Programme

In General Relativity, spacetime is modeled by a four-dimensional Lorenz manifold (M,g) satisfying the Einstein equation, which relates the curvature of the Lorenz metric g to non-gravitational fields. Loosely speaking, a spacetime is spatially homogeneous if “its geometry evolves with time, but is the same at all contemporary vantage points”. More formally, a spacetime (M,g) is called spatially homogeneous if the manifold M is diffeomorphic to the product IxG of a time interval I=(a,b) by some three-dimensional real Lie group G, and the Lorenz metric writes g=-dtdt+h(t) where h(t) is a left-invariant riemannien metric on the Lie group G.

Spatially homogeneous spacetimes have drawn attention due to the combination of three facts :

1. heuristic arguments suggest that a “generic spacetime should behave as if it were spatially homogeneous in the vicinity of its initial singularity”,
2. spatially homogeneous spacetimes form the largest class of spacetimes for which the Einstein equation (which is a non-linear hyperbolic PDE in general) can be reduced to a system of ODEs,
3. the above-mentioned system of ODEs has a very interesting chaotic dynamics, suggesting that the geometry of the Universe should oscillate in a chaotic way as one approaches the initial singularity (the “Big-bang”).

The mini-course will focus on vacuum unimodular spatially homogeneous spacetimes. The purpose of the mini-course will be twofold. First explain how the vacuum Einstein equation can be reduced, in this context, to a polynomial ODE on a 4-dimensional real projective algebraic manifold. Then describe what is known and unknown concerning the (very rich and interesting) dynamics of this ODE.

We will give introductions to the topics of Kahler geometry, geometry on the space of Kahler metrics and geometric quantization. We will then see how the 3 topics interact strongly with each other and will describe some examples. Namely we will describe how quantization in so-called real polarizations can sometimes be related to the (easier to define) quantization in Kahler polarizations. Geodesics for a natural metric structure on the space of Kahler metrics play a central role in this relation.

To a mechanical system S, one associates its configuration space Conf(S), the set of all its states, i.e. its realizations in the ambient space, which may be for instance the Euclidean space of dimension 2 or 3. Examples of rigid systems are given by a pendulum with Conf(S) being a circle or a sphere (according to having 2 or 3 degrees of freedom). The next example is the solid pivoting around a fixed center for which Conf is the orthogonal group. We are interested here in “piecewise” rigid systems, i.e articulated systems, the simplest example of which is a double of more generally multi-pendulum (here Conf becomes a product of circles or spheres). Actually, these spaces also have a geometry given by a Riemannian metric encoded by their kinematic energy. The free evolution of the system is given by the geodesic flow of this metric. The lectures will revolve around the question of what topology, geometry and geodesic flow can occur in this way.

Einstein manifolds are the mathematical formulation of the Einstein equation with a non-vanishing cosmological constant. We introduce Einstein manifolds with skew torsion, explain their occurence in the context of special geometric structures on manifolds, and sketch their physical applications. This is joint work with Ana C. Ferreira.

In this talk, we will consider conformally flat globally hyperbolic lorentzian manifold, with a focus on the 3-dimensional case. I will present what can be expected for their classification, and also a way to produce many examples, through a lorentzian analog of “grafting”.

The talk is devoted to an interesting and rather unexpected relationship between some ideas and notions well known in the theory of integrable dynamical systems on Lie algebras and geometry of pseudo-Riemannian manifolds (holonomy groups, symmetric spaces and projectively equivalent metrics).

We study new examples of translating solitons of the mean curvature flow. We consider for this purpose manifolds admitting pseudo-Riemannian submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers the classical Euclidean examples. As an application, we completely classify the rotationally invariant translating solitons in Minkowski space, obtaining six types.

The geometry, topology and dynamics of globally hyperbolic anti-de Sitter spacetimes of dimension 2+1 are well understood since the groundbreaking work of Mess, through a description of their moduli space.
In higher dimensions, very little is known apart from some of their dynamical properties.
In this talk, we will discuss the possible topologies. Since they are globally hyperbolic, they are diffeomorphic to a product of a manifold M with the real line. In the first examples, M is a hyperbolic manifold.
With Jean-Marc Schlenker and Nicolas Tholozan, we constructed examples for which M is a Gromov-Thurston manifold, which is a class of closed manifolds with pinched negative curvature which are not hyperbolic.

In dimension at least 3, conformal maps of pseudo-Riemannian manifolds have strong rigidity properties. This suggests that manifolds with "large enough" conformal groups shall be quite rare, and maybe classifiable. Whereas a beautiful theorem of Ferrand and Obata confirms this idea for Riemannian manifolds, it appeared later that higher signatures show much more diversity. In this talk, I will give recent global rigidity results when the conformal group is assumed to contain a lattice of a simple Lie group of rank at least 2, such as SL(3,Z), in relation with recent progress on Zimmer's conjectures.

We will give two results on completeness of Lorentzian manifolds which are related to the completeness of Riemannian systems. The first one is that any compact manifold endowed with a linear connection of precompact holonomy is complete, extending a well-known consequence of Hopf-Rinow theorem (arxiv:1511.03605). The second one is a result about the completeness of relativistic pp-waves, which becomes equivalent to the completeness of a dynamical system in R^2. The latter provides a positive answer to the Ehlers-Kundt conjecture under a polynomial assumption (arxiv:1706.03855).

Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, by “transitioning” between different geometries. Danciger introduced a new such transition, which enables to deform from hyperbolic structures to Anti-de Sitter structure, going through another type of real projective structures called “half-pipe”, and provided conditions for a compact 3-manifold to admit a geometric transition of this type. By extending a construction of Kerckhoff and Storm, I will describe examples of finite-volume geometric transition in dimension 4. This is joint work with Stefano Riolo.

Let O(2,n) be the orthogonal group of a non-degenerate symmetric bilinear form <.,.> of signature (2,n) on R^{n+2} and let O_0(2,n) be the identity component of O(2,n). We prove that any Anosov representation of a word hyperbolic group into O_0(2,n) with negative limit set is the holonomy group of a spatially compact globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the spatial compactness needs a particular care. We introduce the space of causal geodesics C consisting of the timelike and lightlike geodesics of anti-de Sitter space and the lightlike geodesics of its conformal boundary: the Einstein universe. We prove that C is a manifold with boundary and that the boundary is homeomorphic to the space of lightlike geodesics of the Einstein universe. The spatial compactness is a consequence of the following theorem : Any Anosov representation acts properly discontinuously by isometries on the space of causal geodesics avoiding the limit set; besides, this action is c compact. This last result is stated in a general algebraic setting by O. Guichard, F. Kassel and A. Wienhard. One can see it as a Lorentzian analogue of the action of convex cocompact kleinian group on the complementary of the limit set in the hyperbolic space.

Can you parametrize the space of isometric embeddings of the hyperbolic plane into Minkowski 3-space? I'll give a partial result and conjectural answer, in terms of, equivalently, domains of dependence, measured laminations, or lower semicontinuous functions on the circle. Using the Gauss map and its inverse, I'll then interpret this result in terms of harmonic maps to the hyperbolic plane. Finally, I'll restrict to the case where the isometric embedding is invariant under a group action, and describe connections to Teichmuller space. This is all joint work with Francesco Bonsante and Andrea Seppi.

To a conformal harmonic map from the complex plane to the symmetric space SL(n,R)/SO(n) one can associate holomorphic differentials q_k of degree k=3, ..., n. We say that a harmonic map has polynomial growth if all such differentials are polynomials and cyclic if only q_n is non-zero . In this talk, we will describe the asymptotic geometry of the minimal surface associated to cyclic harmonic maps with polynomial growth when n=4. Moreover, these surfaces are images under the Gauss map of maximal surfaces in the pseudo-hyperbolic space H^2,2 and our result can be viewed in this context as the solution of an asymptotic Dirichlet boundary problem for maximal surfaces with light-like polygonal boundary at infinity.

The pseudo-hyperbolic space H^{2,n} is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of H^{2,n}, we construct find a unique complete maximal surface bounded by this circle. This construction relies on Gromov's theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.