The pure analysis programme focuses on analysis on manifolds, with a strong geometric flavor. On the one hand this encompasses analysis on Lie groups and on the other hand the study of dynamical systems.
Lie groups are equipped with the structure of a differentiable manifold for which the group operation is smooth. They appear in many situations in mathematics and physics, where continuous symmetries play a role. In such situations one is often interested in Fourier (or harmonic) analysis.The non-ommutative nature of Lie groups requires the description of harmonic analysis in terms of (often infinite dimensional) representation theory. The rich geometric structure of Lie groups allows one to develop a theory which at the same time is amazingly general and surprisingly concrete. Subjects of current research are: Plancherel en Paley-Wiener theorems for symmetric spaces, parameter dependence of representations, asymptotic behaviour of matrix cofficients, Radon transformation, cusp forms for symmetric spaces, symplectic geometry and convexity theorems.
Dynamical systems describe the evolution of deterministic behavior. This can be conservative (think of Hamiltonian mechanics) or dissipative. Typical examples come from physics (celestial mechanics, resonant circuit), chemistry (reaction-diffusion equations) biology (predator-prey systems) or economy (market models) --- this short list is far from exhaustive. In mathematical terms they often can be described by a vector field on phase space, preserving the inherent geometric structure. Given a concrete example (model), the aim is to describe as much of the dynamics as possible. This is helped by theoretical considerations --- one can only find what one is looking for. The emphasis in this specialisation is given to bifurcations, explaining how to pass from one robust regime to another.
Minimal requirements for a successful start in this programme are a good basic knowledge of the following subjects at the bachelor level:
- theory of series, in particular power series and fourier series;
- ordinary differential equations;
- basic theory of Banach and Hilbert spaces;
- analysis of several variables: implicit function theorem, submanifolds of Rn; substitution theorem for integration, a version of Stokes theorem;
- topology, in particular the notion of compactness;
For successful participation it is desirable to also have basic knowledge of differentiable manifolds, complex functions and the Lebesgue integral.
Courses in the Master programme are selected in close contact with a tutor, in order to get tailor made preparation for writing a Master's thesis.
- For specialization towards manifolds and Lie groups, a master course in either Lie groups, or both Lie algebras and differentiable manifolds is required.
- For specialization towards dynamical systems, a master course in dynamical systems is required.
In view of the geometric nature of the programme it is advisable to follow master courses in differential geometry, Riemannian or symplectic. For obtaining a good general background in analysis it is advisable to also follow master level courses in functional analysis, distribution theory, and/or partial differential equations.
The tutor will assist in finding an appropriate specialisation and thesis supervisor. In some cases it may be quite natural to specialise in subjects on the overlap with the programmes in applied analysis, geometry and topology, or mathematical physics.
Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.
For more information about this specialisation, please contact:
Prof. dr. Erik van den Ban