Geometry.
Geometry studies topologic spaces with a supplementary structure: differential, riemannian, lorentzian, symplectic, holomorphic, algebraic, etc. Often based on physics, these structures in turn form the natural framework in which contemporary physics theories are formulated, especially when global phenomena are involved.
From a mathematical perspective, the structures concerned lead to as many branches of mathematics, which interact with each other in a multitude of ways and rely on a wide range of analytical, algebraic and topological techniques.
Thus, could be covered from the Riemannian geometry perspective, complex analysis, harmonic analysis and algebraic geometry. This applies more generally to complex geometry, which studies varieties defined by holomorphic functions.
Here are a few possible ideas for internships in these areas.
- Riemannian geometry: topology and bending.
- Gauge theory: the study of connections on a vector bundle, which constitute the natural framework of the Yang–Mills theory.
- Riemann surfaces:
- Hodge theory:
- Introduction to algebraic bending.
- Holomorphic functions of several variables and pseudoconvexity.
- Lamination and tissues.
Dynamical systems
The purpose of the dynamical systems is the study over a long period of a transformation acting on a space of configurations. Physical systems are often described by differential equations that lead to continuous flows and therefore a study of this kind. As a general rule, a differential equation cannot be integrated explicitly, we try to obtain qualitative information discretizing time. This leads us to consider discrete dynamic systems.
This two main categories are themselves devided into multiple subcategories depending on the nature of the transformation and the geometric structures it preserves (measurable, topological, differentiable, symplectical, holomorphic, algebraic). While this classification is of course very permeable, it does enable us to identify general concepts for each of the classes. But it is often the significant examples of new dynamic phenomena that ultimately guide the research. Here are a few ideas for internship topics.
- Symbolical dynamic. Space is a series of symbols from a finite alphabet, the change, difference of details. The properties of these series involve basic notions of ergodic theory such as entropy.
- Geodetic flow. It is a natural flow on all surface, which is particularly interesting to study when the bending is negative. Its dynamic enable to describe global geometric properties of the surface.
- Hamiltonian systems. These systems appear naturally in classical mechanics. Fine techniques allow either to build periodical orbits () or to describe their stability (KAM theorem).
- Dynamics in small dimension. The study of the family x⇒ax(1-x) on the interval [0,1] shows spectacular cascade bifurcations, whose study provides a good introduction to chaos, and to universality phenomena. This family spreads in dimension 2, and we can then observe strange attractors.
- Holomorphic dynamic. The thoery of iteration of polynomials in the complex plane is particularly rich, combining complex analysis and topology of plane compaxts. This theory is particularly visual, and many images of fractal can be obtained, whose interpretation is formidable.
- Algebraic dynamic. While the iteration of rational applications in any dimension is a field combining several dificult theories, some questions with digital aspects can be covered, such as the study of degree growth.
Course language: French
- Teaching coordinator: Krikorian Raphaël