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### NYGR Seminar

 New York General Relativity Seminar We meet 2-3 times per semester most semesters alternating between Columbia, Stony Brook and the CUNY Graduate Center.   Current organizers: Marcus Khuri, Dan Lee, Henri Roesch, Christina Sormani, and Mu-Tao Wang.   Past organizers include Mike Anderson, Po-Ning Chen, Sergiu Klainerman, and Lan-Hsuan Huang. June 13, 2019 at IAS Simonyi Hall 101 Speaker Marcus Khuri  11 am-12 pm Simonyi 101  Stability of the spacetime positive mass theorem in spherical symmetryWe formulate a conjecture for the almost rigidity statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. Furthermore we establish the conjecture under the assumption of spherical symmetry in all dimensions.  In particular, it is shown that a sequence of asymptotically flat initial data  satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the sense that the bases converge in the volume preserving intrinsic flat sense to Euclidean space. Furthermore, the difference of the second fundamental forms must converge to zero in . This is joint work with E. Bryden and C. Sormani.   Speaker Jim Isenberg  1 pm-2 pm Simonyi 101 Symmetries of Cosmological Cauchy Horizons with Non-Closed OrbitsWe consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago Vince Moncrief and I proved that, if the null geodesic generators of such a horizon were all closed curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of an infinite dimensional, analytic family of such generalized Taub-NUT' spacetimes and show that, generically, they admitted only the single (horizon-generating) Killing field alluded to above. In the work discussed in this talk,  we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon-generating null geodesics densely fill a 2-  torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes, we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the conjectures that every such spacetime with an ergodic horizon is trivially constructable from the Kasner solution by making certain irrational' toroidal compactifications and that analytic vacuum space times containing degenerate compact Cauchy horizons do not exist.   The work I discuss in this talk has been jointly carried out with Vince Moncrief. Speaker Jared Speck 2:30 pm-3:30 pm Simonyi 101 A New Formulation of Multidimensional Compressible Euler Flow: Miraculous Geo-Analytic Structures and ApplicationsI will describe my recent works, some joint with M.\,Disconzi and J.\,Luk, on the compressible Euler equations and their relativistic analog.  The works concern solutions with non-vanishing vorticity and entropy, under any equation of state. The starting point is our new formulations of the equations exhibiting miraculous geo-analytic structures, including I. A sharp decomposition of the flow into geometric wave parts'' and `transport-div-curl parts;''II. Null form source terms;III. Structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the entropy and vorticity.I will then describe a main application: the study of stable shock formation, without symmetry assumptions, in more than one spatial dimension. I will emphasize the role that nonlinear geometric optics plays in the analysis and highlight how the new formulations allow for its implementation.  Finally, I will describe some important open problems, and I will connect the results to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions in the presence of shock singularities.Speaker Iva Stavrov virtual presentation (video will be linked to here)Continuous Matter Distributions as limits of Brill-Lindquist-Riemann sumsWe discuss a method of representing time symmetric, asymptotically and conformally Euclidean initial data for charged dust clouds as limits of point source configurations. Our method relies on the concept of intrinsic flat limit developed by Sormani and Wenger.  Feb 7, 2019 at Columbia Math Bldg 507: Speaker Pengyu Le 3-4 pm Title: Perturbations of Null Hypersurfaces and the Null Penrose Inequality Abstract: The Penrose inequality in general relativity is a conjectured inequality between the area of the horizon and the mass of a black-hole spacetime. The null Penrose inequality is the case where it concerns the area of the horizon and the Bondi mass at null infinity along a null hypersurface. An effective method to prove Penrose-type inequalities is to exploit the monotonicity of the Hawking mass along certain foliations. In his thesis [S], Sauter constructed the constant mass aspect function foliation aiming to prove the null Penrose inequality. The behavior of the foliation at past null infinity is an obstacle for his method. An idea to overcome this difficulty, which is suggested in the end of [S], is to vary the null hypersurface to achieve the desired behavior of the foliation at null infinity, leading to a spacetime version of the Penrose inequality. To formalise this idea, one need to study perturbations of null hypersurfaces. I will talk about my work on the study of perturbations of null hypersurfaces and its application to the null Penrose inequality. References [S] Sauter, J. Foliations of Null Hypersurfaces and the Penrose Inequality, Diss. ETH No.17842.  Speaker Christos Mantoulidis 4:30-5:30 pm Title: Capacity, fill-ins, and quasi-local mass Abstract: (Joint work with P. Miao and L.-F. Tam.) We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown-York mass and the other is new. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them. Nov 29, 2018 at CUNYGC in 3305: Hyun Chul Jang (U Conn) 3:00-4:00 pm Title: Rigidity conjecture for asymptotically hyperbolic spaces with zero mass Abstract: Recently, Huang and Lee proved the rigidity of the spacetime positive mass theorem by using a variational approach. Motivated by this work, we prove that assuming the nonnegativity of the mass of asymptotically hyperbolic Riemannian manifolds (without boundary) with scalar curvature greater than or equal to -n(n-1), the mass is zero if and only if the manifold is isometric to the standard hyperbolic space. By a variational approach, we first prove the existence of a static potential for AH manifolds with zero mass (in fact, AH manifolds with minimal mass), and then conclude the proof by using the certain integral identity which has been discovered first by Wang. Furthermore, by using this integral identity, we prove that for a static, asymptotically locally hyperbolic manifold with a certain boundary condition, the mass is zero if and only if the manifold is isometric to (some portion of) the reference manifold to which it is asymptotic. This is based on the joint work with Lan-Hsuan Huang and Daniel Martin. Jeff Jauregui (Union College) 4:30-5:30 pm Title: Semicontinuity of total mass and low-regularity convergence in general relativity Abstract: In general relativity, the ADM mass captures the total mass of an asymptotically flat spacelike hypersurface in a spacetime. Motivated by the conjectured near-rigidity of the positive mass theorem and by Bartnik's minimal mass extension conjecture, it is natural to study the interaction between the ADM mass and low-regularity convergence. We will discuss joint work with Dan Lee on establishing the lower semicontinuity of ADM mass for pointed C^0 convergence and Sormani--Wenger intrinsic flat convergence, which has shown promise for low-regularity problems in general relativity. We will make use of Huisken's isoperimetric mass concept. Oct. 4, 2018 at Stony Brook: Eric Woolgar (U Alberta) in Room 102 of SCGP 3:00-4:00 pm Title: Bakry-Emery curvature-dimension conditions in relativity Abstract: Many important theorems in Riemannian geometry depend on a lower bound for the Ricci curvature. In general relativity, \emph{energy conditions} play a similar role by providing lower bounds for components of Ricci curvature. In recent years, it has been realized that these lower bounds need only hold up to a Hessian term, or more generally a Lie derivative term. I will discuss new applications to general relativity that have arisen from this realization, including generalized singularity and splitting theorems and new results that constrain extreme black hole horizon topology. This talk is based on joint work with Greg Galloway, and with Marcus Khuri and William Wylie. Siyuan Lu (Rutgers) in Room 102 of SCGP 5:00-6:00 pm Title: On a localized Riemannian Penrose inequality Abstract:  For a bounded manifold with nonnegative scalar curvature, the Brown-York quasi-local mass is nonnegative and equals to 0 iff it's a domain in Euclidean space by fundamental results of Shi and Tam. Moreover, it is shown that the inequality is equivalent to Positive mass theorem. We consider the general setting that the bounded manifold allows a horizon. We establish a localized Riemannian Penrose inequality and prove that the equality holds iff it's a domain in Schwarzschild manifold. Similar to the Shi-Tam case, the inequality is equivalent to Riemannian Penrose inequality. This is based on joint works with P. Miao. No meetings in 2017-2018 except SCGP Mass in GR Workshop and CUNY GR Workshop Previous meetings (this list may be incomplete): April 14, 2017 at Columbia Jeremie Szeftel  Title: Remarks on the nonlinear stability of Schwarzschild Pei-Ken Hung and Jordan Keller Title: Linear stability of Schwarzschild spacetime Oct. 21, 2016 at Columbia: Abhay Ashtekar (IGCP at Penn State) Title: Asymptotically deSitter Spacetimes and positivity of the Bondi mass Henri Roesch (Duke) Title: Proof of a Null Penrose Conjecture using a new Quasi-local Mass. March 24, 2016 at CUNY Chen-Yun Lin (U Toronto) Title: Estimating the Bartnik Mass  Iva Stavrov (Lewis and Clark) Title: Shear Free Conditions on Initial Data Sets May 8, 2015 at Columbia Sergio Dain  Title: Geometrical inequalities for black holes and bodies  Yakov Shlapentokh-Rothman Title: Stability and Instability of Scalar Fields on Kerr Spacetimes Dec 12, 2014 at Stony Brook Lydia Bieri  Title: Analysis of Radiation for the Einstein and the Maxwell Equations Xinliang An Title: Formation of Trapped Surfaces in General Relativity Oct 10, 2014 at CUNY Robert Wald  Title: Dynamic and Thermodynamic Stability of Black Holes and Black Branes Iva Stavrov Title: Hyperboloidal initial data for the Einstein evolution equations and the shear-free condition Philippe LeFloch Title: Weakly regular spacetimes with T2 symmetry Sept 19 2014 at Columbia Steve Liebling Title: On the nonlinearly stability of AdS Frans Pretorius Title: The Instability of 5-dimensional black strings Mar 7, 2014 at Stony Brook   Spyros Alexakis Title: A reconstruction of waves from their emitted radiation and the final states of space-times. Mu-Tao Wang  Title: On the center of mass in general relativity Feb 7, 2014 at Columbia Robert Wald Title: Dynamic and Thermodynamic Stability of Black Holes and Black Branes Piotr Chrusciel Title: The mass of light cone Speakers in 2013 included: Hugh Bray Po-Ning Chen Sergio Kleinerman Igor Rodnianski Mihalis Dafermos Pengzi Miao Stefanos Aretakis Greg Galloway Jim Isenberg