 Thank the organizers for inviting me. It was a wonderful occasion. It was Yvonne, of course. I know her for essentially the beginning of my career, maybe the first year after I finished my PhD. I remember discussing with her the global existence. I remember being very intrigued of one of the examples that she has for non-linear wave equations in three dimensions, she showed that in principle you could not have global existence. Fortunately, it happened that the Einstein equations had better structure, so in the end we could prove the stability of Minkowski space, together with Christodulu. Yvonne, who in a sense indirectly was responsible for getting Christodulu and I together. Christodulu has in fact visited you for a year or two years and I think that's when she started to think about the stability of Minkowski space. I did it in a different way. Anyway, we met and we both felt that our subject started in a sense with Yvonne's great work in the 1952, which opened somehow the whole subject of general activity. Because to me the subject of general activity or at least classical general activity is really the subject of the initial value problem, which obviously started with Yvonne. And in general I should say that it is here in France that the initial value problem was in general activity, was seriously developed. With Loré, Yvonne Chocqueblouin, and I guess Nick Narovitz. And I don't think, I think it was really just in France. I don't think it was any, I mean people are not interested in the initial value problem, at least outside France. And at a personal level of course Yvonne has always been extremely warm and encouraging and I always had a very good time whenever I meet her. And happy birthday Yvonne. Alright, so the subject of my talk is on the reality of black holes. Okay, happy birthday. So here is what I'm going to talk about. First of all I discuss very fast what I mean by reality. Then I introduce black holes and I talk about the main issues that have to do with what I call reality of black holes which are rigidity, stability and collapse. And first of all here is a picture of sort of a cartoon picture of what I mean by reality. So you see here it's what can call perception of reality that's happening here on Earth. Here you have physics which in some sense takes sort of a middle way between what one might call mathematical reality. A lot of mathematical equations come up here which of course are intimately tied to things that happen in the real world. And then of course here you have subjects such as classical mechanics, geometry. Geometry was maybe the first subject within mathematics that clearly started with observations in physical reality or observation in perceptional reality if you want, the Pythagoras theorem and so on and so forth. And then they developed all the subjects developed sort of independently after that within a mathematical framework and that's the way I want to think about general activity also. It's a subject which of course started here, it went there, it was formulated by Einstein in his famous Einstein field equations but then somehow we can think about the subject as being a subject losing mathematics and now just in the same way as geometry was developed independently to some extent from where it started with but nevertheless having deep contact to physics. In particular it led later on to as we know to general activity. I want to think about also general activity as a subject which is purely a mathematical subject but which has intimately connections with the physical world. So this is a cartoon picture of what I mean. Now this is what one might call mathematical reality and you see there are subjects like geometry, partial differential equations, general activity which are intimately interconnected and which develop within, I mean, this would be sort of the image of physical reality and this is how the subjects develop within mathematics. So now more seriously the issue of what I call reality of black holes has to do with the questions whether physical reality can be tested by mathematical means in the framework of a given theory. So in particular in the case of general activity black holes are well known to be special solutions of the Einstein vacuum equations but they also have physical meaning in the sense that as we all know in astrophysics one expects that there are plenty of black holes in the universe in particular in the center of galaxies and of course whether they are really tied to the special solutions of the Einstein equations can be debatable. In any case I won't talk about the astrophysical side of it, I will talk about the mathematics and the point I want to make is that one can formulate the issue of reality of black holes in purely mathematical language. So are these objects which are obviously important in astrophysics, are they real? And the question that I'm asking can be formulated in mathematical language in the following way. So stationary asymptotic and flat solution of the Einstein field equations in vacuum, in other words I'm only interested in, I should say the Einstein vacuum equations refer to equations which are tied to a particular matter field. For simplicity I would just take the matter field to be zero. In other words I'm looking at the Einstein vacuum equations and as we know black holes are simply stationary solutions of the Einstein equations and the definition of black hole or more precisely an extender black hole is asymptotic. So an extender black hole is really an asymptotically flat regular Lorentzian manifold with boundaries. Deferomorphic to the complement of a cylinder in R1 plus 3, I'll make this a little bit more clear in a second. And in addition to being a solution of the Einstein vacuum equations it also satisfies, it's also asymptotically flat and it has a keeling vector field which is asymptotically time-like and the derivative of the metric relative to t is equal to zero. That's the definition of a keeling vector field. So that's the definition of a mathematical definition of a black hole and we have explicit solutions of such black holes so the care family depending on two parameters which is expressed in this border-linked coordinates here. So never mind exactly the precise expression that you have. The important thing to keep in mind is that the solutions have two keeling vector fields, one which is d over dt which is reflected in the fact that all the coefficients of the metric are time-independent, do not depend on t. The coordinates are t, r, t times phi. And a second keeling vector field which is just the derivative with respect to phi. So this is a care solution. And of course a particular case of the care solution is a Schwarzschild solution in which case a is equal to zero, m is strictly positive, and the solution in this case is static and spherically symmetric. So it's a little bit more than stationary and I won't say more than this at this stage. So here is a Penderl's diagram of the care solution. So here you have the two ends of the care solution. This is the horizon r equal r plus. R plus is a solution of delta, which is this quantity here. And of course in the black hole region, the black hole region terminates with the Cauchy horizons, which are r minus, r equal r minus, which is the other solution of the delta equal to zero. External regions, you have these two external regions here, which are asymptotically flat. So you have a scry, scry plus and scry minus. Again the event horizon is r equal r plus. And when we talk about the external black hole, we talk about this region up to the event horizon. So this is sort of the observable region. Of course we cannot see what happens in the black hole region. All right. So again, this is another picture of the external care solution. That's this one. It corresponds to a rotating black hole. Again, this is a solution of a stationary axisymmetric. They have, in the external region, there are no trap surfaces in the sense of Penderl's. Of course, there are plenty of trap surfaces within the black hole region. There is a non-trivial ergo region in which the energy, which corresponds to this vector field T. So this is a Keating vector field, the stationary Keating vector field, which is time-like far away from the black hole. But it can become space-like in the vicinity of the black hole, exactly in the ergo region, in fact. All right. So, and of course this leads to all sorts of difficulties both physically and mathematically. The major difficulty connected to it is that the energy, which is associated to the vector field, is non-positive in the region where T is space-like. And then finally, there are trap null geodesics. In other words, they exist null geodesics. So typical null geodesic, which unfortunately here we cannot see, the only trap null geodesic, sorry, the only null geodesic that we can see in the picture are the radial ones, which can either go to Scribe plus or can go to the black hole. But in fact, there are also more complicated type of null geodesics, which are trapped. For example, in Schwarzschild, they are trapped exactly at the value r equals 3m. And this is another major technical difficulty in connection with the mathematical study of black holes. Okay, so this is sort of a quick description of black holes. What are the questions, which I call reality questions of the black holes? The first one is rigidity. And this is a question of whether, besides the care family, which we described earlier, any other possible black hole, I should say, exhaust all possible vacuum black holes. In other words, there are no other stationary black hole solutions, which are asymptotically flat, and which are solutions of the Einstein-Valkomot equations. So this is a famous issue of rigidity, and I'll say a few words about it later on. The second question, which is even more important from the point of view of reality, is the issue of stability. So here the question is, well, can be formulated in terms of the initial value problem, as we shall see in a second. And obviously, there is an initial value, or initial data set, rather, which is associated to a precise care solution. And you'd like to know what happens if you make small perturbations of that initial data set. Because if the perturbations grow, then this will correspond to the fact that the black hole is unstable. And obviously, if the black hole is unstable, it doesn't have any physical reality either, because, of course, there is no pure black hole. So any perturbation, if a small perturbation of a black hole would lead to something entirely different, the black holes, in the sense we understand in astrophysics, do not exist. So that's an issue of stability. And finally, the last is that of collapse. Namely, typically black hole forms, as we know, form from collapsing stars, for example. And in other words, somehow, you'd like to, in order to formulate the issue of collapse, you'd like to start, again, from the point of view of the initial data, you'd like to start with initial conditions, which are non-trapped in the sense of penrose. So there are no black holes originally. And you'd like to show that the black hole. You'd like to understand what is the mechanism of formation of a black hole, so that the black hole could form from regular initial conditions. So that's the issue of collapse. And I think you'll all agree with me that all these are reality issues of black holes. So any questions about it? So these are the sort of the main problems that I want to talk about. So, of course, initial value problem, again, is one of the most famous results of Yvon Choké-Brua, which started the whole thing. So you start up with an initial data set, which is a three-manifold, together with a Riemannian metric, G0, and the second fundamental form K0, which satisfies the constraint equations. And we are looking at the development of this initial data set. So these are the issues. But each one of them in more detail. So let me start with the rigidity conjecture, which is in some sense the simplest question connected to reality. So again, the conjecture says that the care family KAM, whose A between 0 and M, exhausts all stationary asymptotically flat vacuum black holes. So what's known about it? Therefore, there is a famous result due to Carter Robinson, which says that the rigidity is indeed correct if you also assume axial symmetry. In other words, you look at spacetime, which are not only stationary, but also they are actually symmetric. So in this case, the problem is completely solved. Of course, the original results are due to Carter Robinson, but there are many issues which I think have been completely solved in particular by Piotr. So this is very well understood. Now, the general conjecture, of course, has no axial symmetry in it. So the question is what happens if you don't have axial symmetry. There is a result of Hawking, which I think it's a fake result, in the sense that he makes a huge assumption, which is assumption of analyticity. And there is really no reason. I mean, it's an assumption which is completely ad hoc. There is absolutely no reason that solution of the Einstein equation, stationary solution of the Einstein equation should be analytic. But nevertheless, in this case, the result is true. In the sense that what he does is he shows that under the analyticity assumption, you can construct a second killing vector field, which will be axially symmetric. And therefore, he reduces the problem to the Carter Robinson. But in doing it, he uses analyticity, which roughly, so the reason I think it's a cheat is because it's essentially giving up on the Einstein equations altogether. He's using instead the Cauchy-Riemann equations, which is funny. I mean, all of a sudden, you throw out the Einstein equations. The Einstein equations are used very little in this result. I mean, almost sort of in a trivial way, I would say. And then after that, the problem is purely one of the Cauchy-Riemann equations, which has nothing to do with the Einstein equation. So this I think it's a fake result. So the real issue is what happens if you don't assume analyticity. And the problem becomes infinitely more complicated. And here are the results that we have so far. So for a general Ricci-Flat, again, so Ricci-Flat, the Einstein-Walkam equations. So we're looking at smooth solutions. Then you have this result, which is together with Alexakis and UNESCO, which says that if you have a spacetime, which is stationary, and it's also closed in some very geometric sense, which we can make very precise, is very close to a care spacetime, then it must be cared. So it's sort of a local rigidity result. And there are actually many improvements of this result, more recent improvements in the sense of making it more precise what you mean by closed. So for example, in the original result, we had closed everywhere, and it turns out that we have better results where you have to make assumptions only on the bifurcate sphere. By the way, all these results require non-degenerate spacetime. So we don't have anything in the degenerate case. There is no result in the degenerate case in the category of smooth solutions of the Einstein equations. All right, so this is one result. Of course, then we have some other results which are in some sense connected with it, but of independent interest. So for example, I want to mention two. So we have a very, I should say, here is a sort of a very simple version of uniqueness, of rigidity. Assume that you have a metric, G, which is Ricci flat. It actually doesn't even have to be Lorentzian, and it can be any dimension. So this is a sort of a very general result. It's Ricci flat, any dimension, and it doesn't even have to be Lorentzian. It can be pseudo-reminion. And you have a Keeling vector field, K, which is defined in a domain D, included in spacetime. And you look at the point, so K is keeling here, only here, only inside. And you are looking at the point, P, where a certain condition which is called pseudo-convexity condition. So this condition is essential. If the pseudo-convexity condition is satisfied, then K can be extended past D, in a small neighborhood of that point. And this condition goes back to the Calderon-Harmander sort of pseudo-convexity condition, which is a general condition in partial differential equations. And of course, if you have analyticity, this result is trivial, and you don't need the Einstein equations. But in the smooths category, you need this. It's essential. And you also need the pseudo-convexity condition. And in fact, we have counter-examples. The P-convexity condition is violated. So for example, here is a very simple case. You see, if you look at the black hole region of a care solution, and this is the event horizon, this is the bifurcate sphere. So if you are exactly the bifurcate sphere, you can show that there exists a second keeling vector field. I'm talking about a stationary solution which coincides with the care solution inside here. And the result is that you can always find points on the horizon which are not on the bifurcate sphere. So on the bifurcate sphere, this pseudo-convexity condition is satisfied. But if you are outside the bifurcate sphere, pseudo-convexity is violated. And in fact, you can construct counter-examples to this local extension result in the sense that you can find solutions of the answer to a question which are care here, stationary everywhere in this neighborhood, but for which the second keeling vector field does not extend. So it's not actually symmetric. So this is, I think, another sort of interesting resulting connection with this stuff. Okay, so that's rigidity. Right, so I leave stability for the end, and I'll talk very fast about the collapse. So here the question is, can black holes form starting from reasonably initial data configurations? And it's intimately tied to the issue of formation of trap surfaces. So the trap surface is in the sense of penrose. So I'll talk very fast about Christodoulou's trapping theorem from 2008. This was a major breakthrough in the field. Then improvements by Rodnianski and myself. And finally, recent results of Rodnianski, Luc and myself, which is, I think, an important extension of Christodoulou's theorem. All right, so let me recall very fast the penrose singularity theorem. So you have a space time which doesn't even have to be Ricci flat. All you have to assume that Ricci in the direction of a vector field L, which is null for the metric G, has to be strictly positive. So you have these positive conditions satisfied. M contains a non-compact Cauchy hyper surface. So that's a topological condition. But the most important one is that M contains a closed trap surface, which I'll explain in a second what it is. Then the space time has to be future null geodesically incomplete. So that's the singularity theorem. It's really not a singularity theorem. It's an incompleteness theorem. Because it doesn't tell you anything about any kind of information about how singularity is formed. But it's a very important result. In particular, what's very important is this notion of a trap surface. So what is a trap surface? Well, it's a surface which says the property is that if you look at all perpendicular null geodesics, you have two families of perpendicular null geodesics. One which is typically incoming. Even in Minkowski space, you have one that's going like this. And the other one in Minkowski space will be outgoing. Now, if you have a trap surface, it means that both families of null geodesics are actually contracting. In the sense that the two areas, if I go along null geodesics, or relative to an affine parameter, I see that the areas in both directions are decreasing. Instead, one should be increasing and the other should be decreasing as in Minkowski space. Now, both are decreasing. This has to be a point-wise condition. So this has to happen everywhere. It can be measured in terms of what is called the expansion associated to null geodesics, which are in fact traces of so-called null second fundamental form, which are geometric objects associated to the two surface. Anyway, so I don't want to say much more than this. The question is, can these trap surfaces form in evolution? So in the result of Penrose, you assume that you have such a trap surface. And the question is, can such trap surfaces form in evolution? So in other words, can I start up with initial data, which is free of trap surfaces and form a trap surface later on? All right, so the second question is, does the existence of a trap surface imply the presence of a black hole? So this, again, is a major issue. A trap surface is a local condition where, of course, a black hole is something global that has to be defined from infinity. It turns out that this statement is true if the weak cosmic censorship holds true, which is, of course, an even more difficult conjecture to prove. But it's at least interesting to notice that all these things are consistent. So the weak cosmic censorship is another major conjecture in general relativity. Okay, but I'm not going to worry about this. The more important question that can be answered, and in fact was answered, is this one can singularities form starting with non-isotropic initial configuration. So this is the new result of Jonathan, of Luke Rodnianski and myself. So, again, this is a picture that you must have in mind. Actually, I don't know exactly why I have this picture here. Well, sorry. Yeah, I'll mention it in a second. Okay, right, so again, problems specify an open set of regular initial conditions free of trap surfaces. But now I can specify this ison space-like hypersurface or on a null hypersurface. Okay, so let me make it a little bit more precise. So in the next slide, I'll make it a little bit more precise. Now, the formation of trap surfaces is pretty well known in the case of spherical symmetry. If you have spherical symmetric space times, and, you know, for example, Dimitris Christodulu in his famous results on spherical symmetric solutions of the Einstein equations coupled to the scalar field, you always need, in spherical symmetry, you need to couple the Einstein equations with a field. Otherwise, you cannot have non-trivial solutions, non-trivial dynamics. So in the spherical symmetric case, he has sort of very general conditions for formation of trap surfaces, which were extremely important in his program. Now, if you don't have spherical symmetry, then the dynamics of the Einstein equations are infinitely more complicated. And it's not at all clear what would be the mechanism. In particular, what would be the mechanism in vacuum? I mean, the expectations, as far as physical intuition is concerned, is that trap surfaces and black holes will form as collapse of matter. Now, Dimitris Christodulu was actually able to show that the issue has nothing to do with matter. It can be understood purely in terms of the vacuum Einstein equations. So, now, the major difficulty here is that you need to control the space times, so you need to control the entire dynamics of the space time for a very long time. So it's not a local existence result. You have to go beyond Yvonne's local existence result, go much further and be able to understand the details of a space time for given initial conditions. The initial conditions themselves cannot be too small. They have to be actually quite large in order to produce a trap surface. Anyway, this is a thing that Christodulu has done. So basically, this is a picture. So that's the picture with the fake higher dimensions in it. Otherwise, I have to restrict myself to just two dimensions. Here you have a null hyper surface, which is an outgoing null hyper surface. This is an incoming null hyper surface. And as we know, in order to solve the initial data, the characteristic initial data, you need to put conditions both here and here. So he picks up the conditions here to be Minkowski. So this is trivial. While here he takes a short pulse, which corresponds to some relatively large initial conditions, free of trap surfaces. And then he shows that a space time can be defined and controlled in full details, can be controlled for a long time. You see, if this is of size delta here, and you have a short pulse of size delta, the space time can be controlled up to something of size 1, which is quite far. So this is way beyond the local existence result. But nevertheless, he can do it. So he can prove a semi-global existence result with detailed control. And then if in addition, you pick up special initial conditions, which he can identify very precisely, within the class for which this is true, for which this control is true, then you form a trap surface somewhere here. So even though there are no trap surfaces here, and of course no trap surfaces here, you'll form a trap surface later on. So a similar result is also given with data at personal infinity. So you can imagine going from personal infinity and you get sort of a similar result. In fact, the important part of the result is really the local result. Once you have the local result, you can easily extend it to personal infinity. OK, so the theorem that I mentioned earlier, which is that one of Ronjanski's look and myself is different in the following sense. So you see, it's very important in Christodurus' result that here the data that you pick up in order to form a trap surface is uniformly sufficiently large in all directions. While in this new result, we prove that it suffices to get something which is large along one null direction. So you pick up initial data which is large in one null direction. In fact, it can be flat everywhere else. And you still form a trap surface later on, but the trap surface is more complicated. You see, in the work of Christodurus, the trap surface is obtained by... Well, you define the spacetime, which is this one here. You define it in terms of a doubly null variation. So relative to a doubly null variation, you form a trap surface which corresponds exactly to the doubly null variation. While in the new result, you still have the doubly null variation because you need to control the spacetime. In fact, the result still relies... Our result still relies on the first step here. But now I can take initial data which is concentrated in just in one direction. And you form a trap surface, but the trap surface is not going to be the intersection of the leaves of the doubly null variation, but it's rather going to be a deformation of it. In other words, in order to prove this result, you have to combine somehow the mechanism of Christodulu together with a new mechanism which corresponds to a deformation argument along the incoming null geodesics. So it combines all the ingredients in Christodurus' theorem with a deformation argument along the incoming null hypersurface, in other words, along this. And that's why you can get a very complicated surface like this. And the condition for producing such a sink, of course, has to be put on the initial condition, it's just a differential inequality. In fact, it's an elliptic PD, nonlinear PD that has to be satisfied. So you pick up your initial data to satisfy a nonlinear elliptic PD which is not too different from what comes up in the uniformization theorem. And that will produce a singularity. Are you saying that if you have only one ultra-energy graviton? Yes, you'll form. Yeah, right, it's basic. I'll explain. I don't want to go through the proof now because I want to talk about something else, but I can tell you more about it. All right, so here is the last aspect of reality of black holes I want to talk about, which is stability issues, and this is the most important one, I would say. Because once again, if stability is not satisfied, there's no way you can talk about black holes as physical objects. So this is, of course, a fundamental issue which has been studied quite a bit. The conjecture is the following. So the stability of the external, in other words, I'm looking at stability of this region. So you pick up initial conditions on a space like hypersurface, and you can imagine that the care itself, the restriction of care itself, gives you a set of initial conditions on this hypersurface and you perturb it a little bit, and you like to prove that the perturbation leads to a space time which converges asymptotically to another care solution. Not necessarily the same. It's very important, and in fact, it's one of the difficulties of the problem, that the final state may not be the original care solution you started with. All right. So that's a picture. And so I want to say what... I want to talk a little bit about general stability issues in partial differential equations. So, well, imagine a nonlinear equation. I don't want to make it very precise, but it's n of i0 plus i is equal to 0. I guess I need the bigger one now. This way. All right, so this is... Imagine a nonlinear partial differential equation which has a stationary solution, which is phi0. So phi0 is a solution that you can think is independent of time, right? That's basically what stationarity means. Of course, in GR it's more complicated, but roughly that's what it is. And, well, how do you... And obviously, Psi denotes a perturbation. So you'd like to show that the perturbation is small or converges even to zero for all time, okay? So asymptotically, you look in the direction of future time and you'd like to show that Psi tends to... either tends to zero or maybe it tends to something else. But in any case, you have to control Psi. You have to control the perturbation. So how do you do that? Well, typically, you linearize. So you linearize around phi0 because you are making a perturbation of phi0. You linearize, you get a linear operator plus something which is quadratic in Psi, right? Which has to be equal to zero. And this L operator is really the fresh air derivative of this n, okay? So that's so far nothing special. And then, of course, you have to look at the linearized equations. And in a first approximation, you have to show that the linearized equations are well behaved. In other words, the solution of these equations do not behave badly. And then there is orbital stability. You can distinguish between the various notions of stability, like orbital stability and asymptotic stability. Orbital stability will be that Psi remains bounded for all time. Asymptotic stability Psi converges to zero as t goes to infinity. Now, it turns out actually that orbital stability you can almost never prove. In other words, in order to prove this you will have to prove this. So you always have to prove the stronger version. And the module, of course, is something that I'll talk about later. Whenever you have a quasi-linear equations, in particular, there is no chance that you'll be able to do something like this directly. You'll have to prove that something much stronger which is asymptotic stability. Now, the simplest case is, of course, that of constant phi zero. So phi zero is a constant solution. For example, Minkowski space, if you think in terms of stability of Minkowski space, the Minkowski corresponds to something which is constant, you could say. And you want to perturb around it. In that case, the linearized equations can be analyzed. And in order to prove their asymptotic stability you need not only the size bounded, for example, the perturbation remain bounded, but you also need to actually get decay. So you want to find the rate of decay of how psi converges to zero as it goes to infinity. All right. So in the stationary case, however, the situation is much more complicated. So if phi zero is not a constant, but it's a general stationary solution, like the care solution, then as I mentioned earlier, it's not even clear that you converge to the same phi zero. So you can perturb phi zero and converge to a completely different one. So this is the issue of modulation, which you can analyze in a very simple example. So this is, for example, take the equation, take the simplest possible wave equation, which is the one above. So wave equation equals phi to a p. Or p, let's say, is quadratic. The most natural case would be p equals two. It's very easy to prove perturbations of phi zero equal to zero. Phi zero equal to zero is, of course, a solution. It's very easy to prove perturbations of that, but it's much more complicated to prove perturbation of a larger class of solutions, which are stationary. So it turns out that that equation has solutions which depend only on x, which you can write down explicitly and depend on a parameter a, which can easily be obtained by scaling. And a typical perturbation, actually, maybe p should be cubic. Anyway, it doesn't matter at this point. It should have cubic and it should have the right sign, the wrong sign. In other words, the energy is not positive in that case. Anyway, what I wanted to say is that this is what you want to perturb and the solutions that you obtain to the nonlinear problem when you do this perturbation argument may not converge to the same a zero. So if you start with an a zero, you may converge to a different one or it may not converge at all. I mean, it could be that the problem is unstable, which is actually the case in this case, and then you don't converge to anything else. And that has led to, in recent years, in the last, I don't know, 15, 20 years, this has led to lots of this type of issues. In other words, perturbation around stationary solution has led to a lot of work, which is, I'll call it modulation theory, or it's known under the name of modulation theory. And of course, there are many people who have contributed, in particular, Frank Merlech and his school here in France. Okay, so let me get to the issue of black hole stability. So what are the difficulties? How much time do I have? I don't know when I start it, actually. 10 minutes, approximately? Okay, so I don't have to rush, okay. So what are the difficulties? Obviously, this is a very simple problem by comparison. So here you have just a scalar equation. You have only one phi. The nonlinearity is very simple and already is very complicated. In the case of the stability of the case solution, the difficulties are made much tougher because, first of all, the gauge covariance of the equations. I mean, a solution is expressed in terms of, say, a coordinate system, but depending on which coordinate system you take, the problem may look very, very different. So that's a huge problem. This was a huge problem in the stability of Nikovsky space. Then, of course, wonderful linearization. Linearization is intimately tied to the problem of gauge covariance because if you write down the equation in a certain gauge, you linearize in that gauge, the linearization will be heavily dependent on the gauge. So this is a huge problem and it leads to all sorts of possibilities in terms of how you want to linearize. You can linearize relative to metric efficiency. In other words, you write everything relative to a coordinate system. You can do relative to the connection coefficients. By the way, the connection coefficients turn out to be very useful in the boundary delta curvature conjecture, which was recently proved with Jeremy and Rodiansky. So this can be very useful in some cases. You can do it relative to the curvature components. This was, stability of Nikovsky space was done relative to curvature and there is this issue of modulation because you don't know a priority where you are converging to. So you have to also keep that in mind as a major difficulty. Alright, so linear problem. Let's say once you assume that you have solved this problem and you found sort of a good description of how to linearize, now you have to study the linear problem. For the linear problem, you need uniform decay estimates which are sufficiently strong. If you're not strong enough, you cannot prove asymptotic stability. So for example, in order to prove the stability of Nikovsky space, you have to show that the curvature decay at certain rates which are polynomial rates and they have to be very precise. If you don't have the precise rates, you just don't get the result. Now, in the case of care, you have this issue that you have limited symmetries even for the care solution itself. If the care has only two vector fields which are killing, Nikovsky space has a lot more. So the stability of Nikovsky space you expect to be simple because of this. Limited symmetry. Presence of the Ergo region, this is a major problem because now you don't even have a positive energy to start with. So this is absolutely fundamental. Presence of a Trub region. All the estimates degenerate in the Trub region. I'll say a few words later on. Of course, across the linear nature, in the sense that the characteristics of the equations differ substantially from those of the linearized problem. So even if you understand the linear problem, you still have to deduce a nonlinear problem. So anyway, you see there are lots of difficulties. And here are the results which are known. So first of all, the only true result, nonlinear result is the global stability of Nikovsky space. Of course, I'm talking about the asymptotically flat case here. So in the asymptotically flat regime, the global stability of Nikovsky space, which of course it's only the case when A and M are equal to zero. So it's a very special care solution, right? The trivial care solution. So this is a constant case, which I think we understand very well. So the case of constant perturbation around the constant states is well understood today, I think. I mean, there are still problems, but mainly I think you could say that it's well understood. So the real challenge now is to do perturbations around general stationary solutions. Now, another major result, which is again at the linear level, so these are just the linearized equations, was due to Whiting in 1989, where he proved the most stability of the care family. So no decaying rates whatsoever, just more stability. So you use a separation of variable method, and you show that each mode behaves well. This of course does not mean at all that you can sum the modes and get something which is reasonable. You can not even prove Bonan as a toy, but in any case, it was an important result. Now, the major breakthrough on this linear problem was obtained in recent years by the combined efforts of sulfur, blue, blue-sterbens, the Fermo-Zrodnianski, Mazzuola, Metcalf, Tatarotokhanianu, Bluandesen, Tatarotokhanianu, et cetera, which have led, so all this have led to a well understanding of just the simplest possible linear equation, which is, you look at the care metric, and you look at, so this is the metric, and you look at just this equation in the exterior of the care solution, and you can prove all sorts of things about it. Highly non-trivial, because as I said, you have to solve, this problem has to deal with the Ergo region problem, has to deal with the Trub region, has to deal with the situation at infinity. There are also the fact that there are few symmetries and so on and so forth. This, by the way, was done originally only for small a, recent results which were announced by the Fermo-Zrodnianski in 2010 hold for all a between 0 and m. So that's... All right, so what I want to talk in the last, five or six minutes, if I can, will be the work in progress with Ionescu and myself, which, so the idea is, I think there is no way right now to prove full stability issue. It's just too complicated. So anyway, we decided, and this was the decision made with Ionescu and also with Jeremy, to really concentrate on the axially symmetric case. So let me tell you what one can do. So first of all, how do... How can you formulate the problem? Well, it turns out that if you have a space time which has a killing vector field, which is axially symmetric, you have a reduction procedure. So the reduction is well known. If you have worked on it, the idea is that you can define an Ionescu potential. So this is Ionescu potential associated with a killing vector field which has a real and imaginary part. So it is defined by two functions, x and y. Then the manifold itself can be expressed as a product between an SO2 and the manifold is boundary and one plus two will be a manifold is boundary. And the metric can be decomposed like this. There is an x times d phi. So phi corresponds to this rotation here in SO2. And then you have a two plus one metric. So this would be a Lorentzian metric, two plus one. This is x coming up from there, x minus one. And this is a Lorentzian metric, h alpha beta, is a Lorentzian metric in two plus one variables. We said dx somewhere. Excuse me. Do you mean a square? You mean a square. You mean a square. You mean a righto metric. So it's just the first differential. Oh, d phi. Yeah, sorry. This is out. This should be a square here. Excuse me. Yeah. Okay. And yeah, so you define the, this w is defined by a divergence called system of x and y. And then the important thing is that phi, which is again the sense potential, can be interpreted as a wave map from this manifold's boundary to the Poincare disk h2. And so phi itself satisfies an equation which is tied to the metric h. And the metric h is tied to phi through this equation. So Ricci of the h is equal to that. Okay? So that's sort of a very simple explanation. Now the wave map, in fact, takes this form. So it's x times the dimension with respect to g. Sorry, I should say this wave map phi is a wave map defined from n1 plus 2 with a metric h. But it also can be understood as a wave map defined in the whole spacetime, mg, with values in h2, which takes this form. So this is the metric, the original metric of your axially symmetric spacetime you started with. Right? So, okay. So now this turns out that it's still very difficult to do. So we don't have any claim right now that we can solve this problem. Even this reduced axially symmetric case. But what I think one can do is to look at simpler model problems associated with it. So one model problem is the following. Take the metric g, so take the metric g here. Okay? So sorry, the metric g here. Take the metric g to be the care metric. Okay? Then you have a wave map. And the particular solution of the wave map is going to be given by the nth potential of the care solution. Right? So you have a care solution here, it was an a and an m. And you have the nth potential corresponding to that one. That's a special solution. So you want to protect that one. So you have a perturbation, a true nonlinear perturbation of a stationary solution, but assuming that the metric is kept fixed. Okay? So that's model problem number one. Model problem number two, which is something that I'm discussing with Jeremy, is to prove the stability of the care solution among axially symmetric polarized perturbations. So what that means is that you look for special solutions that have y exactly equal to zero. In that case, you see the equation simplifies quite a bit. And in fact, the stability of, stability in this case really is a stability of the Schwarz's solution in the class of polarized. So in other words, the final state is going to be a Schwarz's solution, not a care solution. So a will be zero. Now this will lead to all sorts of simplifications. And it's a problem that we think is doable. We don't know how to do it yet. But at least we think that maybe in the next 10 years, this will be solved. Now, okay. Now the problem that I think, right, so I'll finish with this. The problem that I think we can solve, and this is work in progress with UNESCO, is a first model problem. So the first model problem, again, you take G to be care, you take X, and you try to perturb the N's potential of the care solution. So this is how it looks in Bore-Linck's coordinates. This is a care solution. The N's potential is exactly A is equal to sigma. So sigma being defined here. Sigma squared times sine squared of T divided by Q squared. And B is this ugly expression here. So this is the second part of the, right? Now, of course, A and B are special solutions, stationary solutions of these equations. And I want to perturb A and B. So what I want to do is to, and this is a conjecture, which we think we can solve, though it's still work in progress. So you look at the whole system, and you take solutions which are small perturbations of that A and B that we saw in the previous slide. So the A and B that we saw here. So the claim is that the perturbation, so if you perturb X0, Y0, the perturbation converges back to phi 0. So this is one of the special cases where you actually converge it back. So modulation is not necessary, which is remarkable in its own right. You converge to the exact same thing. And we have, can I have maybe two more minutes? All right. So here is a, the conjecture is that the final, the solution of the nonlinear problem converges through the solution of the, to the potential of the care solution. And I just want to say a few words about what's interesting here. Well, you see, if you look at the linear, if you linearize, you linearize like this. You take A and B, and you add the perturbation, which is expressed like this. It has an A in it, because the perturbation has to be 0 on the axis of the symmetry, right? So that's why I need an A here. So here is an ugly system that you get. So you see it's a system in which you have, it's linear in psi, derivative of psi in phi. It's very complicated. So a priority, this is very far, as far as possible from the dimensional phi and the dimensional psi is equal to 0, which is a case which was understood. So the linearized problem is much more complicated than the linearized problem which was, which was understood by Daphir Motronyanski and many others. And obviously this is a picture. You start up with the initial condition here for the phi and psi, which is a perturbation, and you want to prove that you can cover all these regions. Now, the reason, so these are the challenges, which maybe I won't say much more. I'll say only one thing is the most important supportive evidence for one is the fact that there exists an energy momentum tensor for the complicated linearized equations, which has these three properties. It's positive. So it's an energy momentum tensor which depends on psi. So psi is the perturbation. So psi is this perturbation of phi psi. It depends, of course, on the derivative of psi. But of course it depends also on the potential of the care solution, which is phi 0 d phi 0. So it's a complicated thing, but it's positive. It has a divergence expression, but with a source term on the right-hand side, which, of course, could be extremely dangerous. In fact, the main point is exactly the source, because in principle you could write the source starting up with just a linear wave equation. But this would be a terrible source. So the idea is that you can find a good energy momentum tensor with a j on the right-hand side, which is perpendicular to the stationery... I'm sorry, this should be z here. It's killing to the... It's killing to the stationery vector field, which is the dt of the care solution. So we are still in a... The metric is still the care metric here. So this is the most important thing, because as a corollary of this, you still have an energy conservation for the linearized system, because how do you prove energy conservation? You multiply q by t, and you take its divergence. But because t cancels the j, you still get a conservation law. So you get a positive conservation law, and this is sort of the beginning of everything. In addition, you can prove more of its estimates. So this is a second proposition that we can prove so far. This is a trub region. It turns out that the trub region, because you are in the axially symmetric case, the trub region is still relatively simple. In other words, it consists on just one surface, r equals r star, and you can do the more of its type estimate, which is a crux of the matter in these linear stability issues of Daphermos Rodnianski and Company. The most important estimate was this more of its estimate. This more of its estimate could be a problem, because again, j is very complicated. There is no reason that j will go for the ride. It turns out that it does. And you can prove that more of its estimate. And as a consequence, you can prove the uniform decay estimates and so on and so forth. So I'll stop here. Are there questions? No questions? Please. So the model problem 2, as I mentioned, so this way has to be solved. This corresponds to proving the stability of Schwarzschild within the set of data which has... Which is polarized. Which is polarized, but if I try to express it in terms of the parameters at the N and A, then I'm trying to get Schwarzschild while keeping A is... A is zero. A is fixed, yes. I should say that Kholzegel, Daphermos, Rodnianski have sort of a different program which is moving it also in that direction. So it's good that we have two different points of view, because these problems are extremely hard. Very high dimensions. So all this formalism with all the composition and stuff like that is not well understood yet, maybe there is something like that. But you expect that wave equation, solution of wave equation decay faster and therefore maybe this stability problem would be easier for black holes? Not necessarily. Not necessarily, because this has a fundamental problem which has to do with Ergo region, which has to do... Uniqueness itself is a problem. Rigidity is much more difficult in higher dimensions. So even... Rigidity obviously has to be an important aspect of the stability problem, because of course if there were other stationary solutions, then who knows where you would converge? So rigidity is very important. Of course rigidity, you don't have in higher dimensions. I don't know, I think there will be other problems in higher dimensions. Decay is easier, it's true, if you can prove decay, which is a major challenge. More questions? Okay, so thank you very much. In some cases one could combine analytical results with numerical results to prove theorems. Now, there are numeric and many numerical experiments showing the stability of curve. Yes. Is it possible to think that one could combine some numerical information with stability with an estimate and then one could get the proof at the end? Well, everything is possible, yeah. I mean, these problems are incredibly difficult, the stability. And in fact, actually, looking at the physics literature, I've never seen a reason. I mean, people look at the linearized problem and they say the linearized problem is somewhat stable. But there is no reason. Also in very energetic situations and they stabilize to curve back home. Yeah, but these are always special initial data, right? I mean, it's a... But there is no... But a mechanism, I want to know a physical mechanism why it should be nonlinearly stable. I mean, I've never seen one. And this, by the way, our first result, I think, gives you something, because it's very much tied to the positivity of the curvature on the target manifold, which is hyperbolic space. It's intimate, if I do that. So at least you can see sort of beginning of a mechanism of nonlinear stability. Linear stability, of course, but what does linear stability mean, right? Because whenever you do linear stability, you pick up a gauge and, you know, it's gauge dependent, so it's not... Unless you prove nonlinear stability, you can't really say anything. Yeah. So how would you prove that it's not stable? I mean, so, assuming that it's linearly stable... I don't know. I mean, just by the fact that you say... I mean, there's tons of work going on on the stability, and if people can't prove it, that doesn't show that it's not stable. I'm wondering, is there some mathematical thing that would argue, yes, we now know that... No, I mean, I think we are starting to see reasons why it should be stable, but of course, only the axiomatric is, at least. I mean, as far as I can tell, of course, the thermosrodnianski have something else, but it's still tied to the Schwarzschild solution, to the actual Schwarzschild solution. So, yeah, I think we are just at the beginning of the game here. In 10 years, I think if you ask the question in 10 years, we'll be able to give you a decent answer. No more questions? Oh, you won't? You are not a shopper. No. We hope so. Provided that we get the money. Thank you very much.