Ali Feizmohammadi (University College London)

Title: Inverse problems for hyperbolic equations with time-dependent coefficients

Abstract: We present two results for recovery of time-dependent coefficients or the wave equation in Lorentzian geometries, from boundary measurements. In the first result we study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold. The second result is concerned with the study of semi-linear wave equations in globally hyperbolic Lorentzian geometries. We show that it is possible to uniquely recover zeroth order terms from boundary measurements of solutions to these semi-linear equations.

Anne Franzen (Técnico Lisboa)

Title: The wave equation near flat Friedmann-Lemaître-Robertson-Walker and Kasner Big Bang singularities

Abstract: We consider the wave equation, \(\square_g\psi=0\), in fixed flat Friedmann-Lemaître-Robertson-Walker and Kasner spacetimes with topology \(\mathbb{R}_+\times\mathbb{T}^3\). We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface \(\{t=0\}\). The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate \(L^2\)-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the \(L^2(\mathbb{T}^3)\) norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

This is joint work with Artur Alho and Grigorios Fournodavlos

Dejan Gajic (University of Cambridge)

Title: Resonances on asymptotically flat spacetimes

Abstract: A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary asymptotically flat spacetimes. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the stationary spacetime, much like the harmonic frequencies of a vibrating string. These frequencies are called resonances. I will introduce a new method for characterizing and studying resonances for wave equations on asymptotically flat spacetimes, based on joint work with Claude Warnick.

Claudia Garetto (Loughborough University)

Title: Hyperbolic systems with multiplicities and non-diagonalisable principal part

Abstract: In this talk I present some recent results, obtained in collaboration with Michael Ruzhansky (QMUL/Gent) and Christian Jäh (Göttingen), for hyperbolic systems with multiple variable multiplicities and non-diagonalisable principal part. We discuss the well-posedness of the corresponding Cauchy problem and we provide a qualitative analysis of the solution.

Gustav Holzegel (Imperial College London)

Title: The non-linear stability of the Schwarzschild family of black holes

Camille Laurent (Sorbonne Université)

Title: Stabilization for the semilinear wave equation with and without geometric control condition

Abstract: In this talk, we will deal with the stabilization of the damped semilinear wave equation. The damping is assumed to be active in a zone satisfying different geometric conditions. We will state some results of stabilization in some situation where the linear damped equation satisfies either the usual geometric control condition or some weaker assumptions where the trapped set is small. The unique continuation will be an important step. This is a joint work with Romain Joly (Grenoble).

Matthieu Léautaud (Université Paris-Sud)

Title: Controllability of linear waves from an interior hypersurface

Abstract: We study regularity, observability and controllability of linear waves from an interior hypersurface. We prove controllability under the assumption that all generalized bicharacteristics intersect the surface transversally. We deduce lower bounds for the Cauchy data of Laplace eigenfunctions on a hypersurface. This is joint work with Jeffrey Galkowski.

Oliver Lindblad Petersen (Universität Hamburg)

Title: Compact Cauchy horizons in vacuum spacetimes

Abstract: Moncrief and Isenberg conjectured in 1983 that any compact Cauchy horizon in a smooth vacuum spacetime is a smooth Killing horizon. They have proven the conjecture (in dimension 3+1), under the assumptions that the spacetime metric is analytic and the generators are “non-ergodic”. In this talk, we prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth (as opposed to analytic) vacuum spacetime is a smooth Killing horizon. The method relies on new energy estimates and Carleman estimates for wave equations close to compact Cauchy horizons.

Bruno Vergara (Universität Zürich)

Title: Carleman estimates and boundary observability for waves with critically singular potentials

Abstract: In this talk I will present a novel family of Carleman inequalities on cylindrical spacetime domains featuring a potential that is critically singular, diverging as the inverse square of the distance to the boundary. These estimates, which we prove using geometric multiplier arguments that generalize the classical Morawetz inequality, are sharp in the sense that they capture both the natural boundary conditions and the natural \(H^1\)-energy. Quantitative uniqueness properties such as the boundary observability for the associated wave equations will be discussed too. This is based on joint work with A. Enciso and A. Shao.