Solutions to general relativity with a negative cosmological constant have received significant attention due to the conjectured AdS/CFT correspondence, a particularly well-understood example of which is exhibited in 2+1 dimensions. I will review known vacuum solutions to general relativity with a negative cosmological constant in 2+1 dimensions and discuss the difficulties in defining mass, which are resolved via minimisation using a positive energy theorem. I will present a gluing theorem for vacuum time-symmetric general-relativistic initial data sets in two spatial dimensions. By gluing two given time-symmetric vacuum initial data sets at conformal infinity, we obtain new time-symmetric vacuum initial data sets. I will sketch the derivation of the mass formulae of the resulting manifolds. Our gluing theorem yields complete manifolds with any mass aspect function, which are smooth except for one conical singularity. Based on joint work with P. T. Chruściel.
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 × Tn-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.
