Mini-courses:

• Minimal surfaces and scalar curvature, by Otis Chodosh (Princeton/IAS).
  Abstract: I'll discuss the link between minimal surfaces and scalar curvature. For example, I will explain how minimal surfaces were used to prove that there is no metric on the 3-dimensional torus with positive scalar curvature by Schoen--Yau. I will also discuss the surprising link between such results and general relativity, as well as some related rigidity results. I'll assume basic knowledge of Riemannian geometry.
  
• Riemannian geometry in general relativity, by Dan Lee (CUNY).
  Abstract: I will explain how questions in general relativity lead to interesting problems in Riemannian geometry. This will involve a tour from the Einstein equations to the constraint equations, to the study of asymptotically flat initial data sets, ADM energy-momentum, the positive mass theorem, and the Penrose inequality.

Invited Speakers:

• Structure of globally hyperbolic spacetimes with timelike boundary, by Luis Aké (UADY)
  Abstract: Globally hyperbolic spacetimes with timelike boundary, $(\overline{M},g)$ where $g$ is a Lorentzian metric and $\overline{M}=M \cup \partial M$, are the natural class of spacetimes containing naked singularities where boundary conditions can be posed; such conditions can be regarded as asymptotic, when the boundary is obtained by means of a conformal embedding. The properties of these spacetimes were introduced and studied in [1]. In this talk, we will study the properties of its causal ladder and we will focus in its top level, that is, globally hyperbolic. This work is based on [2].

• Asymptotically flat extensions with charge, by Aghil Alaee (Harvard University)
  Abstract: The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, a la Mantoulidis-Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of a notion of charged Bartnik mass for suitable charged minimal Bartnik data.

• The hyperboloidal initial value problem in general relativity, by Paul Allen (Lewis and Clark College)
  Abstract: We begin with a discussion of mathematical problems related to the initial value problem for asymptotically flat spacetimes. We then specialize to the question of constructing appropriate initial data and give an overview of known results, techniques, and open questions. Finally, we discuss recent and ongoing progress related to the hyperboloidal initial data. The latter is joint work with James Isenberg, John Lee, and Iva Stavrov Allen.

• Static vacuum space-times from Riemannian points of view, by Lucas Ambrozio (IAS)
  Abstract: A space-time admitting a global notion of time for which each time slice looks always the same, and moreover where to go towards the future is the same as going towards the past, is called a static space-time. As this description suggests, the geometry of these objects is encoded as a Lorentzian manifold of the form (RxM^3,-Vdt^2+g), where (M,g) is a Riemannian three-manifold and V is a function on M. The vaccum Einstein equations (with cosmological constant) translate into a nice system of equations on M relating the metric g and the function V. We will survey what is know about the solutions of these equations (and what is not known yet), focusing on the case of positive cosmological constant.

• The extrinsic curvature in the IMCF , by Eugenia Gabach-Clement (IFEG-Conicet)
  Abstract: In this talk we revise the Inverse Mean Curvature Flow (IMCF) and its application to study geometrical inequalities in General Relativity. The role that the IMCF has played in the proof of the Riemannian Penrose inequality by Huisken and Ilmanen is well known. We study alternatives to introduce a non zero extrinsic curvature into the problem. We also discuss the difficulties that arise when trying to obtain a geometrical inequality for both black holes and material bodies.

• On the geometry and topology of initial data sets in General Relativity, by Greg Galloway (University of Miami)
  Abstract: We present some results concerning the geometry and topology of initial data sets in General Relativty motivated by the notion of topological censorship. The initial data sets we consider model the region of space outside of all black holes. The black hole boundaries are modeled by marginally outer trapped surface (MOTs), which, on the one hand play an important role in the theory of black holes, and, on the other, are natural spacetime analogues of minimal surfaces in Riemannian geometry. After a brief overview of topological censorship, we present results which establish restrictions on the geometry and topology of the initial data sets under consideration. The talk is based on joint work with various collaborators: Lars Andersson, Ken Baker, Mattias Dahl, Michael Eichmair and Dan Pollack.

• Desingularizing positive scalar curvature metrics on 4-manifolds, by Demetre Kazaras (Stony Brook University)
  Abstract: We study 4-manifolds of positive scalar curvature (psc) with severe metric singularities along points and embedded circles, establishing a desingularization process based on work by Li-Montoulidis in dimension 3. To carry this out, we show that the bordism group of closed 3-manifolds with psc metrics is trivial by explicit methods, using scalar-flat K{\"a}hler ALE surfaces recently discovered by Lock-Viaclovsky. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidian geometry. Subsequently, we obtain a low-regularity positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature.

• Bartnik's quasi-local mass, by Stephen McCormick (Uppsala University)
  Abstract: The problem of quasi-local mass in general relativity is the problem of assigning some meaningful notion of the total mass (or energy) contained in a bounded domain (compact Riemannian manifold with boundary). Some motivation and background on quasi-local mass is given before discussing Bartnik's proposed definition.
  We will then turn to discuss some recent progress on understanding the Bartnik mass. We will discuss an approach to estimate this mass both in the CMC case and more generally, and also show that in certain circumstances a known ambiguity in the literature can be resolved. This talk is partially based on joint work with Armando Cabrera Pacheco, Carla Cederbaum, and Pengzi Miao.

• A beauty of General Relativity: Gravitational Waves, by Claudia Moreno (UDG)
  Abstract: Gravitational Waves (GW), the ripples in the fabric of the space-time predicted one hundred years ago in the Einstein's General Theory of Relativity, were finally detected and a new era of multimessenger astronomy has begun. This talk presents an introduction to the fundamental theory of GW, the nature and characteristics of the different astrophysical sources of GW and describes the scientific and technological efforts developed to detect GW using laser interferometry LIGO (Laser Interferometer Gravitational-Wave Observatory). Finally, the works done by our research group in the search and detection of gravitational waves making use of analytical techniques and data analysis will be presented.

• Stability of graphical tori with almost nonnegative scalar curvature , by Raquel Perales (IMATE)
  Abstract: By works of Schoen--Yau and Gromov--Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang--Lee, Huang--Lee--Sormani and Sormani. With A. Cabrera Pacheco and C. Ketterer.

• Mean Curvature and General Relativity, by Dan Pollack (University of Washington)
  Abstract: Spacelike hypersurfaces with constant mean curvature (CMC) have played an important role in some areas of general relativity. Marginally Outer Trapped Surfaces (MOTS) in initial data sets may be viewed as spacetime generalizations of minimal surfaces (which have zero mean curvature) which are a fundamental object of study in Riemannian geometry. We will survey a number of results concerning each of these objects, and draw connections between CMC hypersurfaces, MOTS, spacetime topology and singularities.

• Perspectives on synthetic Lorentzian geometry, by Didier Solis (UADY)
  Abstract: Dating back to the seminal works of Alexandrov and Toponogov, synthetic methods emanating from Riemannian geometry have proven to be a powerful tool in the study of spaces with low regularity. For instance, some remarkable examples can be found both in the context of Alexandrov spaces or CAT spaces. In this expository talk we discuss some alternatives for a synthethic theory in the Lorentzian setting and focus on the notion of Lorentzian lenght spaces —as introduced by M. Kunzinger and C. Sämann— and present some recent developments related to the causal structure of such spaces. This is joint work with L. Aké (UADY) and L. Montes de Oca (UADY)

Short Talks:

• Mass rigidity for hyperbolic manifolds, by Hyun Chul Jang (University of Connecticut)
  Abstract: We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds with scalar curvature $R_g\ge -n(n-1)$ in a following sense: if the energy-momentum vector $(p_0,p_1,\ldots, p_n)$ satisfies the equality $p_0=\sqrt{p_1^2+\cdots +p_n^2}$, then the manifold is isometric to hyperbolic space with constant curvature -1. The result was previously proven for spin manifolds or under special asymptotic conditions. This talk is based on joint work with L. -H. Huang and D. Martin.

• A uniqueness result for higher-dimensional Reissner--Nordström manifolds, by Sophia Jahns (University of Tuebingen)
  Abstract: We consider n+1-dimensional static solutions of the electrovacuum equations which are asymptotic to a member of the Reissner--Nordström family, with a lapse and an electric potential fulfilling some asymptotic conditions. Assuming that we are given such a spacetime whose inner boundary (a priori possibly with multiple connected components) consists of static horizons or photon spheres (which are characterized by a quasilocal subextremality condition), we show that it is isometric to a subextremal Reissner--Nordström spacetime of positive mass. The proof relies on ideas going back to the well-known black hole uniquess thereom by Bunting and Masood-ul Alam and generalizes results by Cederbaum, Cederbaum--Galloway, and Lazov--Yazadjiev.