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Research

Gemeinsamer Forschungsschwerpunkt des Instituts für Mathematik ist die
Modellierung, Analyse und Simulation komplexer Systeme unter
Einbeziehung diskreter und kontinuierlicher, deterministischer und
stochastischer Strukturen. Damit bringt das Institut für Mathematik
Kompetenzen in den Bereichen Modelle, Materialien, Moleküle der Fakultät
ein, wobei traditionell die Bereiche Analysis, numerische Mathematik,
Stochastik und Optimierung besonders sichtbar und durch Neuberufungen
auch in der Zukunft stark vertreten sind, da Anwendungen mathematischer
Methoden und Ergebnisse in den Natur- und Wirtschaftswissenschaften eine immer größere Rolle spielen, auch im Hinblick auf die gemeinsamen
Forschungsinteressen der Fakultät.

Professorship of Algebra

• Prof. Rebecca Waldecker    

Professorship of Geometry

• Prof. Joachim Rieger    

Professorship of Applied Analysis

• Prof. Tomáš Dohnal

Professorship of Functional Analysis

• Prof. Nils Waterstraat

Professorship of Numerical Mathematics

• Prof. Martin Arnold    

Professorship of Numerics of Stochastic Differential Equations

• Prof. Raphael Kruse

Professorship of Variational Methods

• Prof. emer. Christiane Tammer    

Professorship of Applied Stochastics

• Jun.-Prof. Martin Redmann

Professorship of Didactics in Mathematics

• Prof. Kirstin Erath
• Prof. emer. Karin Richter    

Professorship of Optimisation

• Prof. Axel Kröner

Research Areas

Stochastic partial differential equations and differential equations with fractional time derivatives as well as fractional Fokker-Planck equations and theory of stochastic processes are in numerous ways connected to statistical physics, including homogenization techniques for the coarsening of probability measures. Furthermore, stochastic effects are an essential in theoretical polymer physics, especially within the framework of stochastic field theory. Theory of operator groups and operator semigroups, conservative diffusion processes, and diffusion processes on manifolds are particularly important for quantum mechanics, as is work on stochastic optimal control and on the stochastic principle of least action on (pseudo)Riemannian manifolds.

Numerical solution methods for these problems as well as symplectic integrators in molecular dynamics simulations are of great interest. Besides algorithmic aspects, great emphasis is placed on numerical stability with regard to physical conservation variables. Simulation of stochastic processes and systems with uncertain parameters requires research pertaining to fundamental aspects of Monte Carlo simulations for Markov processes and in stochastic approximation theory.

In applied analysis, one area of research concerns wave packets in complex nonlinear structures, such as plasmonic structures or photonic crystals. These can be described using rigorous asymptotic analysis that includes application of evolution equations as well as stationary problems for special patterns, such as solitary waves. Another major research interest lies in fluid mechanics and materials science equations, in particular free boundary value problems for multiphase flows in fluid dynamics.

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