# Research

The research activities of the group *Dynamical Systems, Geometry and Mathematical Physics*
cover a broad and diverse spectrum of subjects in the fields of fundamental, applied and computational dynamical systems theory, classical, statistical and quantum systems and their interfaces in the light of dynamics, and theoretical and applied aspects of geometry with many connections to dynamical systems theory.

## General description

The interest of dynamical systems theory is the behaviour of systems that evolve in time. This first of all concerns the long-term behaviour which comprise stationary, periodic, multi-periodic and chaotic dynamics, but also transient behaviour is of interest. Moreover bifurcations or transitions between asymptotic states -- in particular transitions between regular and chaotic motions -- under variation of parameters are of great importance. We develop mathematical tools using methods from analysis, geometry and measure theory to grasp, study and develop the structures involved. Moreover, we develop methods to detect and understand the dynamics in specific models, employing numerical and graphical tools and computer algebra. Many applications are from the field of mechanics. This concerns the motion of point masses like planets and their satellites in celestial mechanics, and also the motion of atoms and molecules which again can be described as point masses or rigid bodies. Here also relativistic or quantum effects may play a role. This is a wide area with great outreach, also in the direction of life sciences. If the number of constituent particles is huge then such systems are best described by statistical means. Statistical mechanics deals with the question of how global observables, like temperature, can be explained from the microscopic behaviour. There is a close relationship with dynamical systems theory in particular with regard to random and chaotic behaviour and the so-called non-equilibrium systems. Mathematical physics is the encompassing discipline of all the above and still larger areas of theoretical physics. The research is very geometrical in nature. It involves many tools from differential and computational geometry where the research in the field of geometry has many facets also in its own right.