Starts on: July 17, 2017

Abstract: The observation of the merger of a binary black hole system from LIGO in September 2015 has opened up a new window onto the Universe. Thanks to gravitational waves emitted by massive compact objects or cataclysmic events, one can observe the Universe with a probe that is barely affected by matter. Gravitational waves provide also a unique means to test the consistency in the general relativity in the strong field regime. In my lectures I will review the state of the art of gravitational wave searches from ground based detectors, space missions, and pulsar timing arrays. In addition, I will explain the different challenges of such searches and show (with many examples) the richness of the newly emerged field of gravitational wave astronomy.

Starts on: July 17, 2017

Abstract: Initial data sets for the Einstein equation must satisfy a nonlinear elliptic system, the Einstein constraint equations. We formulate the constraint equations in the context of initial data sets, and develop methods to produce solutions of these equations. In particular, we will discuss interesting solutions which can be constructed by gluing techniques. Such techniques can be used for connecting multiple initial data sets, or for understanding the asymptotic structure of isolated systems.

Starts on: July 18, 2017

Abstract: In this lecture, we explore a fundamental observation of R. Schoen and S.-T. Yau which, together with the ideas from the lecture of Lan-hsuan Huang, leads to a proof of the positive mass theorem in dimension three (in fact in all low dimensions): A Riemannian three-manifold with positive scalar curvature does not admit an area-minimizing torus.

Starts on: July 19, 2017

Abstract: The lectures discuss dynamical properties of solutions to the Einstein equations: On a suitable foliation of spacetime the Einstein equations take the form of a geometric flow: a system of hyperbolic PDEs that describes the evolution of a metric and a second fundamental form on a given 3-manifold. We outline one particular approach to resolve the local-existence problem for this geometric flow and apply this to study the global existence problem and stability of some cosmological spacetimes. Eventually, we give an overview on major results and open problems in this area.

Starts on: July 18, 2017

Abstract: An initial data set in spacetime consists of a spacelike hypersurface $V$, together with its its induced (Riemannian) metric h and its second fundamental form $K$. A solution to the Einstein equations influences the curvature of $V$ via the Einstein constraint equations, the geometric origin of which are the Gauss-Codazzi equations. After a brief introduction to Lorentzian manifolds and Lorentzian causality, we will study some topics of recent interest related to the geometry and topology of initial data sets. In particular, we will consider the topology of black holes in higher dimensional gravity, inspired by certain developments in string theory and issues related to black hole uniqueness. We shall also discuss recent work on the geometry and topology of the region of space exterior to all black holes, which is closely connected to the notion of topological censorship. Many of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.

Starts on: July 20, 2017

Abstract:I will review the existing evidence for the existence of a black hole in the center of our galaxy, and discuss the associated theoretical and observational challenges.

Starts on: July 17, 2017

Abstract:We will discuss the analytic background for the positive mass theorem on the scalar curvature deformation. Using those deformation results, one can reduce the general case of the positive mass theorem to the special case of initial data sets that have harmonic asymptotics or can be further compactified. The equality case of the positive mass theorem will be also discussed.

Starts on: July 19, 2017

Abstract: We will introduce positive energy/mass problems in general relativity and discuss methods for solving some of them. The methods will involve minimal hypersurfaces and marginally outer trapped surfaces and require an understanding of existence problems as well as the second variation and stability notions for these hypersurfaces. We will also introduce Penrose inequalities which give stronger versions of positive energy theorems for black hole spacetimes. We will describe the flow approaches which have been developed to prove the Riemannian Penrose inequality.