In this talk, I will discuss our recent result. Either there is a counterexample to black hole uniqueness, or the following statement holds. Axisymmetric, complete, simply connected, maximal initial data sets for the Einstein equations of nonnegative energy density with ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum. Moreover, equality is achieved only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the linearized harmonic map equations.

The set of (smooth) metrics that can be placed on a Riemannian manifold defines an infinite-dimensional "superspace" that, remarkably, can itself be imbued with the structure of a (Fréchet) manifold. The subspace pertaining to (spatially-sliced) Einstein metrics was explored in detail by Wheeler and collaborators back in the late 50s, as it provides a means to describe a collection of spacetimes purely in terms of geometry through the famous words "mass without mass; charge without charge". At least in some restricted contexts, a natural basis relates to multipole moments which provide a tool to decompose a spacetime into a set of numbers. I will describe the construction of such superspaces, how to define inner products and (weak) Riemannian metrics there, and how they may be useful to provide astrophysical intuition. For instance, geodesics can be computed on Met(M) which allows one to define a single number that tells you how "distant" two spacetimes (e.g., two Kerr black holes) are from one another.

In this talk, I explore how internal deformations in spinless extended bodies, such as periodic pulsations or oscillations of the body’s shape, can trigger chaotic motion even when the background spacetime is fixed and the underlying geodesic dynamics is integrable. Using Dixon’s multipolar framework, I show how time-periodic finite-size effects can split separatrices and generate homoclinic chaos, diagnosed via the Melnikov method. I discuss Schwarzschild black holes for nearly spherical bodies with oscillating oblateness, as well as spherically symmetric pulsations in non-vacuum spacetimes, including electrovac/charged black holes and black holes embedded in dark-matter halos. Finally, I highlight a genuinely relativistic boundary of the point-particle idealization: in Ricci-flat vacuum, spinless spherical bodies move geodesically through quadrupole order, yet finite-size deviations can arise at hexadecapole order, where even vacuum Schwarzschild admits nontrivial corrections and chaos under pulsations.

On a background Minkowski spacetime, the Euler equations (both relativistic and not) are known to admit unstable homogeneous solutions with finite-time shock formation. Such shock formation can be suppressed on cosmological spacetimes whose spatial slices expand at an accelerated rate. However, situations with decelerated expansion, which are relevant in our early universe, are not as well understood. I will present some recent joint work in this direction, based on collaborations with David Fajman, Maciej Maliborski, Todd Oliynyk and Max Ofner.

In this talk I will discuss some results obtained in collaboration with Filipe C. Mena and former PhD student Vítor Bessa on the global dynamics of a minimally coupled scalar field interacting with a perfect-fluid through a friction-like term in spatially flat homogeneous and isotropic spacetimes. In particular, it is shown that the late time dynamics contain a rich variety of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.