<i>Ecole Thématique</i><br>Aspects Géométriques de la Relativité Générale<br><i>Thematic School</i><br>Geometric Aspects of General Relativity<br>

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Monday, September 28, 201512:45 PM WelcomeWelcome12:45 PM - 1:15 PMRoom: Building #101:15 PM Introduction to the mathematical theory of black holes (lecture 1) - Piotr Chrusciel (University of Vienna)Introduction to the mathematical theory of black holes (lecture 1)
- Piotr Chrusciel (University of Vienna)

1:15 PM - 2:15 PMRoom: Lecture room SC 10.01, building #10 In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole, and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented. The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf2:20 PM On the notion of quasi local mass in General Relativity (lecture 1) - Mu-Tao Wang (Columbia University, NY)On the notion of quasi local mass in General Relativity (lecture 1)- Mu-Tao Wang (Columbia University, NY)

2:20 PM - 3:20 PMRoom: Lecture room SC 10.01, building #10 Contents:<br> (1) Review of mass and energy in relativity.<br> (2) ADM mass and energy-momentum.<br> (3) Various definitions of quasi-local mass.<br> A set of notes is provided for the 4 lectures in the Abstracts section3:20 PM Coffee breakCoffee break3:20 PM - 3:50 PMRoom: Building #103:50 PM Horizons in General Relativity (lecture 1) - Michael Eichmair (Universität Wien)Horizons in General Relativity (lecture 1)- Michael Eichmair (Universität Wien)

3:50 PM - 4:50 PMRoom: Lecture room SC 10.01, building #10 In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter. -
Tuesday, September 29, 20159:45 AM Introduction to the mathematical theory of black holes (lecture 2) - Piotr Chrusciel (University of Vienna)Introduction to the mathematical theory of black holes (lecture 2)
- Piotr Chrusciel (University of Vienna)

9:45 AM - 10:45 AMRoom: Lecture room SC 10.01, building #10 In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole, and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented. The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf10:45 AM Pause café / Coffee breakPause café / Coffee break10:45 AM - 11:15 AMRoom: Building #1011:15 AM On the center of mass in General Relativity (short talk) - Carla Cederbaum (Tübingen University)On the center of mass in General Relativity (short talk)- Carla Cederbaum (Tübingen University)

11:15 AM - 11:55 AMRoom: Lecture room SC 10.01, building #10 In many situations in Newtonian Gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in General Relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian Gravity, there is a priori no definitive corresponding notion in General Relativity. Instead, there are several alternative approaches to defining the center of mass of a system. We will discuss some of these different approaches for both asymptotically Euclidean and asymptotically hyperbolic systems and present some new ideas as well as explicit (counter-)examples.12:00 PM On the notion of quasi local mass in General Relativity (lecture 2) - Mu-Tao Wang (Columbia University, NY)On the notion of quasi local mass in General Relativity (lecture 2)- Mu-Tao Wang (Columbia University, NY)

12:00 PM - 1:00 PMRoom: Lecture room SC 10.01, building #10 Contents:<br> (1) New definition of quasi-local energy. <br> (2) Isometric embedding into the Minkowski space.<br> (3) The proof of positivity.<br> A set of notes is provided for the 4 lectures in the Abstracts section3:00 PM Horizons in General Relativity (lecture 2) - Michael Eichmair (Universität Wien)Horizons in General Relativity (lecture 2)- Michael Eichmair (Universität Wien)

3:00 PM - 4:00 PMRoom: Lecture room SC 10.01, building #10 In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.4:00 PM Pause café / Coffee BreakPause café / Coffee Break4:00 PM - 4:30 PMRoom: Building #104:30 PM On the notion of quasi local mass in General Relativity (lecture 3) - Mu-Tao Wang (Columbia University, NY)On the notion of quasi local mass in General Relativity (lecture 3)- Mu-Tao Wang (Columbia University, NY)

4:30 PM - 5:30 PMRoom: Lecture room SC 10.01, building #10 Contents:<br> (1) Variation of quasi-local energy.<br> (2) The optimal embedding equation.<br> (3) Solving the optimal isometric embedding equation at spatial and null infinity. A set of notes is provided for the 4 lectures in the Abstracts section. -
Wednesday, September 30, 20159:45 AM Introduction to the mathematical theory of black holes (lecture 3) - Piotr Chrusciel (University of Vienna)Introduction to the mathematical theory of black holes (lecture 3)
- Piotr Chrusciel (University of Vienna)

9:45 AM - 10:45 AMRoom: Lecture room SC 10.01, building #10 In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole, and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented. The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf10:45 AM Pause café / Coffee breakPause café / Coffee break10:45 AM - 11:15 AMRoom: Building #1011:15 AM On the notion of quasi local mass in General Relativity (lecture 4) - Mu-Tao Wang (Columbia University, NY)On the notion of quasi local mass in General Relativity (lecture 4)- Mu-Tao Wang (Columbia University, NY)

11:15 AM - 12:15 PMRoom: Lecture room SC 10.01, building #10 Contents:<br> (1) Minimizing and rigidity property of critical points of quasilocal energy. <br> (2) Quasilocal angular momentum and center of mass and their limits at infinity.<br> (3) Asymptotically hyperbolic initial data sets. A set of notes is provided for the 4 lectures in the Abstracts section12:20 PM Instability phenomena for the Einstein-Lichnerowicz equation (short talk) - Bruno PREMOSELLI (Université de Cergy-Pontoise)Instability phenomena for the Einstein-Lichnerowicz equation (short talk)- Bruno PREMOSELLI (Université de Cergy-Pontoise)

12:20 PM - 1:00 PMRoom: Lecture room SC 10.01, building #10, Science Campus The stability of the Einstein-Lichnerowicz equation is defined as the continuous dependence of the set of its positive solutions in the choice of the background physics data of the conformal method. When the conditions ensuring stability fail, surprising phenomena can arise, such as the existence of an infinite number of concentrating positive solutions. In this talk we will investigate some of these instability phenomena for the Einstein-Lichnerowicz equation when a non-trivial scalar field is present.3:00 PM Horizons in General Relativity (lecture 3) - Michael Eichmair (Universität Wien)Horizons in General Relativity (lecture 3)- Michael Eichmair (Universität Wien)

3:00 PM - 4:00 PMRoom: Lecture room SC 10.01, building #10 In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.4:00 PM Pause café / Coffee breakPause café / Coffee break4:00 PM - 4:30 PMRoom: Lecture room SC 10.01, building #104:30 PM Localized solutions of the Einstein equations: a few words about their geometry and physics (short talk) - Alessandro CARLOTTO (ETH Zürich)Localized solutions of the Einstein equations: a few words about their geometry and physics (short talk)- Alessandro CARLOTTO (ETH Zürich)

4:30 PM - 5:10 PMRoom: Lecture room SC 10.01, building #10 It is a truly surprising fact that the Einstein constraint equations own an overabundance of localized solutions, namely solutions that coincide with arbitrarily-assigned data inside a given solid cone and are trivial outisde of a cone of slightly larger angle. In this talk, I will briefly present them and comment on their physical relevance (related to gravitational shielding phenomena) as well as on their geometric content (which concerns the link with the isoperimetric problem, large outlying CMC spheres and stable minimal surfaces). This is mostly based on joint work with Richard Schoen.5:15 PM5:15 PM - 6:00 PMRoom: Lecture room SC 10.01, building #10 -
Thursday, October 1, 20159:45 AM Horizons in General Relativity (lecture 4) - Michael Eichmair (Unversität Wien)Horizons in General Relativity (lecture 4)
- Michael Eichmair (Unversität Wien)

9:45 AM - 10:45 AMRoom: Lecture room SC 10.01, building #10 In the first two lectures I will describe the basic results on the significance, existence, and properties of apparent horizons (or more precisely “marginally outer trapped surfaces”) in initial data sets. Taking into account the preferences of the audience, I will then sketch the proofs of one or two fundamental results in mathematical relativity that build on this theory. The possibilities include the minimal surface proof of the Riemannian positive energy theorem, the marginally outer trapped surface proof of the spacetime positive mass theorem, and the existence of black holes due to condensation of matter.10:45 AM Pause café / Coffee breakPause café / Coffee break10:45 AM - 11:15 AMRoom: Building #1011:15 AM Introduction to the mathematical theory of black holes (lecture 4) - Piotr Chrusciel (University of Vienna)Introduction to the mathematical theory of black holes (lecture 4)- Piotr Chrusciel (University of Vienna)

11:15 AM - 12:15 PMRoom: Lecture room SC 10.01, building #10 In the lectures I plan to discuss the geometry of spherically symmetric black holes, that of the Kerr black hole, and of the Emperan-Reall black rings. Conformal and projection diagrams will be discussed, and some elements of the theory of uniqueness of stationary black holes will be presented. The lectures will be based on selected chapters of the monograph "Geometry of black holes", available at http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black%20Holes/BlackHolesViennaJanuary2015.pdf12:15 PM Closing of the SchoolClosing of the School12:15 PM - 12:30 PMRoom: Building #10