# Research Areas

### Algebra

#### Representation Theory (Prof. Dr. Rolf Farnsteiner), Geometric Group Theory (Prof. Dr. Richard Weidmann)

The current focal points in Algebra are Arithmetic Algebraic Geometry, Geometric Group Theory and Representation Theory.

### Analysis

#### Functional Analysis (Prof. Dr. Markus Haase), Abstract Harmonic Analysis and Operator Algebras (Prof. Dr. Hannes Thiel), Complex Analysis and Complex Dynamics (Prof. Dr. Walter Bergweiler)

Analysis as a mathematical discipline reaches back to the seminal works of Newton and Leibniz and is based on the idea to reduce global statements about mathematical objects to local (and even "infinitesimal") ones. Techniques and results of analysis are essential for our modern, scientific-technological civilization. In Kiel, one finds the following special research areas: Harmonic Analysis in theory and applications (a generalization of Fourier series), Functional Analysis, in particular its applications to evolution equations and ergodic theory, and Complex Analysis, in particular complex dynamical systems.

### Geometry

#### Differential geometry and Lie groups (Prof. Dr. Jens Heber), Differential geometry, Geometric analysis (Prof. Dr. Hartmut Weiß)

In ancient times geometry had its roots in measurement problems. As a mathematical discipline differential geometry nowadays is concerned with measurable quantities of curves, surfaces and higher-dimensional spaces. Apart from numerous applications it provides the basis of our modern view of the universe. Current research topics in Kiel are the interplay between local and global structure of spaces and their symmetries on the one hand, and the geometry of moduli spaces of geometric structures on manifolds on the other.

### Logic

#### Axiomatic set theory, descriptive set theory (Prof. Dr. Otmar Spinas)

Mathematical logic analyzes the structure of mathematical reasoning. In the 19th century it evolved into an independent subject and then shattered mathematics by the discovery that there is no proof of its own consistency and that the independence phenomenon must be accepted as a third option besides true and false. Moreover, logic provides the theoretical foundations of computer science. In Kiel, research is being done in the area of set theory. A main topic is the construction of models for the mathematical universe, by which it is possible to prove consistency or independence of mathematical propositions.

### Numerical Analysis and Optimization

#### Numerical Mathematics (Prof. Dr. Malte Braack), Scientific Computing (Prof. Dr. Steffen Börm), Discrete Optimization (Prof. Dr. Anand Srivastav)

The numerical mathematics groups in Kiel develop and analyze methods for the solution of differential and integral equations, optimal control problems, and stationary and transient processes. The focus is on discretization methods and the resulting linear or non-linear systems of equations.

The mathematical optimization group in Kiel focuses on discrete optimization, graph theory, and algorithms for big data problems with overlap to complexity theory, probability theory, and harmonic analysis.

In all research groups the development, the analysis, and the implementation of new efficient mathematical methods on high-performance platforms are central topics. The applications cover problems from physics, chemistry, biology, marine science, and engineering.

### Probability, Statistics and Mathematical Finance

#### Probability Theory and Mathematical Statistics (Prof. Dr. Mathias Vetter), Mathematical Finance (Prof. Dr. Jan Kallsen), Stochastic Processes (Prof. Dr. Sören Christensen)

The mathematical analysis of stochastic phenomena dates back to the 17th century when questions about gambling were considered. Nowadays probability theory and statistics are concerned with all kinds of random experiments or uncertain events, and they have become an important tool in the natural and the social sciences as well as in medicine and economy. The research group in Kiel focuses mostly on the evolution of stochastic phenomena in time, both in theory and in statistical or numerical applications, with a particular focus on questions involving mathematical finance.

*Please note:* For the department Didactics of mathematics please switch to Leibniz-Institut für die Pädagogik der Naturwissenschaften und Mathematik (IPN).