# Columbia General Relativity & Geometric Analysis seminar

The Columbia General Relativity & Geometric Analysis seminar takes place **on Fridays from 3:30pm to 4:30pm** in room 307.

If you are interested in the seminar and want to be added to the mailing list, please write to Elena Giorgi, firstname.lastname@columbia.edu

- October 7th, 2022:
__Mario Apetroaie__(University of Toronto),**Linear instability of gravitational and electromagnetic perturbations of Extreme Reissner–Nordström***In a series of papers, Giorgi proved linear stability to gravitational and electromagnetic perturbations for the full subextremal range Q < M of Reissner-Nordström spacetimes as solutions to the Einstein-Maxwell equations. We address the aforementioned problem for the extremal Q=M Reissner-Nordström spacetime, and contrary to the subextremal case we see that instability results hold for a gauge invariant hierarchy along the event horizon. In particular, depending on the number of translation invariant derivatives of derived gauge-invariant quantities, we show decay, non-decay and polynomial blow up estimates asymptotically along the horizon. As a consequence, for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy analogous estimates. It is worth mentioning that more unstable behavior is realized for the negative spin solutions, with the $L^2$-norm of the extreme curvature component $ \underline{\alpha} $ not decaying asymptotically along the event horizon.*

- October 14th, 2022:
__Jingbo Wan__(Columbia University),**The Existence of a Black Hole Due to Condensation of Matter***For an asymptotically flat initial data set (three dimensional), with the mass density large on a large region, Schoen and Yau showed that there is an apparent horizon in the initial data. (SY83’) Their proof is based on a contradiction argument where assuming no apparent horizon in the region will give a global solution to Jang’s equation over the region (SY81’). Furthermore, positivity of mass density and this global solvability of Jang’s equation will give rise to positivity of a certain operator on this region. Finally, the argument is completed by showing such positivity of certain operator on the region will assert that the region can not be too large in a certain sense.**Aaron Chow and the speaker used a slicing technique that was introduced in a recent paper of S. Brendle, S. Hirsch, and F. Johne (BHJ22’) to show that in a $n+1$-dimensional torical band $T^n \times [0,1]$ where $n+1\leq 7$, positivity of a similar operator will assert that the band cannot be too long. If time permits, we will also discuss the possibility of generalising Schoen-Yau existence result of Black hole to higher dimensions, with such torical band width estimate.*

- October 28th, 2022:
__Allen Fang__(Princeton University),**A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter***The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques prove a Morawetz estimate which uncovers a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap argument to conclude nonlinear stability.*

- November 4th, 2022:
__Sam Collingbourne__(Columbia University),**The Gregory--Laflamme Instability of the 5D Schwarzschild Black String Exterior***I will discuss a direct rigorous mathematical proof of the Gregory--Laflamme instability for the 5D Schwarzschild black string. This is a mode instability at the level of the linearised vacuum Einstein equation. Under a choice of ansatz for the perturbation and a gauge choice, the linearised vacuum Einstein equation can be reduced to a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution which has admissible behaviour at the future event horizon and spacelike infinity in harmonic/transverse-traceless gauge.*

- November 18th, 2022,
**from 2:15pm to 3:15pm (NOTICE THE TIME CHANGE)**:__Georgios Mavrogiannis__(Rutgers University),**Relatively non-degenerate estimates on Kerr (de Sitter) spacetimes***I will start discussing how to prove exponential decay for the solutions of the wave equation on a Schwarzschild de Sitter black hole spacetime by exploiting a novel "relatively non-degenerate" estimate. This estimate does not degenerate at trapping. The main ingredient in proving this estimate is to commute with a novel vector field that "sees" trapping. Then, I will discuss the generalization of these methods to the entire subextremal Kerr de Sitter black hole spacetime, by commuting with a pseudodifferential vector field. There are more technicalities because of the elaborate nature of trapping. Finally, time permitting we will discuss how to use this black box estimate to prove stability and exponential decay for the solutions of a quasilinear wave equation on Kerr de Sitter.*

Here are our past seminars:

- October 1, 2021
*(virtual)*:__Sam Collingbourne__(Cambridge University),**Weak Linear Stability of Schwarzschild Black Hole Exterior from Canonical Energy***The canonical energy of Hollands and Wald gives an appealing criterion for stability of black holes based upon conserved current. The method is appealing in its simplicity as it requires one to 'simply’ check the sign of the canonical energy. However, in practice this proves difficult. Indeed, even for the 4D Schwarzschild black hole exterior the positivity was not established. In this talk, I will present a resolution to this issue by connecting to another weak stability result of Holzegel.*

- October 8, 2021
*(virtual)***at 11am (NOTICE THE TIME CHANGE)**:__Stefan Czimek__(ICERM/Brown University),**The characteristic gluing problem of general relativity***In this talk we introduce and solve the characteristic gluing problem for the Einstein vacuum equations. We show that obstructions to characteristic gluing come from an infinite-dimensional space of conservation laws along null hypersurfaces for the linearized equations at Minkowski. We prove that this space splits into an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges. We identify the 10 gauge-invariant charges to be related to the energy, linear momentum, angular momentum and center-of-mass of the spacetime. Based on this identification, we explain how to characteristically glue a given spacetime to a suitably chosen Kerr spacetime. As corollary we get an alternative proof of the Corvino-Schoen spacelike gluing to Kerr. Moreover, we apply our characteristic gluing method to localise characteristic initial data along null hypersurfaces. In particular, this yields a new proof of the Carlotto-Schoen spacelike localization where our method yields no loss of decay, thus resolving an open problem. We also outline further applications. This is joint work with S. Aretakis (Toronto) and I. Rodnianski (Princeton).*

- October 15, 2021
*(virtual)*:__Pieter Blue__(University of Edinburgh),**Linear stability of the Kerr spacetime in the outgoing radiation gauge***This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the "outgoing radiation gauge", a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski's outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski's outgoing radiation gauge.*

- October 22, 2021
*(virtual)*:__Zhongshan An__(University of Connecticut),**The initial boundary value problem and geometric uniqueness for Einstein equations***In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For an initial data set satisfying the vacuum constraint equations, there exists a solution to the vacuum Einstein equations and it is geometrically unique in the domain of dependence of the initial surface. On contrast, the initial boundary value problem (IBVP) has been much less understood. To solve for an vacuum metric in a region with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss obstacles in establishing a well-defined IBVP for vacuum Einstein equations and the geometric uniqueness problem. Then I will talk about some new results in a joint work with Michael Anderson.*

- October 29, 2021
*(virtual)*:__Dejan Gajic__(Radboud University),**Instabilities of extremal black holes***When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes display a variety of interesting phenomena that are not present in more slowly rotating black holes. I will introduce upcoming work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss their connection with previous work on axisymmetric instabilities and the precise shape of late-time power law tails in the emitted radiation.*

- November 5, 2021
*(in-person)*:__Grigorios Fournodavlos__(Sorbonne University, Princeton University),**Stable Big Bang formation***In this talk we will investigate the past dynamics of cosmological solutions to Einstein's equations, containing a Big Bang singularity. More precisely, we will focus on the classical generalised Kasner examples. The celebrated ``singularity'' theorem of Hawking tells us that the past of sufficiently small perturbations of such solutions are causally geodesically incomplete. However, it is not in general known whether such a degeneracy is related to the formation of a curvature singularity. In many cases, unstable dynamics are predicted, which add to the difficulty of the problem. We will discuss a recent joint work with I. Rodnianski and J. Speck that classifies the behavior of perturbed solutions in the so-called subcritical regime.*

- November 12, 2021
*(in-person)*:__Lan-Hsuan Huang__(University of Connecticut),**Existence of static vacuum metrics with prescribed Bartnik boundary data***The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a special Ricci flat manifold, and it is also deeply related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to that rigidity phenomenon, R. Bartnik conjectured that one can find an asymptotically flat, static vacuum metric with arbitrarily prescribed boundary data. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An.*

- December 3, 2021
*(in-person)***(NOTICE THE TIME CHANGE for the two speakers)**

- from 3:30pm to 4:15pm:
__Martin Lesourd__(Harvard University),**Scalar Curvature of Noncompact Manifolds - Geometry, Topology, and Mass***The question "which manifolds admit a complete metric of positive scalar curvature R>0?" has been the subject of a number of breakthrough works by illustrious mathematicians like Atiyah, Gromov, Hitchin, Lawson, Lichnerowicz, Perelman, Rosenberg, Schoen, Stolz, Yau, and many others. The answer for certain classes of closed manifolds is known, but what happens in the context of noncompact manifolds is more mysterious. A related question (recently emphasized by Gromov among others) is "what is the geometry of spaces with R>0?". Both of these questions are related to the Positive Mass Theorem. We describe this connection and some recent work with Ryan Unger and S-T Yau which closes a number of questions on this topic. A score of hard questions remain.*

- from 4:30pm to 5:15pm:
__Maxime Van De Moortel__(Princeton University),**Violent nonlinear collapse inside hairy black holes***A stationary black hole is “hairy” if it admits extra degrees of freedom (“hairs”), in addition to its mass, electric charge, and angular momentum, unlike the well-known Schwarzschild/Kerr/Reissner-Nordström families. Hairy black holes appear naturally in the gravitational collapse of massive matter and, very recently, as a holographic model of a superconductor in the context of AdS/CFT. I will present my recent work on hairy perturbations of the Reissner-Nordström black hole interior, and show it is governed by a novel dynamical instability – violent nonlinear collapse – that destroys the Reissner-Nordström Cauchy horizon and causes a crushing singularity to form.*

- from 3:30pm to 4:15pm:
- December 10, 2021
*(in-person)***(NOTICE THE TIME CHANGE for the two speakers)**

- from 3:30pm to 4:15pm:
__Christoph Kehle__(ETH Zurich, IAS),**Strong Cosmic Censorship for \Lambda< 0***The statement that general relativity is deterministic finds its mathematical formulation in the celebrated `Strong Cosmic Censorship Conjecture' due to Roger Penrose. I will present my recent results on the linear analog of this conjecture in the case of negative cosmological constant. It turns out that this is intimately tied to Diophantine properties of a suitable ratio of mass and angular momentum of the black hole and that the validity of the linear analog of the C0-formulation of SCC depends in an unexpected way on the notion of genericity imposed.*

- from 4:30pm to 5:15pm:
__Rita Teixeira da Costa__(Princeton University),**Hidden spectral symmetries and mode stability for Kerr(-de Sitter) black holes***One of the major open problems in General Relativity is the stability of black holes under the Einstein equations. The Teukolsky master equations describe the linear behavior of perturbations of the Kerr(-de Sitter) black hole family, of which the conformal Klein-Gordon equation is a particular case. As a first essential step towards stability, Whiting showed in 1989 that the Teukolsky equation on subextremal Kerr admits no exponentially growing modes. His method of proof breaks down in the Kerr-de Sitter setting. In this talk, we present a new approach to mode stability, based on uncovering hidden spectral symmetries in the Teukolsky equations. This yields a novel proof of Whiting’s classical result as well as a partial mode stability statement for Kerr-de Sitter. This talk is based on joint work with Marc Casals (CBPF/UCD).*

- from 3:30pm to 4:15pm:
- February 18th, 2022
*(virtual)*:__Gunther Uhlmann__(University of Washington),**Seeing Through Space-Time***The inverse problem we address in this talk is whether we can determine the structure of a region in space-time by measuring point light sources coming from the region. We can also observe gravitational waves since the LIGO detection in 2015. We will also consider inverse problems for nonlinear hyperbolic equations, including Einstein's equations, involving active measurements.*

The

**organizers**,

Elena Giorgi, Mu-Tao Wang