Plenary Talks and Mini-Course
Martin Taylor
The black hole stability problem (Mini-Course, 4 lectures)
I will discuss some aspects of the stability problem for the Schwarzschild and Kerr families of black hole spacetimes in general relativity. I will focus on highlighting the interplay between geometric properties of the background solutions and analytic statements such as dispersion of waves, which plays a central role in the stability analysis.
Roland Steinbauer
Singularity Theorems Beyond C² : A Status Report
With their singularity theorems, Penrose and Hawking have firmly established singularity formation as a generic feature of General Relativity. However, the original proofs rely on the metric tensor being at least C²-regular, which undermines the physical significance of the results: incompleteness might be avoided by a physically harmless drop in regularity, and the theorems might merely predict a rough, but otherwise nonsingular, geometry. While this caveat has long been known, it is only during the last decade that real progress has been made in lowering the regularity in the singularity theorems. In this talk we summarize these developments, which build on (a) extensions of causality theory and (b) extensions of focusing estimates for causal geodesics under (c) distributional or synthetic curvature bounds / energy conditions. Our focus will be on two recently established versions of the Hawking theorem. The first, for Lipschitz metrics, rests on a worldvolume estimate derived from a segment-type inequality, which bounds the volume of the set of points on a spacelike surface from which long maximisers emanate. The second, for W^{1,p}-metrics with L^p-bounded curvature, relies on the RT-regularisation, which uses elliptic theory to raise the regularity of the metric by one derivative. Both approaches employ manifold convolution to regularise the metric, combined with Friedrichs-type estimates controlling the curvature of the regularised metric in terms of the distributional curvature.
Ana Cristina Ferreira
Isometries of 3-Dimensional Semi-Riemannian Lie Groups
Let G be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric g. By definition, the isometry group Isom(G, g) contains G itself, acting by left translations. It turns out that, generically, Isom(G, g) is actually equal to G, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs (G, g) whose full isometry group Isom(G, g) has dimension greater than or equal to four. As a consequence, we determine, for every pair (G, g), up to automorphism and scaling, the dimension of Isom(G, g), which can be three, four, or six. (Joint work with S. Chaib and A. Zeghib).
Eric Ling
Cosmological singularities à la Penrose
Cosmological singularities—characterized by past causal geodesic incompleteness—are largely attributed to the foundational work of Stephen Hawking, beginning with his 1966 doctoral thesis. A critical assumption of Hawking's singularity theorem is the strong energy condition, or timelike convergence condition. Although satisfied for classical matter models, this energy condition is violated in spacetimes dominated by a positive cosmological constant—a regime supported by current observational measurements of our universe. In this talk, I present a cosmological singularity theorem in four spacetime dimensions (joint with G. Galloway) that circumvents this issue. By utilizing Penrose's original 1965 singularity theorem, our result relies only on the null energy condition, which is not affected by a cosmological constant. The proof makes use of the positive resolution of the virtually Haken conjecture (specifically the positivity of the virtual first Betti number) for closed three-manifolds. I will discuss some related rigidity results of this theorem which is joint work with C. Rossdeutscher, W. Simon, and R. Steinbauer. Lastly, I will conclude with partial results regarding the extension of this theorem to higher dimensions.
Lilia Mehidi
Rigidity of locally conformally homogeneous Lorentzian manifolds
The isometry group of a compact Riemannian manifold is a compact Lie group. In contrast, its conformal group need not be compact. In fact, there is only one exception: the standard sphere. This was conjectured by Lichnerowicz and proved by Ferrand and Obata in the 1970s. Equivalently, apart from the sphere, the group of conformal symmetries of a compact Riemannian manifold becomes a group of isometries after a conformal change of the metric. In Lorentzian geometry, one can ask for a similar rigidity principle. The expectation is that if a compact Lorentzian manifold has a conformal group which is not the isometry group of any metric in the conformal class, then its geometry must be very special; in fact, it should be conformally flat. This is the Lorentzian version of the Lichnerowicz conjecture. Generalizing earlier work of Alekseevsky and Galaev, we show that, in the locally homogeneous setting, this conjecture reduces to the study of compact manifolds locally modeled on certain deformations of Minkowski space, known as plane waves. This is joint work with Leistner and Zeghib.
Philippe LeFloch
The theory of torus-symmetric spacetimes with finite energy
I will present a series of papers in collaboration with Bruno Le Floch (LPTHE, Sorbonne), in which we study the Einstein equations under T2 symmetry on T3, allowing vacuum, scalar-field, and compressible-fluid matter models, including isothermal and polytropic fluids. We develop a non-perturbative global existence and stability theory at a weak regularity level where the connection coefficients are merely square-integrable, thus allowing both impulsive gravitational waves and shock waves. In areal gauge, we formulate the Einstein--Euler system as a first-order system of nonlinear balance laws with constraints and entropy structure, leading us to propose the notion of a tame Einstein--Euler flow. We prove global existence for large-data Einstein--Euler flows, together with nonlinear stability for well-prepared initial data and nonlinear instability for oscillatory data. We also obtain global geometric and compactness results in a broader Lorentzian setting under positive energy conditions alone. In the future-contracting regime, the areal foliation reaches a geometric singularity generically, whereas in the future-expanding regime it is complete.
Zoe Wyatt
Cancelled