 I'm particularly honored because I think I'm the only mathematician here. Presumably Christodulu was also supposed to be here, but he didn't come. So really, I'm really honored. Thibaut was very important to me, even though we did not collaborate on any paper. I always enjoyed coming whenever I would come to the institute. I had very interesting discussions with him. He's one of the few physicists who actually showed a real interest in what I had to say about problems in mathematical general relativity. So thank you very much. I mean, it's really a pleasure to honor you here. So let me remind everybody that we're talking about general relativity, the Einstein equations. I'll take the simplest case, which is the Einstein vacuum. Reach equal to zero. Initial data sets with constraints. The concept of maximum future global hyperbolic development, which was very much developed here in France, in particular by Bon Jovi-Blois and Geroch. So this is, of course, a picture. And of course, the most important thing in connection with my talk is that these equations admit a large family of stationary solutions. This is a famous care family, depending on two parameters. I will talk only about the case, reach equal to zero. There are also very interesting results in cosmological constant case, results again at a mathematical level by Friedrich and also Kinsen-Wassi more recently. A few words about mathematical general relativity since I'm, again, the only mathematician here. I should say something about what exactly we want. So first of all, a big theme in mathematical general relativity is to elucidate the mathematical structure of the classical general relativity. But not at random. You really want to connect it to interesting equations, which are central problems in general relativity, which are both physically relevant, but they also satisfy our mathematical sensibilities, which is very important for mathematical relativity, which is beauty, rigor and novel mathematical challenges, which is often something that maybe a physicist doesn't really care about. And of course, establish connections to other problems in PD and geometry and other parts of mathematical physics. In the spirit of this mathematical entanglement that I like to call, which means that mathematical concepts, so this is Wigner, said by Wigner, mathematical concepts introduced for solving specific problems that have unexpected, mysterious consequences in seemingly unrelated areas. Somehow, a problem that we are interested, sorry, a problem that we are interested in, are chosen also because of this possibility of interesting mathematical theories and interesting mathematical ideas have significance in other areas. So I will talk about stability of care, so the conjecture I guess is known by everybody here. You start with, so this is a picture of care and you start, so this is a black hole and this is a domain of auto communication, you start with an initial data set of care, of the specific initial data set and you look dynamically what happens for a large time and the conjecture is that you are going to converge to another care solution, not necessarily the same but another one. And the result that I want to talk about it is that, and this is work in collaboration with Jeremy Zepterki in France, is that the conjecture is true for sufficiently small angular momentum and I'll clarify what I mean by this statement in the fact that it's not completely, yeah, so first of all let me, sorry, before saying anything else let me mention some of the results in this context of the Einstein vacuum equation, the asymptotically flat solutions, also it's important to note the only results on stability at a non-linear level, so I'm talking about regular mathematical results on stability at a non-linear level is the stability of Minkowski in, which was in 1990-1993, so this is my work with Christodulu. And then it took quite a long time to get to stability of Schwarzschild and so this is again work with Jeremy Zepterki, so this was between 2018 and 2020, again we had different papers on this. It's under the assumption that, you have to make an assumption because if you start with a typical perturbation of Schwarzschild you may not go back to Schwarzschild, you may actually converge to a small A-care solution and there are lots of technical difficulties in the case of care which we wanted to avoid, so we look at the simpler case of stability of Schwarzschild in which case we had to make initial conditions which we call polarized, which ensure that the final state is also going to be a Schwarzschild solution. So that's what we did there and then there is a recent work by four people, so that's Daphermos, Holzegel, Rodjanski and Taylor who prove the same result in a more general setting where you look at the co-dimensions free of initial conditions which are not initially, you cannot initially specify, but you show that there exists a co-dimensions free for which you have stability. So in connection with the most recent work, we have a series of papers already, so this is 2019 with Jeremy Seftel, so this is something which is very, very important in the construction that I'm going to talk about it, is a construction of GCM spheres in perturbation of care. So this is one of the essential points in the stability of care result. So these are two such things, I'll maybe say more about them later. And there is another paper which I will mention in which we introduce a general formalism for the stability of care. And finally, this result, which is the most recent work, it still has a piece missing, in fact. So this is most of the steps necessary are in this paper, but there is still one which is missing, which is working collaboration with Seftel, myself and Elena Georgi. So maybe I'll mention a little bit later what that is, but we definitely believe that it's work in progress and we definitely believe it's true. So yes, the care family of solution is stable at least for small a. Now it's probably true also for the whole extremal case, but there is only one place in fact in the analysis that we have which really requires small a. The arguments really are not dependent on the smallest of a. Okay, so let me talk a little bit very fast about the general stability problem. So we look at, in general, you have a nonlinear problem, nonlinear PDE typically. You have a stationary solution, which is this one, and you make a perturbation, so you look at solutions of this type. There are three types of stability that usually you work with. There is orbital stability, in which case you show that size stays bounded. So you make a perturbation and for all time, size stays bounded. So in other words, you stay close to phi 0. This is very rare. It's for very, very simple equations that you can ever prove such a thing without proving much more, which is based on asymptotic stability. So here you show more. You show that not only size stays bounded, but actually goes to 0. So this is one of the, it's a case which would call asymptotic stability. But the more interesting type of result, which is what happens actually in the care stability, in the stability of care, is orbital asymptotic stability, which means that this perturbation converges to something different to another stationary solution. So you start to do this, you make a perturbation, you converge to something else. And that's exactly the situation in the care stability problem. You can see that by looking at linearized equation. That's of course what people, people always do at the beginning. You linearize the equations around the stationary solution. So these are, let's say, think of the Einstein equations here. So n is the Einstein equation. You take the linearization. You get a complicated system. And then because of the symmetries of phi 0, these equations have certain symmetries, which allows you to look at mode stability, to discuss mode stability. So you look at the compositions into modes and you show that every mode is stable. For example, you show that there are no exponentially growing modes. So this is sort of typically what physicists have done from the beginning. But of course, mode stability doesn't even give you boundedness of the solution at the linear level. And I want to mention to go from a linear to nonlinear, you almost always need decay, right? So it's not enough even to have boundedness. You actually have to have a quantitative decay, a way of measuring how the psi decays, which is of course consistent with the asymptotic stability here or orbital asymptotic stability, which is a little more complicated. So you need quantitative decay. So the problem is that typically there are zero modes. In other words, there are modes which are simply bounded, so they don't decay. So if you look for example at the linearized Einstein vacuum equations, there is a huge family of zero modes, which have to do with the symmetries. Well, first of all, the presence of a two-parameter family of solution, right? So you have A and M. You can differentiate that every... In other words, you can take a variation corresponding to the family itself, and that gives you a bounded solution. Or general covariance, for every solution of your equations, you can make a deformorphism. You make a one-parameter family of form, if you take the derivative, you get another bound. You get a huge family of bounded solutions, right? So this is a situation where you have huge kernels. So you don't typically have... I mean, you don't have quantitative decay for all modes. You have to have bound and you have to obviously separate the two, okay? So the hope is of course that all these zero modes correspond to the fact that you don't converge to the original phi zero, but you converge to something else. Of course, this involves only two parameters here. But there is something else which is of fundamental importance when you do the issue of stability, which has to do with decay. Decay is a gauge-covalent fact, right? So you prove decay relative to something in a given coordinate system. So you automatically, if you have to prove something like this, you also have to find the right gauge. So it's not just finding the final parameters. You also have to find the correct gauge conditions relative to which converges takes place, okay? So that's sort of the... Maybe I would say in just a few words, it's maybe the main difficulty of the problem. So relative to formal mode analysis, I go very fast over the history. So there is a metric perturbation analysis, which was done by Reggie Wheeler even in the late 50s. This is done at the level of the metric. So in other words, you do perturbation of the level of the metric. There is this Visha Vashar Zeriri. And then in the Neumann-Penrose formalism, when you actually look at curvature, there is the work of Tchaikovsky and Press Tchaikovsky, but in Press, which are related to this issue of showing that there are no exponentially growing modes. You can have bounded modes, but there are no exponentially growing modes. And there is also something very important which plays a very important role in this story that I'm talking about. There is a transformation discovered by Chandrasekhar that connects these two theories, and it was also continued by Wald. It shows you that somehow the main equations that you derive here, or the main equations that you derive here, you can re-derive equations which look like this one, starting with this one, or vice versa. So in particular, what's important for us is that you can start with these equations and you derive equations which are regular. It just so happens that the regular equation are better in terms of analysis. If you want to get quantitative decay, as I mentioned, it's better to work with, at least in certain aspects, is better to work with regular. On the other hand, the formalism is much better at the level of Neumann-Penrose. So you are stuck somehow with this. The equation which I'm going to talk about, which is a Teokolsky equation, is not that good from the point of view of the analysis, and you have to go to this one, and this will sort of put these two things together, it allows you to really take advantage of both theories. So the full mode stability was done by whiting. So again, this is at the level of mode, so this is more or less what was done in the golden era of black hole physics. Obviously, the story, I mean, lots of interesting ideas, but the story is far from being complete. So what was missing is obviously issues of boundedness and decay. Just boundedness of modes is not enough to even prove boundedness of solutions, of all solutions of the linearized equation. So here, there have been a lot of very powerful messages, which are this time by mathematicians. So this is where mathematical generality comes into play. There is a message to have to do with Minkowski space, which go back to Moravets, myself, the so-called vector field method, and stability of Minkowski, which was this work of Christodule and myself. And then, so this was sort of a big story, which was kind of finished in the early 2000s after stability of Minkowski space. So the new things were to study wave equations, so very simple wave equations on black holes and actually prove again, prove decay, boundedness and decay. So this is work of many, many people. So I mentioned here, software is starting with software and blue. So software blue, I mean, blue software, sorry. Fermos Rodianski played an important role, many new ideas, which played an important role in the analysis. This work here for, so this is in the case of Schwarzschild for care, which the situation is much more complicated. There are very important work of Fermos Rodianski, the Taruto Honianu and Anderson Blue. In particular, Anderson Blue is something that I will mention because it plays an important role in my work with Jeremy and Elena. Scalar wave equation all the way to the extremal case. So this is work of the Fermos Rodianski and Schlappeter-Rodman. It's very important, but it's very much based on multi-comp... It's not just the vector field method, it also uses very strongly. In fact, this one, these two here, also use the modes together with vector field methods. It's only this one that does not use multi-composition and that's the one that is relevant to our work. Linear stability, of this equation, in the case of Schwarzschild, this was a very important work of the Fermos Rodianski. And then in particular, these works have to do with part of linear stability. So again, it's linear stability in the Neumann-Prenzl picture. It's works of Ma, the Fermos Rodianski. So this is a linear Teukovsky equation, which is maybe the most important wave equation part in what you have to do to prove stability in the linear case. And then there are some other more recent works on linear stability of Kerr for small a due to Anderson, Bagdal, Blu and Ma, Hintz and Vassi. So, of course, none of these results necessarily generalize to the nonlinear stability. So to go from linear stability to nonlinear stability, it's a huge stretch, right? I mean, but all these ideas are important. In fact, everything developed in this slide is very important in our own work. So for nonlinear stability, as I said, you need gauge condition which have to be dynamically defined. You cannot define them at the beginning. You cannot impose, this is what people always do, but it doesn't work. You cannot impose gauge conditions at the beginning and hope that they are the ones that will work in the end. So you constantly have to redefine your gauge condition dynamically. You need, of course, a mechanism of defining the final state. So the same thing, these parameters, AF and MF, you cannot define them at the beginning. You actually have to work hard to get them. The issue of the gauge, you need decay. And of course, you cannot only expect to have decay relative to a specific gauge condition, as well as a specific coordinate system if you want. Here are some examples of some very simple equations, like sort of just a scalar wave equation with a nonlinear term on the right-hand side. In general, quadratic nonlinearity here in three plus one dimensions are not stable. So the solution says equal zero is not stable. However, and of course, the vector field method plays a very important role in this analysis. However, if the null condition is verified, so there is a structural condition on the nonlinearity, which turns out to be very important in stability of Nikolsky and also in stability of these care black holes in the nonlinearity. If that condition is verified, then you do have stability of the zero solution. So that was an important step understood in the 80s, essentially in the 1980s. So there is this null condition, which you hear all the time in mathematical relativity plays a very important role in many, many things. So this... Yeah, I mean, again, this is the sort of thing that I mentioned when I talked about mathematical relativity that you work on a problem, you find something interesting, but it turns out that the same thing, and by the way, the vector field method is the same, the same things are important in many, many other problems. All right, so... And of course, you also need what is maybe the hardest thing of all, a strategy to this integral of the nonlinear interdependence of all these things, right? So there are many, many things that come together, and linearly, they are disconnected, but in nonlinear theory, they all are intertangled. Okay, so stability of Minkowski space I mentioned. I want to talk a little bit about the framework, because this is the kind of thing that it's different, the framework that we use is slightly different from what is used by physicists, which is the Neumann-Penrose formalism or variation of the Neumann-Penrose formalism. So I want to say a few words about it. So, like the Neumann-Penrose, you start with an alpère, which is E3, E4, normalized. You look at horizontal structures. So this is the important thing that we do. Instead of choosing specific frames, which are perpendicular to E3, E4, which are the spin frames and so on and so forth, which is done in Neumann-Penrose or variation of Neumann-Penrose, we take this horizontal structure seriously. So we keep it geometrically. It's just a horizontal structure. We don't need any particular choice of EAE2. So, for example, now I can define now second fundamental forms of the horizontal structure in the usual way. Well, there should be a B here. So we take DAE4EB, where EAEB are any frame on edge. So I don't need to take a specific frame. So this define kappa, kappa bar. And now the important thing is that if this horizontal structure is not integrable, which is a case of care. So in care, it's very important this natural horizontal structure is not integrable. Then these things are not symmetric. And instead of having just the trace of the second fundamental form, which are usually called expansions in relativity, you get what I call the asymmetric trace, which corresponds to the asymmetric part of chi, the same thing for chi by AB. And these are the only new components. All the other components, in fact, are written here, are very defined exactly in the same way as in stability of Minkowski's space. But again, it's relative to this horizontal structure. In that case, the horizontal structure is integrable, and here it's not. That's the only difference. So somehow the equations become more complicated because of the presence of these two. Integrability is nothing else but the fact that these two are zero. And now, again, also, of course, define curvature components. So this is very similar to what's done in Neumann pentals, except again that these are two forms. They are not scalars. I never get scalars. And in fact, it's terrible to work with scalars in nonlinear stability because you have to do a lot of differentiations and you get a mess with the frame. The frame is not particularly very good when you have to make certain choices. So the point of this is that I don't have to make any choice of horizontal frames. And therefore, I define this as a two-form, this as a two-form. So this is the usual psi, I guess one is psi zero, one is psi five, maybe in the Neumann pentals. But now there are vectors. Sorry, there are actually tensors. Two tensors, they are symmetric because of the symmetry so far. It turns out that you need to complexify to get the equations to be simpler. Complexification is now very important. Of course, complexification also plays an important role in Neumann pentals. So you define complexified quantities and then you get the usual Cartam-Bianchi equations, which are like this, but of course you have lots of them of course. So the comparison that I made is with my work with Christodulu where we use the thing, but again, in our case, horizontal structure was integral and here it's not. And of course Neumann pentals, that's a usual formula until now I think everybody was using or variations of the Neumann pentals. The care family, of course, I recall very fast. So this is in Bore-Linquist coordinates. You have the t and r coordinates theta and phi. You have these terms here, which are expressed in terms of t, r, and t, theta. And here is a picture. Care is, of course, stationary and axisymmetric. In other words, it has dt as the kinetic vector fields and z is d phi as the kinetic vector field. And again, I mean, this is a story that we want to understand what is the stability of the external part of the care family, of the care solution. So now the important thing about care is that you have a principal null frame. It's known as principal null frame, which has expressed the t to Bore-Linquist coordinates. It looks like this. And the crucial fact is that this quantity is a bar. So these are the curvature quantities and they are all zero with the exception of the middle one, which is equal to minus 2m over q cube. So q is exactly the r plus a equals 90. But I guess I had it before. Yeah, this is what it is. Okay, so in addition, you have that a lot of components of rich coefficients are also zero. They are chilled. And this is important, of course, that this pair is actually integrable. So in other words, the horizontal structure is integrable in this case. Also the imaginary parts of this a-trace-kai, a-trace-kai bar, of course, zero. And also the imaginary part of p is also zero. And finally, in Minkowski, you have all the curvature components are zero. And you can see why stability of Minkowski was the easiest, because it was very hard. Stability of Schwarzschild was much harder, and stability of Kerr is even harder. So I want to give you a sense of perturbations, because this plays an important role. You see, obviously, you want to pick up a particular frame. So let's assume that we have an horizontal frame on Kerr, which is such that it's close to the one of the real Kerr. So how do you measure this? You take the gamma and the R of your perturb solutions and subtract from it. Of course, you have to know what that means, how to subtract. You subtract the Kerr values. For that, of course, you have to have these functions, r and theta, defined, so that you can compare. But anyway, this is something that, in principle, you can do. And obviously, you want an epsilon perturbation. So let's assume that this is all of epsilon. Now, the problem with this, of course, as it's very easy to see, is that this very much depends on which frame you take. So, for example, if I give you a frame e1, e2, e3, e4, e3, e4 being the null pair, I can perform a transformation to a new pair. I can make an exact transformation, in fact. It's not too difficult, which looks roughly like this, with f, f bar, and lambda being like this. So they are all of epsilon. This is, lambda minus 1 is all of epsilon. You calculate this, and then you look at what happens with the new curvature components in the new frame. And you see they are also all of epsilon, so you haven't made any improvement. With the exception, and this is remarkable, with the exception of the a and a bar, the extreme curvature components, which are all of epsilon's going by. And so this is essentially, in essence, the discovery of Teukovsky. So Teukovsky really just observed this fact, that they are all of epsilon invariant, and they verify wave equations, which again, it's pretty much connected down. Okay, sorry, a problem chest. All right, so I will have to go fast. So this is, these are the Teukovsky equation. So it's an equation for a. In a perturbed spacetime, you have error terms. And here is a transformation, Chandrasekhar transformation, that allows you to go from this equation to something that looks more like a regi-wheeler equation. Now, if a is 0, which is the case of Schwarzschild, this is a little bit easier. By the way, the first transformation of this type was done by Daphermos Rodjanski and Holzegell, Daphermos Holzegell and Rodjanski. In the case of Schwarzschild, where m, y is equal to 0, in that case, you get exactly the regi-wheeler equation. Otherwise, it has these additional terms here, and we call it generalized regi-wheeler equation. Yeah, unfortunately, I'll have to go very fast. Okay, so let's strategy, talk very fast about the strategy. Expect to be able to treat a using Grw. So, okay, so the point is that you look now at all possible components that you have of the perturbed metric. So a and a bar are these extreme curvature components, and they are essentially all 5-synons in perturbation. So you separate them from everything else by the fact that they verify this good equation that I mentioned earlier, the Stelkovsky equation. Of course, you have error terms. So in that equation, you obviously have error terms on the right-hand side. So error terms depend on everything. But they are quadratic, so that makes it a little bit easier. In fact, these terms, you treat them sort of similarly to the way they are treated in the Stelkovsky space. You use a null condition and so on and so forth. Because normally, you'll have instability. But the structure of the Einstein equations allows you to really, in principle, control those terms. Provide it, of course, that you know how to control every component, gamma hat. So you have to control these two components and then you control the quadratic terms by the null condition. So now the point is that this a, a bar are quasi-invariant because they are all of epsilon square invariant. But the other quantities are not. They are all of epsilon. So now everything else depends on specific choice of coordinates, or specific gauge choices. So we set up a continuation argument based on a family finite of GCM admissible space-time with specific bootstrap assumptions which we recover at every stage. And maybe this is the last slide as I will be able to do in the next two minutes. So let me try to explain it. So you see, you want to construct your space-time starting with initial conditions. It's not too difficult to show, so these are results of Nikola and myself, which are based on technology of stability of Nikovsky space. It's not too difficult to show that if you start with a space like hypersurface, you can construct the space-time in a way to this kind of... So this is almost like a null hypersurface. It's not null. In Schwarzschild it would be exactly a null hypersurface, but this way it's time-like really, but it becomes asymptotically null. So in principle you can imagine that you have data set given to you in a layer, in a boundary, in an initial data layer, which looks like this. So I know the space-time here, it's constructed by techniques which are well known from the initial data. So anyway, the data is given here in the boundary layer. Sorry, but the boundary layer, you should think about it as being a smaller region like this. And I extend my space-time region to something which looks like this. And this is a finite, what I call a finite admissible space-time. Which I want by a continuity argument to go to infinity. So in particular, for example, this space-like hypersurface here will become SkyPlus. So everything will move towards the picture at infinity. Now, in fact, maybe I can show the picture at infinity. Let's see if I have it. Yeah, so this is the final construction it's going to look like. This is where you pushed everything to SkyPlus and you have a SkyPlus which is complete. So the point is how do you construct such a space-time? So the fundamental importance, and that's the one that's connected to this coordinate that I've been talking about all the time, is the Sigma star. So this is the one that I mentioned will go all the way to SkyPlus. So this is where we impose G-CM conditions. So these are general covariant-modulated spheres or hypersurfaces. This is the hypersurface. But the hypersurface itself starts with this star which is just a sphere. At every stage in the continuity argument, of course you have to push the space-time further and further and you have to reconstruct the surface as star and the hypersurface Sigma star. And this is maybe the most delicate part of the construction because the way you construct this as star is by imposing a certain number of conditions which is exactly connected to the numbers of degrees of freedom that you have in the Einstein equations. So that's where you use a full covariance of the Einstein equations in order to set these conditions here. And then similar conditions, there are additional conditions which are on Sigma star which I obviously will not be able to talk about. Once you have this constructed, everything else is constructed by transport equations. So you transport the equation this way and then you transport the equation in the opposite direction going this way. And then there will be a region here because these are time-like hypersurface. There is a region here which is actually quite small and you can treat it by the local existence here. And the spacetime, as you see, is constantly being upgraded. So, and also important is that the axis mass and angular momentum of the final spacetime is determined on a star. So it's in your choices of a star and coordinates of a star, you use a uniformization which we call an effective uniformization which is uniformization results for the sphere are usually not unique, but there are certain ways of making that unique. We call them effective uniformization conditions and they pick up coordinates, they pick up the axis, they pick up the angular momentum, I mean, you can define the angular momentum and the mass, and then you transport everything like this and then like this and you define it everywhere else. But the most important thing is, of course, this is a star which you are constantly readjusting. So as a consequence, you are constantly readjusting also this notion of axis of symmetry, the A and M and so on and so forth. So this is a final result, the initial data layer that I mentioned which is here and you construct a spacetime which goes all the way. So as I said, the result is not entirely finished because there is this remaining part which... So there is a very natural separation between the major part which is already in our paper and a second part which is just essentially on proving estimates for the generalized Teukovsky equation which we also think we have it, but it's not your part. So this is our stop. Other questions? What are you doing? Okay, so I've seen you first. Can you obtain an expression for the final angular momentum? No. Final mass in terms of... No, no, no. I mean, in principle, no, you can't. I mean, everything is done by a continuity argument and you're constantly readjusting everything so it's very hard to trace it back. Can you bounce over some... Oh, yeah, bounce. You certainly can, yeah. Bounce in terms of the perturbation, yes. Absolutely. And also, there are other things that you... There are other things that you... some monotonicity things maybe... There are some qualitative things that you can determine. But not a formula. Okay, yes. No question. It's a more general question than the specific problem of general relativity. Can you show us with the example of the... the scale up? Yeah. That linear stability is not sufficient to ensure... Non-linear stability. Non-linear stability. But is it nevertheless necessary or there's nothing... Oh, yeah. Well, yeah, linear stability, of course, is an important one. But what you mean by linear stability depends on somehow what you want to prove at the end, right? So you have to have... At least that's the way I think about it. You have to envision a certain type of proof and then somehow you linearize and... I mean, at least conceptually, you never... In our proof, we never linearize anything. We just conceptually... We do something which conceptually looks like a linearization, but we don't linearize anything. Now, I should say, for example, in answer to your question, stability of Minkowski's space at a linear level is totally trivial, right? It's just a way of a question. And yet, it took us about 550 pages to prove it. So it's a huge difference between linear and nonlinear. The definition... Oh, yeah. But you need... Transition or... Yeah, yeah, yeah. You definitely need linear stability. What I'm saying is that how you define... There are many types of linearizations that you can do, right? And depending on, you know, some are maybe not as good as others. That's what I'm trying to say. I'm sure that as a physicist some combination idea may be a meaning. Yeah, I mean, sure. I mean, definitely, you as a physicist, you should continue to do whatever you're doing, because it's great that we have all these insights. And I think it's extremely important to have all these insights. But in the end, obviously, there is more to this issue of passing from linear to nonlinear. But if you have a linear instability, which is mine, like just power of law, and if you had strongly nonlinear things that kill them when they become large, it would be... It would be okay, but I don't know. Do you have an example of this? I mean, where you can actually have... You specifically have something which is growing. Yes. A mold which is not bounded, but growing. And yet we... I actually hate trying to... Do you do something like this? Ah, okay, good. So, yeah. Yeah, no, that's very interesting. Of course, bounded is okay, right? As we saw, you can have bounded modes, and you are still okay. But growing modes, I don't know of any example yet. Okay. I would say no. In principle, I would say that you don't have. Unless there is something really miraculous happening. So, if we go beyond the horizon, do you think that the inner horizon will be replaced by the singularity and probably the space-side singularity doesn't even make sense to ask the question mathematically? Yeah, yeah, yeah, absolutely. I mean, there are people who work on this. What happens beyond the horizon. So, for example, there is a very important work of Jonathan Luke and Daphermus. Exactly on the issue of... So, you take the horizon of the care solution, you make a perturbation of it. Now, the perturbation, obviously in principle, it has to be connected with the perturbation from a space-side, but somehow, if stability of the care is done, then you would think that you know what the reason of the perturbation of the horizon that you can make. So, you start up with those perturbations and you show that either you form a space-side singularity in the future or on the country, you stay null. And, in fact, they show, at least in perturbation, when you make this small perturbation of the care horizon, you stay null. In other words, the singularity is stay null. So, there is a stability, in other words, of the Cauchy horizon. So, it's called... Well, right. They cannot prove that it becomes singular, but it stays there. In other words, it's stable, at least in the kind of perturbations that they have on the Cauchy horizon. But the belief now is that the Cauchy horizon is stable in perturbation starting from care. So, for the initial data, do you always consider the data on a space-side surface or, I mean, sometimes maybe a time like to work you around the source, something like that? Well, that's very ill-posed. I mean, from a mathematical point of view, that's a no-no. It's very, very ill-posed. The Cauchy problem outside is ill-posed even for the wave equation. So, if I just take the wave equation in Minkowski's space and I set up initial conditions on the cylinder and I look outside, that probably is ill-posed. I mean, for example, a space-side surface and then running... Oh, it was boundary conditions, you mean. It was boundary conditions. I mean, there are boundary conditions that you look outside, right? And you also wonder about... Yeah, I mean, you can certainly ask that question. Yeah. But that's not the initial value problem, of course. I mean, you cannot put the initial value problem on a time-like surface. Okay, that's another question. Yes? My question is actually not so much, I mean, only restricted to the rotating case. What can you say about space-times and dimensions different from before? Well, so the belief is that... Okay, so there is a lot of linear... There are many more solutions in higher dimensions, right? There is a PERI and I don't know about the ring solution. Anyway, I don't... I never worked in higher dimensions, so I can just refer to the fact that there are linear stability results in that case. Both stability and instability. So some of the things are unstable. I think the ring solution is unstable, right? Minkowski's space is shown to be stable in all dimensions. And Minkowski's space is in all dimensions, and it's easier. The higher the dimensions, the easier it becomes. Yeah, right. Yeah, it's much easier in higher dimensions. So, for some reasons, 3 plus 1 is the critical one. There's nothing that singles out 4-dimension in Minkowski. Nothing that singles out 4-dimension? 4 plus 1, you mean? 3 plus 1, I mean. 3 plus 1, it's singled out, right? Well, I thought it was stable in there. Well, it's singled out because it's... It's complicated. It's complicated and you have... Stability, stability... I mean, we have no linear stability results of black holes. I mean, this is pretty remarkable. Yeah. And in higher dimensions, I don't know. I mean, I'm not sure. Be careful. Again, yeah. So, you can ask, of course, the same kind of question. If you have linear stabilities, then maybe you hope to have also non-linear stability. I presume that's true. Yeah. Are there further questions? Yes, there's another question. Yes. I don't know if it's a question for you, but you mentioned that you cannot fix the gauge from the beginning, you have to adapt it. Yes. Wouldn't this be a problem also when you do numerical simulations of the other? It's a good question. I don't know. I think it's probably people do something like this, right? I don't know. Yeah. Yeah. Right. But I guess you make probably a finite number of adjustments in that case. Right. It is evolved during the evolution. So, you actually continuously evolving? Okay. So, then that's quite similar. But that's, yeah. There is one, excuse me, just one thing. So, there is maybe one difference. So, I guess you allow it to evolve starting from the initial data, right? Yes. Right. So, here it's very, very important to think that you are coming from infinity. So, you are always adjusting at the last boundary, so to speak. Right. So, you are adjusting in a sense from infinity. Good question. That's why we talk about this layer. Right. So, yeah. So, you have to compare that itself. You get new data coming from infinity and you have to compare with the other. That's a tough part, in fact. Actually, it's not easy. And it can be relatively large deviations, in fact. So, from your initial data and the one that you come from infinity, there is a large deviation, which presumably corresponds to, I don't know, something physical, right? There is a boost of the black hole. Yeah. So, you said at some point that there was only a small part of the tooth where you had to constrain two small speeds. Yeah. It's easy to explain where it comes from. It's because of the trapping. The trapping makes life very complicated in terms of getting estimates for the Tukolsky and the disgeneralized Regi Wheeler equation. So, it's a technical issue, but, of course, it's an important technical issue at this stage. So, people can... So, this has been solved in linear theory, however. The issue has been solved in this work that I mentioned by... No, I don't understand. I don't understand. It's still for small a. No, this was still Holzegerl... I mean, Daphirmus Holzegerl-Rodiansky and Ma... Sorry, no, excuse me, and Jakov Schlappent of Rotman. I'm sorry, I'm kind of tired. Jakov Schlappent of Rotman, they treat the case of this Tukolsky equation. So, it bounds for the K and bounds for the Tukolsky equation in the full sub-extremal case. But they use very, very strongly the structure of the care solution. You know, the fact that you are in care. So, in conversion of care, it's much more difficult and we'll see how it... Yeah, I suspect that it will be done in the next five years, yes. There's an online question. Maybe... There's a question online, which is... Ah, there's a question online. Can you give it free? Well, I can try to read it out. Yeah, I don't know whether we can ask the person to unmute and ask it. Yes, that is my question. But maybe it's easier if I read it out. So, the question is whether Minkowski's space-time is stable against De Sitter's space-time. I'm just reading, I can't give more information. But the question is especially... But they see that space is really the case of non-zero cosmological constant, it's not really here at all, right? So, it's a totally different story. There are other results. There are many results. Hints and varsity have interesting, very beautiful results on that case. But it's easier, usually, when you have positive... So, this kind of stability problems are a lot easier if you have a positive cosmological constant. They become much harder if you have a negative cosmological constant. That's a different story. Yeah, they're instant stable, right? Exactly. Then you have answered all questions. Thank you very much. We'll see you again. Thanks.