The general theme of this school lies at the intersection of geometry and dynamics, focusing on the interactions between Riemannian and Lorentzian geometries on the one hand, and dynamical systems on the other. For instance, a Lorentzian, Riemannian, or more generally semi-Riemannian metric naturally gives rise to dynamical objects such as:
• the action of its isometry and conformal groups,
• the geodesic flow (a differential equation on the tangent bundle), and
• the Ricci flow (an evolution PDE on the space of all metrics).
In recent years, the field has witnessed several important advances. We mention only three representative examples:
1. Conformal groups of compact Lorentz manifolds. Lichnerowicz’s conjecture asserts that if the conformal group of a compact Lorentz manifold is essential (i.e. strictly larger than the isometry group, even after conformal rescaling), then the metric must be conformally flat. K. Melnick and V. Pecastaing proved this conjecture in the analytic, simply connected case (J. AMS, 2022).
2. Completeness of compact Lorentz manifolds. The geodesic vector field defines a differential equation on the tangent bundle. Compactness, combined with suitable geometric hypotheses, may imply geodesic completeness—that is, solutions of the ODE exist for all time. This was recently established by L. Mehidi and A. Zeghib (J. Diff. Geom., to appear) for Brinkmann spacetimes, a class of polarized Lorentzian structures that generalize, among others, the physically significant gravitational wave spacetimes.
3. Chaos in cosmology. F. BĂ©guin and T. Dutilleul provided a (partial) proof of chaotic dynamics in Bianchi cosmology (Comm. Math. Phys., 2023, 190 pages). Bianchi dynamics is a kind of geometric flow, similar to Ricci flow, in a homogeneous setting.
The aim of the proposed courses is to equip participants with the foundational knowledge needed to understand these and related results. We believe the program offers both unity and diversity, through the following contributions:
• Suhr’s course: Introduction to Lorentz geometry.
• Mehidi’s and Sanchez’s course: Study of certain Lorentzian spaces (e.g. Plane waves) with a focus on completeness.
• Monclair’s course: Investigation of special globally hyperbolic Lorentzian manifolds modelled on anti-de Sitter space, and the hyperbolic dynamics of their fundamental groups on boundary spaces.
• Béguin’s course: Analysis of globally hyperbolic spacetimes admitting a free isometric spacelike action of a 3-dimensional group. In this case, the Einstein equations reduce to a polynomial ODE on a 5-dimensional space; the course emphasizes its qualitative analysis and the proof of chaotic behavior.
• War’s course: Rigidity properties of the geodesic flow on Riemannian surfaces. Here Lorentzian geometry is implicit, since Anosov geodesic flows preserve a Lorentz metric (albeit with limited regularity).
• Melnick’s course: Cartan connections as powerful tools for the study of conformal groups and, in particular, for addressing Lichnerowicz’s conjecture.
• Pecastaing’s course: Introduction to tools that will be particularly useful in connection with Melnick’s and Monclair’s lectures.
The courses are designed to be accessible to both Master’s and Doctoral students, and can be adapted, if necessary, to match the average level of the participants. While we emphasize some recent advances related to the theme of this school, the core topics — such as conformal transformations, completeness problems, and the Einstein equation — are fundamental to geometry and dynamical systems.
Our instructors, who are not only experts but also contributors to these recent developments, will present these themes through the broader (and long) history of the field.