The theoretical mathematicians in Münster have a long and successful record of joint research within two Collaborative Research Centres. More recently, applied mathematics in Münster has been developing strongly, building close interactions with the life sciences.
Several collaborations have been established within and between our research areas. We plan to enhance existing and spark new interactions in a systematic way. This will lead to a much greater transfer of knowledge, viewpoints and techniques between the different mathematical fields.
Great mathematical challenges we plan to address are a p-adic version of the Langlands programme relating number theory and representation theory, the conjectures of Baum–Connes and Farrell–Jones as central tools to gain geometric information about manifolds, structure-preserving approximations and asymptotics in mathematical modelling, and geometry-based model reduction in non-linear spaces with important applications in optimisation and computational geometry.
Our research focuses on three major areas, namely Arithmetic and Groups (A), Spaces and Operators (B), and Models and Approximations (C), which are unified by three approaches:
emphasising the underlying structure of a given problem, taking the geometric viewpoint when looking at problems and studying the relevant dynamics of group or semigroup actions.
Research Area A - Arithmetic and Groups - combines projects from arithmetic geometry, representation theory and model theory. Methods from group theory and algebraic geometry are an important common theme in all these areas. The major research objectives in this area are to discover hidden structures in arithmetic geometry and representation theory and to export methods of model theory and mathematical logic to other disciplines, and vice versa.
In Research Area B - Spaces and Operators - algebraic topology blends with differential and non-commutative geometry. Fundamental research topics are curvature, Ricci flow, rigidity, automorphisms of manifolds and C*-algebras. The major research objectives are to establish new structural results in differential geometry using evolution equations and to resolve central open questions in topology and C*-algebras.
Research Area C - Models and Approximations - combines projects from analysis of partial differential equations, differential geometry, calculus of variations, numerical analysis, model reduction, optimisation, and the theory of stochastic processes. The research in this area is mainly driven by applications from biology, medicine and physics. The major research objectives are to analyse structures in mathematical models, their asymptotics and dynamics and to utilise and control geometry in mathematical models and in their approximations.
The most important scientific objectives of Mathematics Münster are:
Discover hidden structures in arithmetic geometry and representation theory.
Export methods of model theory and mathematical logic to other disciplines, and vice versa.
Establish new structural results in differential geometry using evolution equations.
Resolve central open questions in topology and C*-algebras.
Analyse structures in mathematical models, their asymptotics and dynamics.
Utilize and control geometry in mathematical models and their approximations.