We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal emebeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a non-degenerate boundary condition is used to construct metrics with nonpositive Einstein constant.

The Friedman-Lemaître-Robertson-Walker (FLRW) spacetime is often called the Standard Model of modern cosmology. In this talk we discuss small data global existence of solutions to a family of nonlinear wave equations propagating on a fixed FLRW spacetime. The initial value problem here takes the form \begin{equation} \label{semiwave}\begin{cases}-\partial_t^2\phi-n\dot a({a}^{-1}\partial_t\phi) +a^{-2} \Delta \phi = \mathcal{N}(\partial\phi) , \\ \phi(t_0,x)=\phi_0(x), \quad\partial_t\phi(t_0,x)=\phi_1(x),
\end{cases}\end{equation} where $n\gt 2$, $t_0\gt 1$, and $\mathcal{N}$ is a nonlinear function depending (at most) quadratically on derivatives of $\phi$. We consider cosmologies undergoing an accelerated expansion in the direction of positive time \[ a(t)\gt 0 , \qquad \dot a(t)\gt 0, \quad {\text{ and }} \int_{t_0}^{\infty} \frac{1}{a(s)} ds\lt \infty.\] By pairing weighted energy estimates with favorable structure on $\mathcal{N}$, we show how one can establish global wellposedness and sharp decay for solutions to a suitable class of equations of the form \eqref{semiwave}. We also discuss how this method can be extended to the case of nonaccelerated expanding FLRW spacetimes.

In general relativity we don't have any preferred time function, so we can not have infinite propagation speeds. Thus, parabolic systems like the Navier-Stokes equations are not possible. To overcome this several proposals for admissible theories have been put forward in the past. Notably, the first ones were ill posed, namely a small variation on the initial data produced huge differences on the solutions. Fortunately new proposals have arisen which are well posed. Here we discuss one of them. In particular we shall concentrate on one describing conformally invariant fluids, the limit when the particle's mass is much smaller than their thermal speeds. In this case the universe of possible theories collapses to a four parameter family, only one of them related to the propagation speeds of the system. This case is well posed. We shall further discuss some simulations exploring these free parameter variations and some efforts to construct algorithms to automatically generate computer codes for solving them.

I will give an overview of numerical evidence that codimension one-sets of smooth initial data for matter coupled to GR form naked singularities – from scalar fields and fluids in spherical symmetry, where the evidence is strong, to vacuum GR, where the evidence is still confusing, but at least now beginning to be agreed by independent codes. Can one hope to prove some form of this naked singularity conjecture? In the last ten years or so, there has been considerable mathematical progress on the existence and stability of smooth self-similar blowup solutions in dispersive PDEs, and on naked singularity formation in GR. I will highlight some of these papers that particularly interest me as steps on the way to a future proof of naked singularity formation from codimension-one smooth initial data.

For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity can be related to those at future null infinity via an antipodal map at spatial infinity. We analyse the validity of this conjecture using Friedrich’s formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at the critical sets where null infinity meets spatial infinity unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at the critical sets are fully determined by the initial data and that the relation between the charges at past null infinity and those at future null infinity directly follows from our regularity condition. This is joint work with Mariem Magdy Ali Mohamed and Kartik Prabhu.