I will discuss joint work with Pei-Ken Hung on the Horowitz-Myers conjecture in dimension at most 7. Our approach relies on a new geometric inequality. For a Riemannian metric on B² × T^n-2 with scalar curvature at least -n(n-1), this inequality relates the systole of the boundary to the mean curvature of the boundary.

In recent years the theory of optimal transportation has made an impact on Lorentzian geometry, mainly through synthetic definitions of timelike Ricci curvature bounds for Lorentzian manifolds and Lorentzian length spaces. But there are also other aspects of Lorentzian optimal transport relevant to general relativity. I will give an overview of the basic ideas of Lorentzian optimal transport and recent results. Further I will indicate some open problems in this context.

The principles of Relativity and Quantum Mechanics form the pillars of our understanding of Nature, and are extremely constraining when we compute observables in fundamental physics. But over the past few decades, there has been growing evidence that this standard physical picture obscures astonishing simplicity and hidden symmetries seen only at the very end of complicated calculations. This has led some physicists to seek radically different ways of conceptualizing physics, that leads much more directly to the final answer, involving the discovery of interesting new mathematical structures. In this talk I will describe some emerging ideas along these lines, and present a new formulation of some very basic physics — fundamental to particle scattering and to cosmology — not following from quantum evolution in space-time, but arising from new ideas in combinatorics, algebra and geometry.

Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_eps of the Einstein vacuum equations which is approximately equal to g far from the geodesic but near any point along the geodesic approximately equal to the metric of a Kerr black hole with mass m_eps. As an application, we can construct spacetimes which describe the merger of a very light black hole with a unit mass black hole, followed by the relaxation of the resulting single black hole to its equilibrium (Kerr or Kerr-de Sitter) state.

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.