Faculty of Mathematics Research

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Polyhedral combinatorics

Polyhedral combinatorics

In polyhedral combinatorics, combinatorial objects are described using geometric methods so that we can better understand or optimize them. This is one of the topics of the research group Discrete Optimization.

Changepoint-Analysis

Changepoint-Analysis

In many cases, data are collected in order to analyse the time dependent dynamic behaviour of an underlying stochastic process. It is particularly interesting to detect changes - so-called structural breaks or change-points - in the time series. Current examples can be found in neurosciences and climatology, but also in quality control and financial engineering.

Efficient designs can be used to increase the precision and meaning of statistical analyses of experiments, and to use available resources as effectively as possible. We investigate the optimal design problem with analytical, combinatorial and algorithmic methods.

Partial differential equations

Partial differential equations

Many processes and equilibrium states in geometry, engineering and the natural sciences are described by differential equations. These are investigated by the research group "Non-linear Partial Differential Equations and Geometric Analysis". To illustrate this work the figure shows a numerical simulation of a model from elasticity theory.

Finite Element Methods

Finite Element Methods

Many processes in nature, engineering and life sciences can be described by differential equations. These can be solved analytically only in special cases. The research group "Numerical Analysis" is concerned with the analysis, development and implementation of stable finite element methods for the discretization of partial differential equations.

Combinatorial methods play an important role in modern algebra and geometry. In addition, sophisticated methods for counting lattice points also have applications in optimization and statistics.

The research group "Numerical Mathematics" deals with solving partial differential equations numerically. The focus here is mainly on hyperbolic conservation equations and population balance. Problems such as image reconstruction in computer tomography are also investigated.

Algorithmic optimization

Methods of mixed-integer non-linear optimal control developed by us help to solve challenging problems in road traffic, engineering, business and medicine. We are currently working intensively on improving our understanding of disease dynamics, for example in cardiac rhythm disorders and blood cancer.

Many Optimal Objects in Discrete Mathematics are constructed using finite bodies.

Fluid-Structure Interactions

Fluid-Structure Interactions

In order to describe the dynamics of an aircraft, a coupling of flow equation and elastic structural equation must be treated. The overall model is a nonlinear system of partial differential equations. These are investigated and solved numerically in the working group Numerical Mathematics in Applications.

Uncertainty quantification

Uncertainty quantification

Statistical inference is composed of two steps, namely estimation, and empirical uncertainty quantification. The latter, which amounts to the construction of a confidence set around the estimated parameter, is crucial for quality assessment and decision making. In high or infinite dimensional models, there is often the paradox that while model adaptive estimation is possible, model adaptive uncertainty quantification is impossible.

Research

The Faculty of Mathematics is home to four institutes.

Institute of Algebra and Geometry

Head:
Prof. Dr. Nill

Research Groups:

Algebra, combinatorics and their applications

Didactics of Mathematics

Discrete Mathematics

Discrete, convex and toric geometry

Institute of Analysis and Numerics

Head:
Prof. Dr. Simon

Research Groups:

Non-linear Partial Differential Equations and Geometric Analysis

Numerical Analysis

Numerical Mathematics

Numerical Mathematics in their Applications

Institute of Mathematical Optimization

Head:
Prof. Dr. Sager

Research Groups:

Algorithmic optimization

Discrete optimization

Geometry of optimization

Institute of Mathematical Stochastics

Head:
Prof. Dr. Carpentier:

Research Groups:

Mathematical Statistics and Applications

Mathematical Statistics and Machine Learning

Statistics and their applications

Last Modification: 01.03.2018 - Contact Person: Dr. Burkhard Thiele