# Abstracts

**Invited speakers**

**Linear stability of the non-extreme Kerr black hole**

Felix Finster

Universität Regensburg, Germany

After a general introduction to black holes and wave equations in black hole geometries, I will report on recent results on the linear stability of Kerr black holes under perturbations of general spin. This is joint work with Joel Smoller.

**Understanding isolated system dynamics in General Relativity**

José Luis Jaramillo

Institut de Mathématiques de Bourgogne, France

The work by Sergio Dain in mathematical relativity provides a brilliant chapter in the theoretical developments of General Relativity over the last two decades. Sergio left us last February, at the age of 46. The relevance of his contributions in different areas of the theory shape him as an outstanding relativist and a crucial figure among the researchers of his generation. This talk aims at presenting a perspective on his work, focusing on those aspects related to the dynamics of isolated systems in General Relativity. This topic offers a natural frame to illustrate the richness, soundness and astonishing inner consistency of his variate contributions. It also underlines a key feature of Sergio as a researcher: his most singular combination of exceptional geometrical and analytical skills with a deep intuition and understanding of the physics modelled by the theory. Lust for Understanding, fresh enthusiasm and elegance of spirit, distinguishing imprints of our colleague and friend.

**The Global Nonlinear Stability of Minkowski Spacetime for Self-Gravitating Matter**

Philippe G. LeFloch

Université Paris 6, France

This lecture will review recent results on self-gravitating massive matter, modeled by the Einstein equations of general relativity. A new vector field method, the Hyperboloidal Foliation Method developed in collaboration with Y. Ma (Xian), is now available in order to establish global-in-time results for the class of systems of coupled wave-Klein-Gordon equations posed on a curved spacetime. This method has recently been used to investigate the global dynamics of massive matter fields.

Pawel Nurowski

Uniwersytet Warszawski, Poland

The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime.

The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very definition of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein's death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the Red Light for gravitational waves research.

In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the `Green Light' for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery.

In this lecture we tell the story of this theoretical breakthrough. Particular attention will be given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts.

**Conformal methods in general relativity**

Tim-Torben Paetz

Universität Wien, Austria

Friedrich's conformal field equations substitute Einstein's field equations in conformally rescaled vacuum spacetimes. They provide an extremely powerful tool to construct (semi-)global solutions to the vacuum equations and to control the asymptotic behavior of the gravitational field. In this talk I will review the conformal field equations and discuss some striking results obtained via these equations for vacuum spacetimes with both positive and vanishing cosmological constant. Although a main focus will be on the construction of spacetimes which admit a smooth null infinity, the issue of constructing spacetimes with smooth timelike and spacelike infinity (using the cylinder representation for the latter one) will be addressed as well.

**Calabi-Bernstein type problems in Lorentzian Geometry**

Rafael M. Rubio

Universidad de Córdoba, Spain

The Calabi-Bernstein theorem states that the only entire solutions to the maximal hypersurface equation in the Lorentz-Minkowski spacetime are the spacelike affine hyperplanes. The present work review some of the classical and recent proofs of the theorem for the two dimensional case, as well as several extensions for Lorentzian warped products and other relevant spacetimes. On the other hand the problem of uniqueness of complete maximal hypersurfaces is analized under the perspective of some new results.

Miguel Sánchez

Universidad de Granada, Spain

Recently, a link between the geometric properties of some classes of Lorentzian and Finslerian manifolds has been developed, including the correspondence between the conformal geometry of stationary spacetimes and the geometry of (Finslerian) Randers spaces, as well as between several notions of convexity and boundares in both fields. In particular, a connection between two classical variational problems have been found:

- Relativistic Fermat's principle: among the lightlike curves joining a point and an observer γ
_{0}, pregeodesics are the critical points of the arrival proper time functional. - Zermelo's navigation: the fastest path between two points for the movement of a plane in windy air or a ship on a current, are geodesics for a certain Finslerian metric
*F*constructed by using a Riemannian metric*g*and a vector field_{R}*W*such that*g*._{R}(W,W)<1

Such a link allows to extend and solve both problems beyond the classical scope, namely, when γ_{0} is not timelike and when the wind/current *W* is not *mild* (that is, the restriction *g _{R}(W,W)<1* is not imposed). Moreover, the latter problem requires an extension of Finslerian Geometry with a natural interpretation from the classical Lorentzian viewpoint. Along this talk, based on joint work with E. Caponio and M.A. Javaloyes (arxiv: 1407.5494), a review on the topic will be provided.

**Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces**

Kotaro Yamada

Tokyo Institute of Technology, Japan

A certain class of spacelike maximal surfaces in Lorentz-Minkowski 3-space can be extended to timelike zero-mean-curvature surface. We explain such a phenomena and construct new examples. In particular, several examples of entire zero-mean-curvature graphs are introduced. Such a phenomena does not occur for minimal surfaces in Euclidean 3-space because Bernstein's theorem.

**Mini course**

**On the geometry and topology of initial data sets in General Relativity**

Gregory J. Galloway

University of Miami, USA

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*. 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.

**General interest talk**

**Ondas gravitacionales: el amanecer de una nueva era**

José M.M. Senovilla

Universidad del País Vasco, Spain

Después de una larga espera de decenas de años estamos de enhorabuena: el 14 de Septiembre de 2015 una colaboración científica internacional (LIGO/VIRGO) logró detectar, en sus interferómetros más avanzados en funcionamiento, lo que llevábamos esperando con tanta ansia: una onda gravitatoria. La primera de la historia. Poco después, el 26 de Diciembre, se detectó la segunda.

El logro científico-técnico es imponente: medir una variación de longitud equivalente a la del tamaño del radio atómico en la distancia entre la Tierra y el Sol. ¿Se puede medir eso? ¡Se ha hecho! Simultáneamente, hemos podido observar, por partida doble y directamente, un sistema binario de agujeros negros. ¡Sensacional!

Lo mejor, con seguridad, está por llegar. Estos hechos excepcionales e históricos demuestran que la humanidad se ha dotado de un nuevo "sentido" para observar el Universo, una nueva ventana por la que escudriñar lo que hay ahí fuera. Hasta ahora éramos insensibles a la radiación gravitatoria, a partir de ahora ya podemos "gravi-sentir" el Universo. La nueva era de la "Astronomía por gravedad" ha empezado. En esta conferencia se explicará, de forma amena y asequible, qué es una onda gravitatoria, cómo y con qué aparatos se mide, la información que porta, sus diferencias con otras ondas cotidianas. Veremos así que todo lo que a partir de ahora se descubrirá superará todas nuestras expectativas, cambiará nuestra cosmovisión radicalmente, nos aportará sorpresas impensables. Los cielos están henchidos de "luceros gravitatorios", ignotos hasta ahora, inenarrables. Podremos conocerlos y estudiarlos. Aprender acerca de, y comprender, el firmamento. Todo lo que existe, sea visible o invisible, gravita. Podremos por ello observar, e indagar, todo el Universo, sus más recónditos rincones y hasta su origen.

Explicaremos, en definitiva, por qué nos encontramos en los albores de una nueva etapa para la humanidad, un momento único y apasionante. ¡No se lo pierdan!