The mathematical models that arise in physics, chemistry, biology and economics, often have an interesting geometry. Differential equations may for example exhibit symmetries, a variational or Hamiltonian structure, or a particular spatial coupling. The aim is to investigate the impact of such geometric properties on the behavior of the model.
Our expertise is in reaction diffusion equations, spatial probability, lattice models, integrable systems, Hamiltonian ODEs and symplectic maps.
A prime example of a geometric approach occurs when one applies the topological theory of braids and knots in low-dimensional dynamics. A typical result that can be obtained with these tools, is that the presence of certain types of periodic orbits of a surface map, implies that the map is chaotic. A related question concerns the famous Weinstein conjecture: under what conditions does the Hamiltonian dynamics on an energy manifold contain periodic behavior? Using the theory of homological linking, we have studied this question on unbounded energy surfaces.
To understand the complex orbits of high-dimensional maps and the spatially extended solutions to PDEs, it is more than desirable that the above topological methods are supported by numerical calculations. To that end we combine interval arithmetic with strong topological tools such as Conley index theory and Floer homology.
As an example of such high-dimensional models, we are interested in lattice models that occur in theoretical physics, including twist maps and ferromagnetic crystals. It is important to understand how these systems transfer energy between degrees of freedom. To answer this question, we use for instance Aubry-Mather theory and bifurcation analysis.
In recent years, the problem of classifying integrable nonlinear PDEs has been reformulated in the language of algebraic structures. Automorphic Lie algebras form one of these structures that we now study in detail.