During the Spring 2021 Semester, a weekly seminar will be held on General Relativity. The seminar will take place at on **Fridays at 9:35am virtually. **Please email the seminar organizers to obtain a link. This seminar is organized by Aghil Alaee.

To learn how to attend this seminar, please contact Aghil Alaee (aghil.alaee@cmsa.fas.harvard.edu).

The schedule will be updated below.

Date | Speaker | Title/Abstract |
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2/12/2021 | Jonathan Luk (Stanford University)Video | Title: Nonlinear interaction of three impulsive gravitational wavesAbstract: Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations for which the curvature tensor has a delta singularity supported on a null hypersurface. Explicit, highly symmetric, solutions featuring the propagation of one impulsive gravitational wave as well as the interaction of two impulsive gravitational waves have been constructed since the influential works of Penrose, Szekeres and Khan-Penrose. In this talk, I will discuss some mathematical results regarding the propagation and interaction of impulsive gravitational waves. In particular, I will present a recent joint work with Maxime Van de Moortel (Princeton), which gives the first construction of a large class of spacetimes featuring the interaction of three impulsive gravitational waves. |

2/19/2021 | Annachiara Piubello (University of Miami)Video | Title: Mass and Riemannian PolyhedraAbstract: We show a new formula for the ADM mass as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large polyhedra. In the special case of coordinate cubes, we will show an integral formula relating the n-dimensional mass with a geometrical quantity that determines the (n-1)-dimensional mass. This is joint work with Pengzi Miao. |

2/26/2021 | Abraao Mendes (Universidade Federal de Alagoas, Brazil)Video | Title: Initial data rigidity resultsAbstract: In this lecture we aim to present some rigidity results for initial data sets that are motivated by the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces (MOTS) are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature. |

3/5/2021 | Martin Reiris (Universidad de la República, Uruguay)Video | Title: New results on compact Cauchy horizons of smooth vacuum spacetimes.Abstract: Cauchy horizons are rather unique and peculiar objects that have been studied for decades. In this talk I will first review old and new breakthroughs on the subject by Isenberg and Moncrief, and by Petersen and Rácz, and then discuss joint recent work together with I. Bustamante where it is proved that non-degenerate Compact Cauchyhorizons on smooth vacuum spacetimes (shortly CHs) have indeed constant non-zero temperature. It then follows by the work of Petersen and Petersen-Rácz that CHs are Killing horizons, and by the work of Beig-Chrusciel-Schoen that such objects are non-generic on the initial data, (in agreement with the Cosmic Censorship conjecture). A null-orbital and topological classification of CHs will also be commented. |

3/12/2021 | Stefano Borghini (Università Milano Bicocca)Video | Title: Static vacuum spacetimes with positive cosmological constantAbstract: We discuss the problem of the classification of static vacuum spacetimes in the case of positive cosmological constant. To this end, we develop some new or improved tools to study extremal points of real analytic functions. Building on this, we deduce a new characterization of the Schwarzschild– de Sitter solution based on the geometry of the maximum set of the lapse. This is a joint work with P. T. Chrusciel and L. Mazzieri. Time permitting, we will then show how similar techniques can be applied in other frameworks, such as in the study of the negative cosmological constant case. |

3/19/2021 | Gustav Holzegel (Imperial College, UK)Video | Title: The non-linear stability of the Schwarzschild family of black holesAbstract: I will discuss recent work with M. Dafermos, I. Rodnianski, and M. Taylor proving the full finite codimension asymptotic stability of the Schwarzschild family of black holes in the exterior of the black hole region. The proof is expressed entirely in physical space and based on our previous understanding of linear stability of the Schwarzschild family in a double null gauge. |

3/26/2021 | Yiyue Zhang (Duke University)Video | Title: Harmonic level sets and positive mass theoremsAbstract: The positive mass theorems are fundamental results in differential geometry and general relativity. Roughly speaking, it asserts that the positive local energy density gives the positive total mass. Schoen and Yau gave the first proof in 1979. Another proof was given by Witten in 1981.In this talk, I will introduce the harmonic level set method developed by Stern in 2019. This technique has been used to prove the positive mass theorems in various settings, for example, the Riemannian case, the spacetime case, the hyperbolic case, and the positive mass theorem with charge. I will focus on the positive mass theorem for asymptotically hyperbolic manifolds. We give a lower bound for the mass in the asymptotically hyperbolic setting. In this setting, we solve the spacetime harmonic equation and give an explicit expansion for the solution. We also prove some rigidity results as corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri. |

4/2/2021 | Ryan Unger (Princeton University)Video | Title: The positive mass theorem with arbitrary ends Abstract: We prove a Riemannian positive mass theorem which allows for incompleteness and negative scalar curvature. The manifolds are assumed to have one asymptotically Schwarzschild end, but the complement of this end is otherwise arbitrary. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau. This is joint work with Lesourd and Yau. |

4/9/2021 | Hari Kunduri (Memorial University, Canada)Video | Title: Asymptotically flat gravitational instantonsAbstract: An asymptotically flat gravitational instanton is a 4d Riemannian manifold (M,g) that is complete, Ricci flat, and approaches a particular quotient of R^4 with flat metric at infinity. Roughly, these are Riemannian analogues of black holes. In analogy with the classic black hole uniqueness theorem, it was conjectured that the Kerr instanton on R^2 X S^2 was the unique instanton invariant under a local T^2 action. However, Chen and Teo recently explicitly constructed a new family of such instantons on CP^2 \ S^1. I will discuss the classification and explicit construction of L^2-harmonic forms on this space. I will also discuss progress on the existence and uniqueness of gravitational instantons admitting a T^2 isometry. |

4/16/2021 | Po-Ning Chen (University of California, Riverside)Video | Title: Supertranslation invariance of angular momentumAbstract: LIGO’s successful detection of gravitational waves has revitalized the theoretical understanding of the angular momentum carried away by gravitational radiation. An infinite-dimensional supertranslation ambiguity has presented an essential difficulty for decades of study. Recent advances were made to quantify the supertranslation ambiguity in the context of binary coalescence. In this talk, we will present the first definition of angular momentum in general relativity that is completely free from supertranslation ambiguity. The new definition of angular momentum is derived from the limit of the quasilocal angular momentum we defined previously. This talk is based on joint work with Jordan Keller, Mu-Tao wang, Ye-Kai Wang, and Shing-Tung Yau. |

4/23/202110:30am | Georgios Moschidis (UC Berkeley)Video | Title: The instability of Anti-de Sitter spacetime for the Einstein–scalar field systemAbstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. I will also discuss possible paths for extending these ideas to the vacuum case. |

4/30/2021 | Gilbert Weinstein (Ariel University, Israel) | Title: A static gravitational vacuum soliton in 5-d and related solutionsAbstract: Generalizing a construction by Myers and Korotkin-Nicolai to 5-d, we construct space-periodic solutions of the vacuum Einstein equations. This gives an affirmative answer to a conjecture of Myers concerning the existence of 5-d regular static vacuum solutions that balance an infinite number of black holes, and exhibit Kasner asymptotics. An added benefit in 5-d is the possibility to construct solitons, i.e. complete non-flat solutions devoid of any black holes. Using these, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number. The topology of the domain of outer communication is analysed. This is joint work with Marcus Khuri and Sumio Yamada. |

Date | Speaker | Title/Abstract |
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9/11/2020 | Dan Lee (Queens College, CUNY) Slides | Title: Bartnik minimizing initial data setsAbstract: We will review some facts about Bartnik minimizing initial data sets in the time-symmetric case, and then discuss new results on the general case obtained in joint work with Lan-Hsuan Huang of the University of Connecticut. Bartnik conjectured that these minimizers must be vacuum and admit a global Killing vector. We make partial progress toward the conjecture by proving that Bartnik minimizers must arise from so-called “null dust spacetimes” that admit a global Killing vector field. In high dimensions, we find examples that contradict Bartnik’s conjecture, as well as the “strict” positive mass theorem, though these examples have “sub-optimal” asymptotic decay rates. |

9/18/2020 | Martin Lesourd (BHI, Harvad) Video | Title: Construction of Cauchy data for the dynamical formation of apparent horizon and the Penrose InequalityAbstract: We construct a class of Cauchy initial data without (marginally) trapped surfaces whose future evolution is a trapped region bounded by an apparent horizon, i.e., a smooth hypersurface foliated by MOTS. The estimates obtained in the evolution lead to the following conditional statement: if Kerr Stability holds, then this kind of initial data yields a class of scale critical vacuum examples of Weak Cosmic Censorship and the Final State Conjecture. Moreover, owing to estimates for the ADM mass of the data and the area of the MOTS, the construction gives a fully dynamical vacuum setting in which to study the Spacetime Penrose Inequality. We show that the inequality is satisfied for an open region in the Cauchy development of this kind of initial data, which itself is controllable by the initial data. This is joint work with Nikos Athanasiou https://arxiv.org/abs/2009.03704. |

9/25/2020 | Cancelled – Math Science Literature Lectures | |

10/2/20209:00am ET | Sharmila Gunasekaran (Memorial University) Video | Title: Slow decay of waves in gravitational solitonsAbstract: Gravitational solitons are globally stationary, horizonless asymptotically flat spacetimes with positive energy. Typically they arise as classical solutions of the supergravity theories which govern the low energy sectors of string theory. They have generated attention within this context as possible classical microstate geometries of black holes. A natural question to consider is whether they are stable. In this talk, I will address the stability at the simplest level by investigating solutions to the linear wave equation in a particular soliton spacetime. I will describe a methodology, introduced by Holzegel-Smulevici to prove that massless scalar waves in a particular family of soliton spacetimes cannot decay faster than inverse logarithmically in time. The proof involves the construction of quasimodes which are approximate solutions to the wave equation. This slow decay can be attributed to the stable trapping of null geodesics and is suggestive of instability at the nonlinear level. This is joint work with Hari Kunduri. |

10/9/2020 | Carla Cederbaum (University of Tübingen)Video | Title: Explicit minimizing sequences related to the Riemannian Penrose InequalityAbstract: Following ideas by Mantoulidis and Schoen and further developments thereof by Cabrera Pacheco, McCormick, Miao, Xie (in alphabetic order) and the speaker, we construct a sequence of asymptotically flat Riemannian 3-manifolds of non-negative scalar curvature with minimal, strictly outward minimizing inner boundary. The ADM-mass converges to the minimal value permitted by the Riemannian Penrose Inequality along this sequence, yet the manifolds themselves do not converge to the Schwarzschild manifold arising as the rigidity case of the Riemannian Penrose Inequality. Instead, they converge to an explicitly given non-smooth manifold (in a suitable topology). The existence of this sequence and the precise form of the limit also have consequences for Bartnik’s quasi-local mass functional. This is joint work with Armando Cabrera Pacheco. |

10/16/2020 | Pei-Ken Hung (MIT)Video | Title: Stability of modified wave maps in spacetimes near SchwarzschildAbstract: In this talk, I will discuss a wave equation for vector fields in Schwarzschild spacetimes. The equation behaves like a damped wave equation. In particular, it allows us to show the stability of modified wave maps in spacetimes near Schwarzschild, including the Kerr spacetime. This is on-going joint work with S. Brendle. |

10/23/2020 | Jeff Jauregui (Union College)Video | Title: Scalar curvature, mass, and capacityAbstract: In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes of small geodesic balls. We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. Complementary to this, we show that the capacity of large balls can be used to detect the ADM in an asymptotically flat manifold (as inspired by Huisken’s isoperimetric mass). These results give new interpretations of scalar curvature and mass from the point of view of capacity and may be useful for low-regularity convergence problems in general relativity. |

11/6/2020 | Henri Roesch (Columbia University)Video | Title: The Isometric Embedding Problem in a Null ConeAbstract: We start by observing that the openness part of the continuity argument, as applied to the Weyl problem by C.Li-Z.Wang, holds in an arbitrary ambient geometry. We also partially generalize the argument to the n-sphere, showing that an arbitrary metric perturbation can be isometrically embedded up to a solution of the homogenous Codazzi equation. We then consider these results within an ambient three dimensional Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theorem for a path of isometric embeddings. Then, after imposing asymptotic decay conditions on the Null Cone, we show that any metric on the 2-sphere can be isometrically embedded up to a scaling factor. Finally, we show the existence of a foliation of any desired metric in a neighborhood of infinity. |

11/13/2020 | Alessandro Carlotto (ETH Zurich) | Title: Constrained deformations of positive scalar curvature metricsAbstract: What manifolds support metrics of positive scalar curvature? What can one say about the associated moduli space, when not empty? These are two fundamental problems in Riemannian Geometry, for which great progress has been made over the last fifty years, but that are nevertheless highly elusive and far from being fully resolved.Partly motivated by the study of initial data sets for the Einstein equations in General Relativity, I will present some results that aim at moving one step further, studying the interplay between two different curvature conditions, given by pointwise conditions on the scalar curvature of a manifold and the mean curvature of its boundary. In particular, after a broad contextualization, I will focus on recent joint work with Chao Li (Princeton University), where we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. We can also refine our methods so to construct continuous paths of non-negative scalar curvature metrics with minimal boundary, and to obtain analogous conclusions in that context as well. In particular, we thereby derive the path-connectedness of asymptotically flat scalar flat Riemannian 3-manifolds with minimal boundary, which goes in the direction of investigating (from a global perspective) the space of vacuum black-hole solutions to the Einstein field equations.Our work relies on a combination of earlier fundamental contributions by Schoen-Yau and Gromov-Lawson, on the smoothing procedure designed by Miao to handle singular interfaces, and on the interplay of Perelman’s Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with closed manifolds. |

11/20/2020 | Yuguang Shi (Peking University)Video | Title: On Gromov’s conjecture of fill-ins with nonnegative scalar curvature (II) View a PDF of the abstract here. |

Information about last year’s seminar can be found **here.**