This research is inspired both by the central role of dynamical systems in modern mathematics and by the fact that they arise throughout the sciences as models for such diverse phenomena as the spreading of diseases, the weather system and satellite positioning. Dynamical systems are usually described by ordinary differential equations (in finite dimensions) or partial differential equations (in infinite dimensions), but more general dynamical systems (maps, delay equations, stochastic aspects) are also considered.
A non-exhaustive selection of current, thriving research topics is:
1. Dynamics on networks: the dynamics of systems with a network structure, such as neuronal networks, metabolic pathways and power grids, display extraordinary dynamical behaviour such as synchronization. We study the relation between the architecture of a network and its dynamics.
2. Applications in systems biology: in a thriving collaboration with the systems biology group, we study bifurcation theory for metabolic pathways, optimization problems, stochasticity in signal transduction and community microbiology.
3. Computational dynamics: we develop methods to capture rigorously important aspects of dynamical systems based on computer calculations, using both functional analysis and computational Conley index theory. This leads to computer-assisted theorems uncovering properties of dynamical systems not accessible to traditional pen-and-paper analysis only.
4. Emergence of patterns: the interaction of a large number of entities, each of which behaves according to rather simple but nonlinear rules, often lead to unexpected emergent patterns. We develop methods to study the dynamics of such patterns, which range from atomic lattices in a composite materials to eddies in the atmosphere and oceans.
5. Topological methods: Morse-Conley-Floer theory uncovers deep relations between dynamical systems and partial differential equations on the one hand, and geometry and topology on the other. We develop this theory further, in both analytic and computational settings, and study its implications for specific applications, such as traveling waves in PDEs, and finite- as well as infinite-dimensional Hamiltonian dynamics (including braided and knotted orbits) inspired by mathematical physics.
6. Dynamics and data: data can be used to calibrate and improve dynamic models (data assimilation). On the other hand, ideas from dynamical systems can be deployed to extract information from data sets (e.g. time series). One example is the use of topological data analysis (which studies the "shape of data") to describe how the topology of a data set changes quantitatively over time.