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This talk will explore Hitchin functionals and the Einstein coindex of cohomogeneity one nearly Kähler 6-manifolds. Nearly Kähler manifolds are Riemannian 6-manifolds admitting real Killing spinors. They are the cross-sections of Riemannian cones with holonomy G2. Like the Einstein equation, the nearly Kähler condition has a variational interpretation in terms of volume functionals, first introduced by Hitchin in 2001. The existence problem for nearly Kähler manifolds is poorly understood and the only currently known inhomogeneous examples were found in 2017 by Foscolo and Haskins using cohomogeneity one methods. For one of their examples, we establish non-trivial bounds on the coindex of the Hitchin-type and Einstein functionals. We do this by analysing the eigenvalue problem for the Laplacian on coclosed primitive (1,1)-forms under a cohomogeneity-one symmetry assumption.

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The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. Among other works, the talk describes contributions by H Friedrich, P Hintz and A Vasy, and my recent joint work with G Fournodavlos.

Our goal consists in developing a commutation vector field approach for the solutions to the massless Vlasov equation on the exterior of a Schwarzschild black hole. We would like an approach that verifies two properties. First, it has to provide decay estimates for the energy flux induced by the energy-momentum tensor of Vlasov fields and their derivatives. Secondly, we would like an approach that is compatible with the ones used to study wave equations on black hole spacetimes. For this, we make use of a weight function that captures the concentration phenomenon of future-trapped null geodesics in phase space. Moreover, we consider the associated symplectic gradient $V_-$, which turns out to enjoy good commutation properties with the massless Vlasov operator. By using a well-chosen modification of $V_-$, we construct a $W_{x,p}^{1,1}$ norm for which any smooth solution to the massless Vlasov equation verifies an integrated energy decay estimate without relative degeneration. This is joint work with Renato Velozo. The lecture will take place as part of the Imperial Day in Muenster.(https://www.uni-muenster.de/MathematicsMuenster/events/2024/imperialday.shtml).

In this work we consider the Lagrangian properties of a random version of the Arnold-Beltrami-Childress (ABC) vector fields in a three-dimensional periodic box. We prove that the associated flow map possesses a positive top Lyapunov exponent, and its associated one-point, two-point and projective Markov chains are geometrically ergodic. For a passive scalar, it follows that such a velocity is a space-time smooth exponentially mixing field, uniformly in the diffusivity coefficient. For a passive vector, it provides an example of a universal ideal (i.e. non-diffusive) kinematic dynamo. This is a joint work with M. Coti Zelati (Imperial College London)

In General Relativity, the wave equation is typically introduced as mere a toy model for gravitational perturbations. While it is true that solutions to Einstein?s Equations are fundamentally more complex, understanding the propagation of waves within such solutions provides insight about some of their geometric features. In this talk, I will look at the case of asymptotically anti-de Sitter black holes and discuss how their conformal boundary at infinity (and boundary conditions imposed there) influences the dispersion of waves.