 So let me, it's a great pleasure to welcome Sergio Klinerman today, a professor of mathematics at Princeton University and one of the foremost experts in the mathematical study of general relativity. So he's a frequent visitor to the IHES and he's on our board of Friends of IHES. As many of you know, Friends of IHES is the American support and public charity that supports the Institute's vision and philosophy in the United States for more than 20 years. So this should be a very interesting talk because it's often astrophysicists and physicists who talk about black holes and there's many, many interesting questions from many points of view and the point of view of mathematicians such as Sergio is quite interesting and different. Without further ado, let's begin, Sergio. Well, thank you very much for the introduction. I'm very glad to be able to give this talk. First of all, I wonder if everything is OK. Can you see the screen? We see it well. Yes. You see it well. Very good. OK. So well, first of all, I apologize. This is not going to be a talk for specialists. I'll try to be quite introductory for the vast majority of people who are attending today. The question that I'm going to try to answer is a question of what is a relationship somehow between the physical reality of black holes and the mathematics of it. So with this, let me start. First, let me talk a little bit about the basic geometric tools or basic geometric objects that are used in general activity. So of course, they are exactly the same as the one in geometry. So to start with, we talk about manifolds. So manifolds really think of manifolds as collections of open sets and coordinates on them and the way of transitioning from one coordinate to another. And the concept, which is a great concept in mathematics, which was introduced by many people starting with re-manifold. The metric, which you see here, which is this kind of expression, is expressed relative to a system of coordinates. So it's again, it's a local definition of a metric. It's a tool of measuring distances on the manifold. So there is a very simple, once I have this kind of tool, I can easily define a distance between points and so on and so forth. We'll see that there is a difference between Riemannian metrics and Lorentzian metrics. I'll say a few words in a second. Once I have the metric, I can talk about the connection, so-called Levy-Civita connection, which is really just an expression involving derivatives of the metric G relative to the coordinate. So again, relative to a system of coordinates, that's what it is. By the way, this kind of formulation here allows, it's a very simple way, which allows you to pass from one coordinate to another. In other words, these coefficients that I see here, they correspond to a particular coordinate system. If I want to change coordinates, I change it very easily according to this kind of expression. I don't want to get into details, of course. But just remember that the metric really is given here relative to a system of coordinates, but of course I can always change coordinates into something else. The connection represents derivative again in the system of coordinates, derivative of the metric relative to the coordinates. This, by the way, is not a tensorial quantity. It doesn't change well relative to system of coordinates. However, you can use it to define what is called the curvature of the manifold. So this is a concept introduced first by Gauss in dimension two. The manifold here was a dimension two manifold, a sphere, a surface, I'm sorry. And it's really a derivative, at the level of derivatives of the gamma. So there is some kind of algebraic relation between the curvature and derivative of the gamma. The remarkable thing about the Riemann curvature tensor is that it's invariant. It doesn't depend on the particular coordinate system. So even though this was depending on the coordinate system, this doesn't. Once I have the Riemann curvature tensor, I can take traces. So again, everything is done relative to coordinates. But again, in reality, these expressions do not depend on coordinates. Of course, it takes some time to see it, but believe me, it's true. So here I'm taking a trace, what is called a trace. You see there are four indices in this Riemann curvature tensor. I can take a trace and I get what is called the Ricci curvature, which is now a two tensor. And because of the symmetries, this Riemann curvature tensor has lots of symmetries. And because of them, in reality, this tensor here is a symmetric two tensor. And by the way, using the so-called summation convention, which in fact was introduced by Einstein from what I understand, which is summation of a repeated indices. So whenever you see beta and beta here, it means you are actually summing and the same thing with the other index. All right, so these are the basic objects. And of course, the simplest manifolds and the simplest geometric structure are the one of Euclidean geometry where you just have Rn and the distances are measured using the Pythagoras theorem. And the generalization of it is called Riemannian geometry. So generalization simply means that the metric is much more complicated and is defined on a manifold. And then there is a Minkowski space. So this is a concept which was introduced in the wake of special relativity. After Einstein, Minkowski is the one who saw the geometric structure of special relativity. These are called, as a consequence, Minkowski and metric. And the generalization of this Minkowski geometry is called Lorentian geometry in the honor of the physicist Lawrence. All right, so let's connect now all these things with physics. So this is a remark we think about general activity is that there is a very, very clear connection between objects which are mathematics and objects which are physical. In fact, mathematical objects are interpreted in physical language. So for example, the so-called inertia, the fact that you can cancel your gravity, you cancel the force of gravity on yourself locally, it's expressed by the fact that the tangent space to the manifold, in other words, that the tangent's been syncopated to something really very intuitive, as in normal Euclidean geometry, the tangent to a surface. Here's a tangent to a manifold. You can define this concept even if the manifold is abstract. So on the tangent space, you have a Minkowski structure. In other words, the metric at the given point can be expressed as a simplest possible Lorentian type of metric, which is a Minkowski metric. So you see the Minkowski, geometrically, you see how the Minkowski metric looks like in the tangent space. You have a light cone. You have what would be the time and spatial coordinates. And the metric is such that it varnishes on this light cone. It's negative inside here, and it's positive outside. In fact, in Lorentzian geometry, if you go back to the definition here, if the coordinates are, say, t from time and then x1, x2, xn, then this metric here, the Lorentzian metric, is nothing else but 1 in which you have minus 1, 1, 1, 1 on the diagonal and 0 everywhere else. So it's a very, very simple. It's a simplest possible metric. Of course, Euclidean in the Euclidean metric you have 1 everywhere on the diagonal and 0 everywhere else for the Euclidean metric. And here it starts with minus 1 and then 1, 1, 1. So it has signature minus 1, 1, 1, which is typical to Lorentzian geometry. OK, so the inertia, which is a physical concept, can be expressed purely by saying that what you are measuring is this kind of Lorentzian metric, which becomes trivial at the point. Events represent points in M. So any point on a manifold is, in fact, an event. Observers correspond to time-wise curves. In other words, curves that really move in such a way that the tangent space at every point on the curve is time-like. Time-like meaning, if you look at the light cone passing through the corresponding point, the tangent is in the direction of the light cone. So inside the light cone, in other words. It's null if it's along the null cone. And this corresponds exactly to light rays. So light rays, again, which are a physical concept, are nothing else in Lorentzian geometry, nothing else than null geodesics. In other words, geodesics, which have this kind of form. So in the Kowski space, they're just straight lines at 45 degrees. The equivalence principle, which is a fundamental concept which was used by Einstein to write down general relativity, is nothing else but the general covariance in mathematical expression, which simply means that all the objects which are used to define your equations, the physical objects, in other words, are invariant relative to any system of coordinates. So you can change coordinates, everyone. You please. Here is a simple mathematical way of saying that for any deformorphies, any map from the manifold to itself, you can take the pullback of the metric by phi. And you get a new metric here. And somehow, all the physical equations should be independent on this kind of transformations. So this is general covariance. And it's really equivalent at the physical level, the equivalence principle. Tidal forces, which are physical correspond to curvature, nothing else but curvature. Isolated systems, which are systems in which something happens outside the particular region, there are no other things become Minkowski in a sense. So in other words, the manifold that we are interested in, it's called asymptotically flat, meaning it may be complicated in some region of space. But as you go to infinity, you are talking about open manifold, as you go to infinity, things become Minkowski. So again, everything you want can be expressed in geometrical language. Every physical concept can be expressed in mathematical language. The equations themselves, which are the famous field equations of Einstein, of this type, you have on the left-hand side, you have something purely geometric. This is a Ricci curvature, which we have just discussed. Minus a half. This is the scalar curvature, which is obtained by just taking another trace, taking g alpha beta, alpha beta up, or alpha beta. This gives you r. So this difference is equal on the right-hand side with a T, which corresponds to the energy momentum tensor of matter fields in the spacetime. So in other words, it's not just this m and g, which is all we study in physics. You also have fields, additional matter fields, that one can put here. And somehow, each of these additional fields have energy momentum tensor. And this is what comes up in the right-hand side. So this is expressed purely in terms of the fields that are present on our spacetime. The simplest case is exactly the one in which there are no additional fields, no additional matter fields, in which case T is 0. And you can get actually read it. So you will have 0 here. And from there, it's easy to see that actually the Ricci curvature has to be identical equal to 0. So from a physical point of view, this is called the Einstein vacuum equations, because it's nothing present on our spacetime. And it corresponds to pure propagation of gravitational waves. So it's not that the spacetime becomes trivial. It's high, no trivial. But even in the case when you don't have matter fields, and this corresponds from a physical point of view to gravitational waves. OK, so as I said, the general covariance, the fact that this every quantity that appears in the main equations is tensorial does not depend on the particular coordinate system that you choose. All right. So a few words about initial value formulation. This was something that took a long time for people to understand. In fact, the Einstein equations originally did not even seem to have hyperbolic character, we say. It took some time for people to understand it. And in fact, it's to a large extent a product of the French School of Mathematical Physics. Yvon Chorke-Brua, in particular, is the one who really formulated the first result of this type. So the result says the following thing. First of all, you can talk about initial data sets in general activity. So what are they? It's very simple. I mean, think of the fact that you already have a spacetime and you take a hyper surface, which is sigma 0. So let's say we're in C plus 1 dimensions, which is the standard physical theory. And the spacetime is here. And you take a hyper surface, sigma 0, which is such that if you look at the light on every point on the hyper surface, the normal to the hyper surface is time-like, which means it points in the interior of the light cone. So this is called the space-like hyper surface. It just happens at every point. So if you take a space-like hyper surface, it's very easy to see that the induced metric, so the spacetime metric induces a Emanian metric here, which is G0. Emanian, again, just simply means that it's positive definite, so it has signal per 1, 1, 1. And this will be the time derivative of the metric, which is a symmetric two-tensor. Never mind exactly what it is. The important thing is that this corresponds to position in some sense, and this corresponds to velocity. And these are the data that you give plus some kind of constrain equations. So this is very much like in Maxwell's theory, when you give initial data, you also have to give some constraints. Of course, the constraints in general activity are more complicated. They are nonlinear, and so on and so forth. Anyway, the theorem of Eman-Bruha, and then complemented by some result with Garridge, is that smooths initial data sets, so these are the initial data sets plus constraints, admit unique, smooth, maximal future global hyperbolic development. Never mind exactly the word. The important thing is that there is always a space time that you can construct, at least locally. And this maximal future development represents sort of the maximal extension of the space time as far as you can go. So this is a remarkable result in the sense that it associates to any initial data set. It associates a space time. And in a sense, the problem, and it's maximal, so it's a maximal possible development of it, and in a sense, general activity becomes just a study of maximal future hyperbolic development in terms of understanding their character, right? So you want to understand the global character of such things. Does they end in singularity? Can anything else happen? Can they be extended, maximal global hyperbolic development extended for all time in the sense of completeness? In other words, completeness means that if an observer moves in such a space time, does it stop, does it time stop at some point because some horrible things happens, falls into a black hole, singularity, or you can go for all time. So all these things are really the issue of general activity, classical general activity studies. You can say studies this maximal future hyperbolic development, right? So this is a remarkable result proved essentially in the 1950s based on a lot of developments in partial differential equations that were done in the first half of the last century. All right, so let me go on unless there are some questions. So if there are no questions, let me go on and try to talk a little bit about mathematical general activity. Well, this is what I do. I'm a mathematician interested in general activity, so I like to explain what this means. So what does it mean to do study mathematical general activity? First of all, the basic question is to elucidate the structure of classical general activity. What is the mathematical structure of general activity? Second type of questions is to formulate and address the central problems of general activity and we'll discuss some in what happens. And I should say that when we choose problems and this makes us different from physicists. We are interested of course in physically relevant problem but they also have to satisfy our mathematical sensibilities like beauty, rigor and mathematical challenges. In other words, you want problems where you really have to develop new mathematics. It's not just the physics of importance which of course it is but you also want to attack the problems that really are exciting from a mathematical point of view. They bring something new. As far as mathematics is concerned. So in that sense we are very different. This is quite different from what the way physics to look at general activity. And finally, another thing which is typical mathematics I would say is to establish connections to other problems in other fields for other equations or in partial differential equations or in geometry. And I would like to say that this is some kind of mathematical entanglement and Wigner expressed this very, very well in this formulation. Mathematical concepts introduced for solving specific. So it's an observation that he made which he thought was very mysterious. And mathematical concepts introduced for solving specific problems have unexpected, mysterious consequences in seemingly unrelated areas. And this is a beautiful thing about mathematics which I would say is part of what they call mathematical GR. Okay, any questions about it? All right, so let me go on. So, okay, so I want to talk about black holes as we said. So I have to say at least a few words about what black holes are. So first of all, there are explicit solutions of the Einstein-Valcom equations, right? So they remember Ricci of G. So the Ricci part of the Riemann curvature of G has to be identically equal to zero. This is the Einstein-Valcom equation. So there are examples of this solution. The simplest one, of course, is the Minkowski space itself which was Minkowski 1907. So this corresponds to this family, right? So the care family is a family involving two parameters of explicit solutions. So this is what I said. Minkowski corresponds exactly with the case equal M is equal to zero. So it's a trivial solution of Ricci equal to zero. Schwarzschild is the one discovered immediately after the general relativity was formulated in 1915. So this corresponds to A equals zero M different from zero and the full care family for all powers of A. The Minkowski space. So here is an expression of Minkowski space in usual variables in the variables T, X, Y and Z. So you see the metric, as I said earlier, is minus dt squared plus dx squared plus dy squared plus dt squared. The light comes from a point is like this and so on and so forth. So this is in physical variables. And here you see a picture which is very common for people in general relativity. It's sort of a, it's a Penrose diagram where you do a conformal compactification. You see this space time of course is infinite. And in order to have a picture which is finite, we compactify. This is of course done very often in geometry also. You compactify the entire Minkowski space by a conformal transformation. In other words, you keep the angle. So for example, the angle is at 45 degrees which correspond to null to propagation of light remain at 45 degree in this picture. So that's this boundary here, it's a boundary at infinity. So this is a boundary infinity in the sense that any null ray in Minkowski space, so this is a line at 45 degrees, any null ray ends at the point on this I plus which is a, it's called SCRI. It's called, this is called future null infinity because you are going to the future past null infinity if you go this way. Okay, so this is how the Minkowski space looks like in a Penrose diagram. It's interesting, by the way, this corresponds to R equal to zero. So R in polar coordinates, so you pass to polar coordinates. So this is R equal to zero. And of course you have to think about the fact that actually the picture rotates around R equal to zero. Right, so that's, so as well as this is not really a boundary. R equals zero is not a boundary, it's just an artifact of the picture. Okay, so let's look at the care family now in coordinates. Again, I just want to give you a sense of the fact that these care solutions are extraordinarily explicit. They correspond to something which are very, very interesting from a physical and mathematical point of view. And yet they have these formulations. So relative to coordinate systems, so you can write down a system of coordinates, T, R, theta and phi. So these are like polar coordinates. The Minkowski metric in polar coordinates will have just minus dt squared. And then there will be plus dr squared. And then a term which will involve d phi and dt tau, which corresponds to just polar coordinates. Anyway, so the coefficients in that case are very simple, but here the coefficients of course are more complicated and they're expressed in terms of A and M, which are the parameters here, and the coordinates R, theta and phi as you see here. So anyway, checking that it's a solution of the Einstein equation is of course what Kerr did. So Kerr discovered this in 1963. It's one of the great discoveries of the last century, both in mathematics and physics. It's really a remarkable family. You can read a lot of things from it, like for example, the fact that they are stationary. Stationary simply means the coefficients of the metric that you see here do not depend on t or probability. They also don't depend on the variable phi, right? Which means that they are not only stationary, but also axisymmetric. Another way of saying it, which is typical to mathematics, is to say that the vector fields t, which is d over dt and z, which is d over d phi are killing vector fields, okay? For those who don't know, it doesn't matter, but they're killing vector fields, they correspond to symmetries of the Kerr spaceland. They're also asymptotically flat in the sense that if you let R goes to infinity, then you can immediately see that all these coefficients trivialize to the coefficients of Minkowski space. So in other words, this become minus one, the whole thing here, and you'll get minus dt squared plus dr squared plus so on and so forth, which is a line element, the metric of Minkowski space, all right? So this is now the Schwarzschild black hole. So in the case of Schwarzschild things simplify because you just take a to b zero, that's all. You just take a to b zero, you get a simplification. The metric now looks like this, where you have a delta over r square is one minus two m over r. And this is a little bit misleading picture of what's going on, but it tells you something. It gives you a feeling of what is a black hole because at r equal to two m, if you look for r larger than two m, the space time becomes asymptotic, so it becomes more and more Minkowski. So in particular, the light cons will look like this, right? So light comes to the future, looks like this. Exactly on the event horizon, this light con looks like this. And if you're inside, it gets tilted and it looks like this. If you're inside r less than 12. So what that means since light propagates inside the light cons, it means that the light starting from any point, any event from this region will never get outside, can never cross the event horizon. So this is called the event horizon. Event horizon in some kind of, it's a boundary which separates the black hole from the exterior region. This picture is not very good. It's much better to make a penderos type diagram, which means a conformal diagram, in which case this gives you the maximally extended conformal picture of Schwarzschild, right? So let's look a little bit what it says. So you have the black hole region where r is less than two m, you have the exterior r larger than two m, and you see the exterior looks more and more like Minkowski space. This is a now future null infinity. So remember I said that null race end up here. So the future null infinity is actually corresponds to end points of this kind of null geodesics. The external region is again expressed here. You see on the event horizon, you see that the light con becomes tangent to the horizon. You see two region, two external region, this one and this one. This is an artifact. It's not a physical fact. It's an artifact of the explicit solution that it has two ends. It's topologically non-trivial. It has two ends. Of course, of interest is just this region, the external region and the black hole region. So the black hole region is of course, where you cannot escape and you have r equal to zero, which is a singularity. You can show that the curvature at r equal to zero becomes infinite in all possible senses. So this is a real singularity. And of course, singularities are terrible from a point of view of physics. You want to avoid them. So people, we'll talk a little bit about this, right? What is the nature of singularity? What do they mean? So this is Schwarzschild manifold. It's important, what I say Schwarzschild manifold, it's important to say that you don't have just one system of coordinates. You cannot view the entire manifold just using these coordinates, which I wrote here. In fact, these coordinates even become singular at r equal to two m, which is exactly on the horizon. So you have to change coordinates and different regions are expressed relative to different coordinates. For example, in order to pass in this region, I have to use a different system of coordinates, which is well understood now, but people had a lot of problems at the beginning of the theory, even Einstein himself, many people had lots of problems with this singularity at r equal to two m, which is not a singularity, it's just the horizon. But this was difficult at the beginning of the theory for people. Now it's considered triviality. The space time looks slightly different, but not much. I'm not even looking at this part. I'm looking, this is the most important part, as I said earlier, that's a black hole region. That's again the exterior region. The horizon is given simply by the roots, the larger root of this polynomial, r squared plus a squared minus two m over r. You have two roots. You have r equal r plus and r equal r minus. This is called the Cauchy horizon. And maybe I'll say a few words about it later on. It's something annoying. It's not a singularity in the real sense. The curvature does not become infinite here, but something terrible happens nevertheless, which is that past this r equal r minus, I can make infinity many extensions of my space time, which means that causality breaks down. So this is awful, of course, again from physical point of view, causality is an important part of physics. So this has to be seriously discussed. It's almost as bad as a singularity. So for Schwarzschild, we had that singularity at r equals zero. Like here, r minus is larger than zero. So you don't have that kind of singularity at all. In fact, these things are null. So these are null hyper surfaces. And again, you can extend the space time in infinity many way, which is awful. So this is one thing that one needs to understand. So again, external region, event horizon, which is exactly at r equal r plus, black hole region, which is this one, which ends up in this Cauchy horizon, which I wrote here, and null infinity, which corresponds to r equal infinity, right? Which are the future and the past null infinities. Okay, so let's go now to the real issue is our black holes real. So, well, black holes, since we cannot really see them inside because nothing escapes. How can we even talk about the reality of a black hole, right? So let's analyze it a little bit. So is an object physically real, if even if we don't have in principle, direct access to any form of detection? Mathematically, of course, they are perfectly real. But some of the reality is defined by the sub-consistency of its objects. There is such a single as mathematical reality. Okay, so that everything is fine from the point of view. We know that the black holes are mathematical real, but are they physically real? In other words, well, for that, you have to have some definition of physical reality. The one that I learned as a kid in Romania based on dialectic materialism was this one, which is my opinion an awful definition, all encompassing, including our minds, sorry, our word is missing here, our perceptibles through our senses, but completely independent of them. So everything that is perceptible by our senses, but it's completely independent of them. So it's transcendental. There is no, we don't affect this reality, but that reality affects us. And of course, it just doesn't fit with this sort of object. It doesn't fit with quantum mechanics also. It doesn't fit with uncertainty principle and so on and so forth. So here's a sort of a better definition, a contingent definition of reality or physical reality. An object is real. If it is mathematically self-consistent, plus it lists observable, measurably effects consistent with all other facts of an acceptable theory. So I hear sort of a question. This reminds me of the Plato's cave. Somehow there is mathematical reality. And physical reality has to have all sorts of other things. It's not just mathematical self-consistent. Okay, so this is again, the kind of reality that we want to talk about is in this sense. Okay, so there are indirect tests. So this makes sense, right? We can talk about something real, only even if it's not directly observable because it leads somehow to observable effects. Indirectly. So there are indirect tests of black holes and there are many Nobel prizes which are obtained on that. So there are astrophysical observations in particular the recent Nobel Prize of cancer and gas. Gravitational wave detectors by Barisch, Sohn and Weiss. So this was a few years ago. Numerical simulations, unfortunately, there is no Nobel Prize for numerical simulations, though maybe they should be. And there are mathematical theorems proved by Penrose for which he also got the Nobel Prize, right? So you can see already here the theme that I'm going to develop later, which is can you test physical theory using mathematical theorems? So I just want to, how much time do I have by the way? I'm a little bit worried this time. Can you finish in, if you can finish in five minutes, we'll have plenty of time for questions. If you need 10, I suppose we can go 10. Yeah, okay, I'll try. So maybe I will stop. I just mentioned the fact that Penrose got the Nobel Prize essentially for proving the theorem, which is a beautiful theorem. As theorems are today, this is very simple, but very penetrating. And if something to do is a notion of trap surface, so spacetime cannot be, so spatial has some kind of singularity. This is a way of saying what I'm saying here, if certain conditions are verified. So this condition, for example, when reach is equal to zero is automatically verified. M contains a non-compact hypersurface. So there's some kind of hypersurface which is infinite and M contains a closed trap surface. So here is a trap surface relative trap surface. The definition of trap surface according to Penrose, this was one of the great definition of Penrose. In fact, the theorem is more or less the definition. So once you understand the definition, the theorem is not very hard to prove. So you see, you compare the way a surface looks in Minkowski space where it generates light cons in the incoming direction and light cons in the outgoing direction. You see the volume is decreasing in the incoming direction and increasing in this direction. While in a trap situation, both directions, the volume in both directions is decreasing. So this can be expressed purely in terms of invariance that are defined on the surface itself. They are called null expansions and both have to be negative. So the null expansion in this direction is negative. The other one is positive in Minkowski space for a trap surface, both have to be negative. So this is a very, very, obviously something which is very far away from Minkowski space. It requires large curvature and so on and so forth. The remarkable thing that Penrose did is to show using this theorem to show the Schwarzschild singularities are stable. So this is actually very simple. Maybe I can explain it in two lines. The important thing is that if you are in a black hole, every point here is in fact a sphere and you can easily see that every such point is a trapped sphere. So the question was, does a singularity of Schwarzschild, does it stay if you make a perturbation on Schwarzschild? And many people saw that this is just an artifact of the special symmetry of Schwarzschild. So if you perturb, it is not going to happen. Well, Penrose theorem tells you that that's not the case, that this trap surface is of course, through that definition, they clearly survive under perturbations and therefore the singularity will survive under perturbation. Of course, the Penrose theorem is not very precise. It doesn't quite tell you what kind of singularities you get but already it tells you there's something terrible happens. Okay, another thing that he did for which he got an overpriced, I believe he deserved an overpriced for this. I'm not sure that he got it for this but in any case, this is a conjecture that he made which is a cosmic censorship projecture which is again connected with the situation that we see in Schwarzschild. You see in Schwarzschild, if you are inside the black hole, you'll encounter singularity if you go forward in time. While if you are outside, you are free of singularities. The conjecture generalizes this to the general initial conditions and it says that somehow you can never have a singularity without having a black hole. So in other words, there are no naked singularity. There are no naked singularity in the exterior of a black hole. All singularities have to be inside the black holes. So there are also asymptomatic results but they won't have time to talk about it that tell you how you can actually form a trap surface. You see, Penrose tells you that if you have a trap surface, you'd necessarily have a singularity. These type of results tell you that trap surfaces can form from regular initial conditions where you don't have trap surfaces to start with. All right, anyway, this is just to give a sense of the kind of things that people do. Here are some mathematical, the broader tests of reality that I call mathematical tests of reality which are the issue of collapse. Can black holes form starting from reasonable initial configuration? This I just talked about, rigidity. So the rigidity is kind of an interesting thing. We know that the care solutions are stationary. In other words, they are in some sense time-independent. Of course, this requires a definition because time is not an absolute notion in general activity. So anyway, but it can be defined what stationary it means. And then the question is, are there any other besides the care funding? Are there any other stationary solutions? There are some results of Carter Robinson and Hawking in the 70s which require an analyticity and which is very restrictive, unfortunately. And there are some other results by Alexakis, myself and Yonescu where we actually show that if you are sufficiently close to care in the smooth category, which is much more realistic, the smooth situation is much more complicated than this situation. If you are in the smooth category, then it's still true that if you are close to care, then you have to be care. In other words, there are no other solutions close to care which are stationary. And then there comes a major issue of stability. So this is maybe the most important thing that mathematicians are working on now. It's a question, is a care family stable and the general asymptotically flat arbitrary small perturbations? So this is a, I mean, this would take me another full hour to describe. Particular, there's been an immense amount of activity in recent year in the mathematical community. Of course, using some of the things that people have done, physicists or mathematical physicists have done earlier, but in the last 20 years, really this was a big problem for mathematicians. And I'm glad to say, to announce that Seftel and I have just been able to prove the general stability of the care family if a sufficiently small. In other words, you can go for a close to M. So the general conjecture would be to do it for A over M strictly less than one. We can do it for the case when this is sufficiently small. It's also very conceivable that in the near future we will be able to do the whole thing. But this remains to be done. So this is just something that the paper will appear recently within the next few days. It's a, unfortunately, it's a huge paper. So I mean, the problem is that these kind of results are extremely difficult and they require a lot of work. So our result is not even complete yet. I mean, the paper which will appear, I should say, is just one part of three papers. And that paper is a main paper, it has about close to 800 pages. So it's very unfortunate that the results like this are take a long time to prove, but the hope is of course that the time things do simplify. So for example, stability of Minkowski space, I simplify to, I mean, there are other results in other words that have originally been very long but have been simplified. I wanted to talk about final state conjecture, which is a conjecture that puts together many steps, in particular stability of Minkowski, which I just mentioned, problem of collapse, problem of rigidity, stability, cosmic censorship conjecture, the two and many more body problem. This is connected with the interaction of black holes. But I won't have the time. I should just simply say that this is, mathematical generativity is really a remarkable part of mathematics today where there is a immense amount of activity. There are lots of very, very interesting problems in particular, if you look for example at this final state conjecture, you see how many extremely difficult problems appear. People simplify things, you assume certain symmetries and then you simplify things become a little bit easier. But nevertheless, the big problems are the ones which I mentioned. There is also, and I'll finish with this one, there are also issues that have to do with the singularities in GR. I mean, in a sense, this final state conjecture relates the mathematics of what happens outside black holes. Right, so you go all the way to black holes. This has to do with what happens inside black holes. It's connected with the fact that this horizon, this question, are these Cauchy horizons real? Of course, they are mathematical real. Are they physically real? In other words, are they stable under transformations? And there is a conjecture, which is a conjecture by Penrose also, that should tell you that this does not survive. If you perturb the care solution a little bit, this is not going to happen. Penrose thought that you'll get a singularity just like in Schwarzschild, but things are more complicated. There are results of the fairness and blue, but I won't have time to talk about it. So I'll stop here. Okay, great. Thanks, Sergio, this is a very enlightening talk. Everyone, we're open for questions, which you can put in the chat or the Q&A. And while you post your questions, let me ask Sergio the first. So you said that singularities are bad for physics, but from the point of view of string theory or any extension of generativity, in a sense singularities are good for physics because you need to go beyond a general activity test to test your theory. And so one would like to know whether cosmic censorship is truly valid so that we do not have access to singularities or if there are exceptions and possible observable consequences of singularities. So has the conjecture been settled? And what's your opinion about the conjecture? Okay, so there are two, unfortunately I had to go too fast. There are two cosmic censorship conjecture. One, which is this weak cosmic censorship, so-called weak cosmic censorship conjecture, which has something to do about the external of black holes which says that outside black holes don't have singularities. And then there is this other one, which I mentioned here, which is inside black holes. Inside black holes, you should have singularities. So you are not going to avoid singularities no matter what. It's just a matter of the nature of singularities. Well, actually here, there is also this issue of Cauchy-Horizon, which is the issue of causality. So in that case, there are specific proposal of what should happen with the singularities inside. Of course, we are very, very far from any of these two conjectures in fact. There's no evidence against the weak one or- So there will be the issue of naked singularity outside the horizon and the inside. Of course, you'll have singularities. The question is what exactly is the nature of singularity. But you're absolutely right. I mean, quantum mechanics would like to, I mean, people would like to introduce quantum mechanics with general activity, string theory, it's such a proposal. But this, of course, is still a dream, right? So we don't know what happens. Okay. Well, I'll continue with a question from attorney A. Do you think your methodology could be used in the arena of quantum physics? Yeah. I mean, look, I mean, this is exactly what I meant when I talked about this entanglement, right? That mathematics, which is developed for certain problems turn now to be relevant in others. I mean, this has happened all the time through history, right? The kind of physics, very often physics that was used by physicists at a given time was developed for a completely different problem by mathematicians a hundred years earlier or maybe 50 years earlier or something like that. That has happened many, many times. And so the hope is that though we don't really work on quantizing general activity, we hope that some of the things which are developed here will be used for other things. I don't know where, of course, but I mean, in other words, again, I think this is very important and maybe many times people don't realize that this is the major difference between mathematics and physics, right? In mathematics, you really, you work on a specific problem, you try to solve it, you push the boundary of what's known mathematically. And as a consequence, you bring new tools which may be, which very often are used in other things. And concepts, not just tools, but you bring concepts very often. Right, thanks. An anonymous question. Do black holes move in space and change position? And if yes, what is the cause and can they change form? Good question, yeah. So this is a issue of dynamical black holes. Obviously I was talking about stationary, the stationary black holes are really the end, the final, they are sort of the asymptotics of more general black holes, which are still dynamical, which are changing things are, there is radiation that moves inside and the black hole itself is changing. And of course, you can have interactions of black holes, two black holes, again, dynamical black holes, they can coalesce even, right? They are very close to each other, they can coalesce. Of course, we have Thibaud Amour, I understand, who is one of the experts in this. So there are very, there's a lot of dynamics with black holes. I mean, so the kind of black holes that I was talking about, these care solutions are the final states of dynamics. In fact, this cause me, sorry, this final state conjecture states exactly this, right? I didn't spend time on this, but initial data sets behaving the large, like a finer number of care black holes moving away from each other. In other words, asymptotically, if you start with initial conditions, you can have some very complicated dynamics. And, but asymptotically, as time goes to infinity, you are going to see just a finer number of care black holes moving away from each other. The reason for this would be that black holes, again, would have complicated dynamics. They can coalesce and they form only one black holes or they can move far away from each other. And therefore asymptotically, if this conjecture is true, then you are only going to see care black holes moving away from each other and some radiating decaying term, in other words, some radiation moving away to infinity. Thanks, I'm going to ask one last question from John Thorella, who in fact, this question is prompted by an article today in the New York Times by Dennis Overby, who asked the question, what should one call a bunch of black holes? And of course the analogy would be to groups of animals for which there are all sorts of whimsical terms in the English language. Of course, there's a pack of lions or a pride of lions or pods of whales. And given that we observe now these collections of black holes in space, perhaps we need a collective noun for the black holes and the suggestions that were, there's even a poll given suggestions such as a crush of black holes or a enigma of black holes. And I wonder, Johnny, have you thought about this question or what would come to your mind for a collective of black holes? Collection of black holes. But again, this finite state conjecture really gives you a story, right? It gives you the fact that in reality, in any finite regional space, you can only see one eventually. So they tend to spread apart. And so the collectibles form only one, right? Okay, okay. So what would be the right term? So we'll think about that. And if you have any suggestions, the poll is going on now, I understand from the New York Times article. And so with that note, let's thank Sergio again for a really excellent talk. I'm going to clap and represent all of us in applauding this talk. Let me now announce that the Friends of IHS will be holding its biennual gala this year on November 16th. And this year will be in a hybrid format. So of course, in part because of the pandemic, but in part to give more of us the opportunity to attend this very illuminating and entertaining event. So the topic this year will be women in fundamental research. So we have honorees and we have quite a spectacular program plan. So mark your calendars November 16th and I remind you all that the IHS relies on the support of all of us, public support, private donors, there's a website friendsofihss.org and I encourage you all to give what you can. And I thank you all for attending our virtual event and I look forward to seeing everyone again soon. So long. Bye-bye. Thanks.