Time Thursday Friday
9.45-10.00 Gerhard Huisken
10.00-10.50 Roger Penrose
Conformal boundaries and cyclic cosmology
Juan Valiente Kroon
On the structure of spatial infinity: a rigidity result
11.00-11.30 coffee break coffee break
11.30-12.20 Sergio Dain
Axisymmetric evolution of Einstein equations
Rafe Mazzeo
Singular Einstein spaces
14.00-14.50 Gregory Galloway
Aspects of marginally trapped surfaces in 2+1 and higher dimensional gravity
Oscar Reula
On the boundary value problem in general relativity
15.00-15.30 coffee break coffee break
15.30-16.20 Michael Anderson
On the Bartnik static extension conjecture
Richard Schoen
On the geometry of stationary and static spacetimes
17.00 Get-together

Michael Anderson
In connection with his definition of quasi-local mass, Bartnik raised a conjecture on the existence of complete AF static vacuum solutions to the Einstein equations with prescribed metric and positive mean curvature function H on an inner boundary 2-sphere. We will discuss a proof of this conjecture for boundary data satisfying a non-degeneracy condition: H has no critical points where the Gauss curvature K is nonpositive. (For example, K > 0 everywhere). This is joint work with Marcus Khuri.
Sergio Dain
In this talk will analyze the evolution of axially symmetric, vacuum, spacetimes. In particular, I want to present new ingredients that I hope will open new possibilities for controlling the evolution of such systems in problems related with the linear and non linear stability of black holes in axial symmetry.

These new ingredients are related with the total mass in General Relativity.� In axisymmetric evolution of isolated systems there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution.� Moreover, extreme Kerr black hole is a global minimum of this integral.

I will also present recent work in progress regarding the linearized Einstein equations in this gauge.

Gregory Galloway
Marginally outer trapped surfaces (MOTSs) have long been associated with the development of singularities in spacetime and the existence of black holes. In recent years a number of mathematically rigorous properties of MOTSs have been established. We will discuss some of these properties in connection with the existence of black holes in 2+1 dimensions, as well as the topology of black holes in higher dimensions. The work in 2+1 dimensions is joint with Kristin Schleich and Don Witt.
Juan Valiente Kroon
I will review some of Helmut Friedrich’s contributions to the understanding of the structure and properties of asymptotically simple spacetimes. In particular, I will focus on the analysis of spatial infinity. I will show that a representation of spatial infinity introduced by Helmut Friedrich in the 1990’s allows to prove a certain rigidity result. More precisely, given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat near spatial infinity, the solution to “the regular finite initial value problem at spatial infinity” is smooth through the sets where null infinity “touches” spatial infinity if and only if the data is exactly Schwarzschildean in a neighbourhood of infinity. Possible generalisations of this result will be discussed.
Rafe Mazzeo
I will discuss the notion of (Riemannian) Einstein metrics on stratified spaces. There are some general problems about the best way to formulate this, as well as some very interesting examples in low dimensions, and these will be the main focus of the talk.
Roger Penrose
As Helmut Friedrich has demonstrated, when a positive cosmological constant Λ is assumed – as supported by current observation – smoothness for spacelike conformal infinity is consistent with full freedom for outgoing radiation. At the other end of the temporal scale, conformal smoothness for the big bang provides a reasonable restriction, consistent with observation, as Paul Tod has proposed. Conformal cyclic cosmology builds upon these findings, positing a succession of universe “aeons”, the remote future of each aeon continuing as a smooth conformal manifold to the big bang of the next. Equations governing a conformally smooth transition from aeon to aeon will be described in this talk.
Richard Schoen
We will discuss recent results which provide rigorous restrictions on the geometry of static and stationary n-body solutions in relativity for various matter fields. Much of the talk will describe joint work with Robert Beig and Gary Gibbons.