In our consecutive master's program in mathematics, you will be introduced to current topics from the research of our professors, building on the basic knowledge from your previous bachelor's program. Depending on your interests, you can combine courses from different areas and set focal points. In addition, you have the opportunity to take a substantial number of courses from other subjects and thus further individualize your studies.
The master's programme in Mathematics consists of:
- Main modules (18 ECTS)
- Specialization modules (14 ECTS)
- Mathematical elective modules (27 ECTS)
- Mathematical or non-mathematical elective modules (27 ECTS)
- Seminar (4 ECTS)
- Master's thesis (30 ECTS)
Main modules focus on the fundamentals of the corresponding specialization areas. Specialization modules are more advanced courses that build on the main modules and prepare you for writing your master’s thesis. The thesis content will consequently be connected to the chosen specialization area. All other modules in the master's programme can be counted as electives. It is also possible to have advanced modules from the bachelor's programme credited towards your master's degree (if they have not been credited towards your bachelor's degree). This option allows you to catch up on courses and content you may have missed during the bachelor's programme.
On a regular basis, we offer main and specialization modules in the following
areas
Algebra
The main modules in "Algebra" are usually from the field of real algebraic geometry. You study geometric objects using mainly algebraic methods, which can be described in "concise", but sufficiently general language. Unlike classical algebraic geometry, real algebraic geometry covers inequalities as well as equations. You also look for real solutions instead of complex ones. These aspects are of great advantage for the modelling of application processes.
The specialization modules in "Algebra" vary widely: from representation theory to invariant theory to algebraic model theory and logic. In contrast to typical lectures on these topics, you may operate in real fields, too. Real solid figures are also used if possible. While the algebraic methods used in the main module "Real algebraic geometry" are of a general nature, the focus is now often on aspects that allow the methods to be customized for specific applications. Such aspects are, for example, "sparse matrice" and "symmetry".
Analysis
In the analysis master's courses, you analyze partial differential equations from different points of view and with different methods, with a focus on well-posedness, different solution terms, stability and long-term asymptotics.
The main module "Theory of partial differential equations II" deals with evolution equations using modern methods, such as semigroup theory, Sobolev spaces and energy methods. Building on this, you can attend specialization modules that focus on the research area of the respective lecturer. Typical examples include lectures on fluid and electrodynamics, nonlinear waves (Freistühler), coupled thermo-elastic systems (Racke), stochastic partial differential equations and parabolic boundary value problems (Denk). These are complemented by more methodologically oriented specialization modules such as dynamical systems, hyperbolic boundary value problems, evolution equations in semi-infinite domains, pseudo-differential operators and quadratic forms.
Differential geometry
The goal of the courses in differential geometry is to study evolution equations such as the mean curvature flow, for which you also need courses on partial differential equations.
The lecture "Differential geometry II" covers abstract manifolds. You can also attend analysis lectures on partial differential equations on L2-theory, maximum principles, Schauder theory and Krylov-Safonov estimates. You can continue both areas with lectures on the graphical mean curvature flow and on Monge-Ampère equations related to Gaussian curvature. We also offer lectures on differential topology, algebraic topology as well as calculus of variations. These lectures take place on an irregular basis.
Geometry
The main modules in geometry are currently mostly identical with the algebra main modules, i.e. from the area of real algebraic geometry, which is described under "Algebra". The more specialized geometry lectures build on these main modules and include: tropical geometry, toric varieties, tensor geometry, geometry of linear matrix inequalities, positive polynomials and tame geometry. So you will use convex sets in an algebraic-geometric context in many ways, with connections to convex optimization and a wide variety of applications.
Numerics
For information on more in-depth content in the area of optimization, please visit the homepage of the WG Numerical Optimization.
Statistics
You can choose between different main and specialization modules in statistics, for example:
Mathematical statistics II (4+2 hours, 9 ECTS):
The first part of the lecture is an introduction to weak convergence on metric spaces, with applications to stochastic processes and functional limit theorems. The second part introduces the theory of statistical estimation. Based on decision theory, we discuss methods for assessing the quality of estimation procedures and methods for constructing estimators.
Time series analysis (4+2 hours, 9 ECTS)
You will be systematically introduced to the theory of time series analysis. Special emphasis is placed on understanding the mathematical principles and the importance of modelling for data analysis. The spectral representation of stationary processes leads to an elegant theory in Hilbert space of square-integrable random variables. We discuss parametric and nonparametric inference and prediction methods at the time and frequency levels.
Multivariate statistics (2+2 hours, 5 ECTS)
The lecture gives an introduction to the theory of multivariate normal distribution. Having learnt the distribution theory, we discuss problems of estimating and testing the actual parameters, as well as problems of variance and regression analysis. The second part introduces the theory of functional data analysis.