, preferably at the end of the first semester.

## Doctoral programme

In principle all advanced courses for the master programme can be chosen as courses for the doctoral programme privided that they have not been credited for the master programme. The recognition of courses for the doctotal programme will be specified individually in an agreement ("Dissertationsvereinbarung").

Topics for a doctoral thesis are usually assigned from the areas of research of the supervisors. Students who are interested in writing a doctoral thesis should get in touch with a possible supervisor as soon as possible, i.e., before completing the master programme.