 Thank you for the introduction. Also, I'd like to start by thanking the organizers for the invitation to IHS. It's my first time here, and also for this opportunity to give a talk at this wonderful conference. So the subject of my talk would be global popularity and scattering properties of two nonlinear wave equations, which you can think of as generalization of the Maxwell equation. So we'll start from there. So Maxwell equation is a system of linear equations, which concern a real-valued one form on a space time, which is supposed to be the electromagnetic potential. In my talk, my space time will always be the Minkowski space, d plus one dimension Minkowski space. From this, you define what's called the electromagnetic field, which is nothing but the curl of A. So this is the anti-symmetric part of the gradient of A. The Maxwell equation reads as follows. It's the space time divergence of the electromagnetic field is equal to the charge current factor. So this is the enogenous term for the Maxwell equation. The first nonlinear equation, there will be the subject of my talk, is what's called the Maxwell Klein Gordon equation, which is a nonlinear generalization of the Maxwell equation by coupling it to a scalar field, which evolves via a Klein Gordon equation. So how this looks as follows. So this is an equation for an electromagnetic field, sorry, potential, and a complex-valued scalar field, which I call phi. So the electromagnetic potential acts on phi as a potential in the Klein Gordon equation. So in order to make sense of that, I introduce what's called the covariant derivative, which is a first order operator, which is in the mu direction, it's just the partial derivative, plus the potential term given by the electromagnetic potential. And then the phi equation, the phi satisfies the Klein Gordon equation but with respect to this covariant derivative, a covariant derivative. And then the A evolves by the Maxwell equation, where the source is given in terms of phi by the following expression. So it's the imaginary part of phi times the complex conjugate of the covariant derivative of phi. So the other, the second nonlinear wave equation that would be subject to my talk is the Yang-Miller's equation, which can be also thought of as a nonlinear generalization of the Maxwell equation, but it's a generalization of sort of geometric nature. So actually in order to properly formulate this, I wanna come back to our Maxwell equation and Mexican Gordon equation and I'll give you point out a important property of these, which is called the gauge invariance, okay? Because that's, so I wanna get to here, talk about that first before I wanna talk about Yang-Miller's. So in both of these equations, there's inherent ambiguity in the description of the solution. So in the case of Maxwell equation, it's clear by what I mean. Or what I mean is if you look at this F, then this value of F doesn't depend on, is independent of, does not change under the change. If you change A by adding a gradient of some real value function, right? So because the curl kills the gradient, right? So this is the ambiguity. So it's called the gauge invariance of the system. And the Maxwell Klein Gordon has a similar invariance where A transforms in the same way and of Fee transforms by multiplication by e to the i chi. But then once you write like this, you realize that actually this lens, this phenomenon of gauge invariance lands on different interpretation of these equations, right? So what you can think is in fact, the Maxwell Klein Gordon equation, the complex field here can be thought of as a section of a complex line bundle, okay? Where, which can be identified with a complex value function by choosing a frame. So frame by frame, I just mean a choice of basis of the fiber at each point. In this case, it would just be the choice of which vector should be deemed as one in the complex plane, right? And then this gauge invariance is nothing but the rotation of the frame at each point. And this derivative is nothing but the covariant derivative on sections of a complex line bundle. The reason why I set this all up is because Yang-Mills equation is a generalization of the Maxwell equation. In the case when the vector bundle has a gauge group which is more general than just the group given by multiplication of this, okay? So in fact, this can be formulated in a more general way but let me stick to the most concrete, yet non-trivial case, okay? Which is when the gauge group is the group SU2. So in this case, the Yang-Mills equation will be equation for a one form which takes values not in the real numbers but in the matrix algebra little SU2 which is the onset of traceless anti-permission two by two complex matrices, okay? And then you can formulate a similar connection. So this gives you a connection on vector bundles with SU2 as the gauge group. And then it turns out that the analog of this electromagnetic field is nothing but the curvature two form, okay? Which is the commutator of the two current derivatives. And you can explicitly compute how it looks like. It looks like the following. D mu A mu minus D mu A mu. So so far it's the same as the Maxwell case but then we have an extra term, the matrix commutator between A mu and A mu, okay? So you see that the formula from A to F is now non-linear and non-linearity is due to the non-competibility of the matrix algebra. And the Yang-Mills equation now reads is the analog of the Maxwell equation with zero source, which is this. So it's the covariant derivative of, so covariant spacetime divergence of F is equal to zero whereby this acts on, you view F as a section, or a tensor of, which takes value in SU2. So in fact, this, what I mean by this is it's, and in this case there is the same, there is a gauge invariance as well, which is given as follows. So if you take A mu and transform it by the following function. So take any function from the spacetime, which takes value in now the group largest SU2. If you make the change like this, then the equation remains covariant. And the interpretation is of course that these are the same objects but just the frames are rotated by this view. So we'll be concerned about large data theory for our initial value problem for these equations. And of course the starting point is the conserved quantities for this equation. So the, both equations have conserved energy. So in the case of Maxwell-Plan Gordon, the energy measure that time T of a solution, which consists of A and F, has to form spatial integral over the slice constant T of basically the sum of squares of the each component of F. Yes? So I just chose it so that it's the simplest example of a non-Abelian group. So that it should be the simplest, non-trivial example of the equation. So of course you can formulate the equation for other D groups but for completeness I'll just exit to be asked. Any other questions? All right, so the conserved energy for the Maxwell-Plan Gordon equation is basically just the L2 norm of F and the covariant derivatives of phi. And the conserved norm energy for Yang-Mills equation is basically just the L2 norm of F. So I can write it in this case as just like this because now these are matrix valued. And both systems also have, okay, so, okay. So these in the massless case here and always in the Yang-Mills case you have scaling invariants. And the both systems are, so the systems are invariant under the following scaling. So in the Maxwell-Plan Gordon case with mass equal to zero, it's invariant in the scaling where you map A and phi into something like this. So in other words on both A and phi scale like inverse of length, okay, which is natural because A is at the level of a connection coefficient. And in the Yang-Mills case, it's also, A also scales, has a dimension of inverse length. And you can see how the conserved energy behave under the scaling variance of the system and realize that for both equations, the critical energy critical dimension is when D is equal to four. So when the space dimension is equal to four. So on this with the case, there will be concentrated from the rest of the talk. And also I should say that in fact, our real motivation would be to try to understand the Yang-Mills equation, okay. But then, and Maxwell-Plan Gordon equation, the has been traditionally sort of, as you will see shares a lot of analogy with the Yang-Mills equation, okay. So the viewpoint we'll take is that the Maxwell-Plan Gordon equation would be a model problem, simpler model problem for the Yang-Mills equation. And in that sense, because the Yang-Mills equation is massless, there's no mass in the Yang-Mills equation, I'll take the mass parameter in the Klein-Gordon to be zero, okay. So if we're properly speaking, one should call this the Maxwell scalar field equation, but unfortunately in the literature, even the massless case is called the Maxwell-Plan Gordon equation. I'll just stick to the mention, is it okay? All right, so I'll erase the, this here, then now I'm not lying about anything here. Okay, so in the, so dimensions below four are so-called energy sub-critical dimensions, and in those cases there were results about the regularity of these equations from early on in the 80s and the 90s by the work of early Monkreef, Monkreef and Klein-Gordon and so on. But the understanding of the critical dimension and at the energy regularity has been obtained even in the perturbative regime, very recently. So in the case of Maxwell-Plan Gordon equation, this is a theorem due to Krieger, Sturbins and Tauro in 2012, that small data global existence told at energy regularity in D equals four, okay. So let me just say the initial value problem for the Maxwell-Plan Gordon equation with zero mass. Okay, on R1 plus four, under some additional gauge condition, I'll explain, I'll get back to this some soon, is globally opposed and scattering holds if the energy of data is sufficiently small. Okay, and I added this condition. So actually, so this condition is important half in order to actually have even a form of the well-posed system. So this is a, and this has to do with the gauge invariance that I explained earlier here. So by the presence of gauge invariance, if you just write down the equation, then you realize that it's a under-determined, you have too few equations to have a well-posed system. So you have to, the way you fix it is you impose one more condition that you want your connection to satisfy. And this is one way of doing it, and this is called the cool-long gauge condition. And the choice here is crucial because it's in the cool-long gauge where the fine, low-structure of the Maxwell-Plan Gordon equation is apparent. This is an observation that goes back to Clinton and Macedon in the 90s. There's a corresponding theorem for Yang-Mills equation. And this was proven only last year. Let me just say, same result hold. So with this small data result in hand, our goal would be, is to address the large data problem for these equations. Okay, that's a good question. So, okay, so it would be a whole story. So let me not get into too much details. But scattering here would just mean that under this gauge the solution to the problem would converge asymptotically to a simpler equation of, yeah. But the linear system would be actually not exactly the linear wave equation. But I'll get back, yeah. Let me not say too much about this. Okay, so the question for large data, the large data question for the Maxwell-Plan Gordon equation was a subject of a result that approved in collaboration with Daniel Tataru in last year. And we can prove, we can show that the result is that the global regularity in scattering, whatever that means, hold for arbitrarily finite energy data. So for the problem arbitrarily finite energy. So one goal of today's talk would be try to give you a overview of the proof of this theorem. Okay, I should remark that basically at the same time and independently, there was an internal proof of this theorem by Krieger and Neumann who followed the Kenick-Murray constant compactness and rigidity approach to prove this theorem. Okay, and more precisely, they followed the development of the scheme due to Krieger and Schlagg which were done earlier in the wave map case. But as you'll see in this talk, the approach that we take is different. And on the one hand, as you'll see, it is a little bit more direct. And on the other hand, the proof that we use relies more on the nonlinear structure of the particular nonlinear structure of the Maxwell-Plan Gordon equation, I guess then the other proof. But okay, so this is the picture for Maxwell-Plan Gordon equation. And then you might ask what would be the corresponding result for the Yang-Mills equation. And of course, in the Yang-Mills equation, you cannot hope for such a unconditional global postage result because there are time independent solutions to the Yang-Mills equation which have finite energy and therefore it doesn't scatter. So for Yang-Mills equations, these are nothing but instantons which are a solution to the elliptic four-dimensional Yang-Mills equation which have been studied in the 80s in the geometric analysis community. And because, and moreover, it's also known that instantons can lead to, can serve as a profile for finite and blow-up. So there are explicit constructions of finite blow-up known due to Raphael and Lonezki in this case. So in this case, the proper problem, the large data result would not be a conditional result, but rather the question until which energy does global postiness and scattering hold. So this would be the problem for Yang-Mills equation. And the problem would be as follows. Is the initial value problem for Yang-Mills global post and you have scattering for energy up to some threshold. And the natural threshold, as I'll try to explain, to a conjecture would be that of the first, the Brown state by which I mean the lowest energy non-trivial solution, time-dependent solution to Yang-Mills equation. So the goal of my talk will be as follows. So in the bulk of my talk, I'll try to explain the proof, strategy of the proof of this theorem. And it will serve two purposes. On the one hand, of course, it will be an expression of the proof of this theorem. But on the other hand, the deeper motivation is I wanted this would be, this would be a roadmap to a possible proof of this problem for the Yang-Mills equation. And then if time permits, I'll report on the recent progress on this problem, which is obtained by collaboration with Daniels Tarr. Okay, any questions? Okay. All right, so let me describe to you the strategy of the proof of the global opposeness and scattering theorem for the maximum climate equation. And this is a, this builds upon the earlier strategy due to Sturban's and Tarrou, which was carried out in the case of the way map equation. And so for the purpose, for concreteness, I'll focus on just the case of global opposeness, okay? I'll try to tell you how global opposeness can be proved. So the whole argument is a, of course, contradiction argument, okay? So we'll assume that global opposeness fails, or in other words, finite time blow up occur, and then we'll try to derive a contradiction. And as is well known by the by the combination of the small data result and finite speed or propagation, you can immediately say that if you have a finite time blow up, then it must happen by energy concentration at finitely many points, okay? So that's what I mean is to find. So there will be some time before infinity where your solution breaks down. So your finite time blow up. You can identify a point on the final slice, such that if you look at the domain of influence of the point to the past, so it will be like comb emanating from this point. And you measure the, so let's call the cross section of the intersection of the constant T slice and this cone as T. Your energy measured on this as T does not go to zero as you, as T approaches the final time, okay? And of course you should contrast it with the case when this point is a part of the regular space time, in which case by continuity, this would, of course, go to zero, okay? But then it turns out that you can say much more, and this goes in line with actually the picture that Piotr described yesterday, which is that you can say that blow up this constant energy happens in a particular form. So what we prove is that you can extract a bubble from this finite time blow up with a particular profile, okay? So what I mean by this is the following. So the claim is that you can find a sequence of times and space coordinates and scales lambda n, okay? So it's that, so I'm sure you should have in mind is somehow there are these space time coordinates which approach this tip of like cone. So Tn approaches T plus these Xn remain in the cone and there's sequence of scales which actually go to zero faster than on the radius of the cone. So it's little o of this, such that if you blow up this picture and make this on scale lambda one to be unit, okay? So which means that we use the scaling to think about the following rescaled objects. So we rescaled everything by lambda n centered at Tn and Xn. Then the sequence converge to something to another solution to the vector plane Gordon equation in a strong local H1 sense. So more precise on a sort of unit time interval, okay? Where this object solves the Maxi-Clinic Gordon equation and what's more, it is a stationary solution to the Maxi-Clinic Gordon equation. And in fact, I can even give you in which direction is stationary, okay? So by passing to some subsequence, you can assume without loss of generality that these points are asymptote to some time like curve, which I'll denote like this. And its generator I'll denote as Y, okay? And the solution turns out to be stationary with respect to that Y variable, okay? Direction. Why does it have to be a time like null? Well, a short answer is because nothing can travel, nothing with mass can travel at the speed of light because of finite energy. Yeah, yeah, yeah. But this is an important point. Actually, it'll actually come up, so yeah. And by stationarity, I just mean the following. So the contraction of Y with F formed by B is zero and covariant derivative of phi with respect to the Y variable is zero, okay? And then you say the following. Okay, well, we can always apply a Lorentz transform to Y to make it a DT, right? And then the claim is this. The claim, the second claim, this is one claim. Second claim is that there does not exist any time independent or actually stationary, non-trivial, finite energy solution to the Maxwell-Flanders equation, okay? And this is very simple. You apply the, I can even give you a proof. You apply the Lorentz transform to make it DT, then you realize these will obey the following equation, okay? Where it looks formally like the Maxwell-Flanders equation, but this is an equation on R4. And you just realize that you can multiply this equation by psi, integral of parts. This tells you that covariant derivative of psi is equal to zero if you can multiply parts. And this tells you that the equation decouple and therefore F must be trivial. So you can take B to be zero and then psi also has to be zero. And of course, the reason why you can do this in this part is because you have finite energy assumption. So it all boils down to showing this bubble extraction on a claim. Actually, let me, I saved this board for this purpose. So, and of fundamental importance to this bubble extraction is a very beautiful monotonicity formula for the Maxwell-Cline-Gordon equation which I'll now try to explain. So the key monotonicity formula, which is responsible for this is something like this. Okay, so but I need to first on fix some patients, okay? So let me for convenience translate the, let me for convenience translate the singularity to the origin of my space time, okay? And moreover, let me flip the time direction so that I look at something that blows up to the past, okay? I just wanna work with sort of this half space. And let me also assume for simplicity that I've cut off everything that happens outside on this region so that essentially the energy outside is very small, okay? So that we only concentrate on what happens here, okay? This can be done by some gluing of initial, excision and gluing of the initial data. And in order to state the monotonicity formula, let me set up some notation, okay? So let me introduce the vector field which I call X which is the normalized scaling vector field, okay? So this is the scaling vector field, but then multiply by a factor so that it's Minkowski norm is always equal to minus one, okay? So this row is square root of p squared minus X squared. And if you plot this vector field on here, it will just be something like this, right? So it'll always point in the scaling direction, right? But then you'll see some degeneration as you approach the boundary of Lycone because there this vector field becomes null, okay? Let me also introduce some null variables, U, which is, according to my convention, T minus epsilon of X and V, which is T plus epsilon of X, okay? So in particular it's chosen so that this boundary of Lycone is the U equals to zero on hypersurface, okay? And with this I'm ready to tell you the key monotonic state formula for bubble extraction. This is actually, okay, so let me just first state it. So let's take two time slices which are called ST1 and ST2, okay? So the monotonic state formula is something like this. So morally, the following monotonicity formula holds. So there exists some weighted non-negative energy quantity P such that its integral on this slice bounds from below the integral of the same density on the other slice closer to the singularity. And in fact, the difference is given by a spacetime integral of certain quantity density which are called Q, which is also non-negative, up to some error, which goes to zero as you approach the singular time, okay? And here, this density of the spacetime integral is given as follows, is one over rho times the interior, the contraction of this expected field to the Maxwell field squared plus this self-similar derivative of P squared. And this P takes the following form. P is some weighted energy quantity where up to constant factors, I'll probably mess up. It's this weight times the R inverse dv R phi squared plus the reverse weight times the U, the same quantity R inverse dU R phi squared plus these weights. Times the angular derivative plus a hardy weight term, okay? And furthermore, okay, and actually, in fact, there is also a contribution of some terms which involve A, which is non-negative. So I'll not talk about this, okay? Okay, so a few remarks before, okay, so I don't have much time. It's not as big as the rest of that. Right, so yes, so that was the first mark I'm gonna say. So of course you might be puzzled how, okay, it's curious that you see a monotonicity formula in a time-reversible wave equation, right? And of course the magic lies in this error term, okay? Which is actually the contribution of the flux. And in fact, the reason why you have this decay so that you have this sort of, this sort of monotonicity formula modulo, this error is that your priory have decay of flux. And the decay of flux just comes from the fact that the flux density, energy flux density is non-negative. So you have finite integral, so if you take your, yeah, it's small, then it's gonna go to zero. So let me just say that this error is due to flux decay, energy flux decay, which is a nice feature about this equation. And second, let me also say that I lied a little bit. So it actually goes to zero, yeah, okay, sure. Yeah, yeah, yeah, yeah. Okay, so I should also say that, let me not write it out because we don't have much time. I should say that this formula is actually only worldly true because the way I wrote it, the weight you see actually goes to infinity at the boundary, okay? So you need some regularization procedure, but let me stick with the prettier lie rather than the messier truth, okay? You hope you believe me that that's a technical thing that can be worked out, okay? But very nice. So what you get out of it, first of all, is the following. That you have, in fact, by independent means you can prove that this integral, these integrals, is bounded from above by some absolute constant times the energy, okay? So this is something that's always bounded. At least, you know, at least as you approach the similarity. And what this tells you is that this space-time integral is finite. Let me write it out. And this signifies the fine. So look, there is a weight here, which goes to zero basically linearly as you approach the tip, okay? So this signifies that the fact that this integral is finite tells you that there is some logarithmic decay of this integrand as zero to zero, right? Otherwise, this cannot be finite. So this is actually the key crucial ingredient for the bubble extraction, as I'll try to explain now. So what this tells you is that, worldly, the derivative, this sort of time-like, derivative on decay in some integrated sense as you approach the tip, right? So you wanna make use of that. You wanna say that you can extract a solution, extract a limit of sequence of solutions and say that that must be stationary like that due to the decay of the integrated decay which comes from this, right? But then, you have to divide into two cases. One, okay, but then, what you also realize is that there is degeneracy of this vector field as you go to the boundary of the cone, right? So this is only effective in a fixed time-like cone, not in the whole cone. So you have to divide into cases. One, when there exists a fixed time-like cone, which I call C tilde, such that some non-trivial amount of energy remains in the time-like cone, okay? So this would be our case one. And you also have to think of the case when this doesn't happen. And by time-like cone, I just mean a cone with opening which is strictly smaller than this cone. Inside ST, but it's not the case because of your intersection. Sorry, so, wait, wait. So I'm only concerning energy in this region. So what this is trying to say is that this is the case when there's some energy remaining in a fixed time-like cone, okay? And this is the case when the energy exits all fixed time-like cones. And let me just say that, so what happens is that in the case of one, the decay of such a time-like vector field is effective, okay? And this allows you to find the desired sequence of space, time, and scales, which would give you the bubble extraction. You see, this explains why you have this particular direction of stationarity, right? Because this is nothing, so you imagine that you found these sequences, right? And imagine that you can do it in such a way that the same integrant decays, okay? In these boxes. And then you realize that that X vector field along asymptotically becomes this Y vector field, right? The direction, time-like vector in the direction of this asymptotic time-like ray that these approach. And, but then, so is this, okay? So the smaller cone is at a given distance from the other one, right? Right. But then, you know. Make sure that you actually get something time-like. Right, right, but then the point is, in fact, you can actually extract such a sequence. I can't go, I don't wanna go into details, but you can extract such a sequence if this holds for any fixed time-like cone. So the case where you cannot use that monotonicity formula is exactly when you cannot find such a cone. So in other words, the energy exits every time-like cone as you go toward the singularity. And in fact, this is precisely the case where you have to think about, this is precisely the case when everything sort of is traveling at the speed of light, right? And this is the case where your prior will think that there cannot be any non-linear bubbles, right? Because if there were any bubbles, then, you know, because of finite energy assumption, you cannot, you know, travel at the speed of light. And in fact, what you can do is you can now look at this statement that this weighted quantity is always bounded and realize that it has a very good weight on certain components of the field phi, right? So in particular, the dv derivative of phi decays and the angular derivative of phi decays. And this tells you that, in fact, there cannot be any non-linear bubbles and more quantitatively, you can say the following. In fact, one way to quantify this, we're fine. So you, one way to find a concentration, a point of my concentration would be to look at phi and, you know, test this or integrate this on some ball of radius lambda, okay? But then await it so that it's a dimensionless quantity. So you wanna take something like, I think, this, okay? And say that this is very small. Well, this is a wave equation, so I need to take a time derivative as well. So I also take something like this. And this, the heuristics that I told you basically translates to the statement that such a quantity uniformly for all lambda and all x goes to zero as you approach the singular time, okay? And of course, what's happening is that, you know, outside we cut off anything that's happening. Inside, you know, there's no energy in any time like home and where the energy lives, there is no, not near bubbles. You have smallness of dv derivative and the angular derivatives which give you on this. And the key hyperbolic, the wave regularity theorem that we prove precisely enters at this point. So the result is that if this is small enough, then we can continue the solution on past zero. So there's a contradiction. If this holds, so this is a contradiction. And this all ends the summary of the proof. So I think I ran out of time, so I can't say anything about, you know, so I'll just stop here, thank you. If I understand correctly, this monotonicity formula, which is not a monotonicity formula. Yeah, well, yeah, it's hidden, yeah. It doesn't give you any prior estimate on any solution. You have to take different cases and use it differently in different cases, is that correct? Yeah, I mean, well, it's a- Because by itself, it doesn't tell you much, right? Yeah, it's a weak, it's a very weak decay statement that it tells you, right? All it tells you is that there's some, this, you know, this integral quantity is bounded, right? So this tells you that sort of on the integrant actually decays in some logarithmic sense. But even that, I don't understand because you don't have any information about this, I think that I think. Oh, no, so what you can do is, because you know that this is, so I was saying that by separate means, you can show that this is always bounded. And this means that you can take, you can take t2 to zero, right? That's all, it's uniformly bounded. So that will tell you that this space-time integral is bounded. But the space-time integral comes out of the cube, I think it's part of the cube. Yeah, yeah, but it's on the, it's on the, it's on the left-hand side. Right, but, okay, by the way, I have to ask you about the, I keep on telling you something. This is uniformly bounded for any t2, that's something that you can prove on, you can prove. So you can take just t2, yeah, yeah, yeah. Two questions, a little bit short. This, that's the line, you want to use three cards? Oh, that's a, so of course, but we also have to use sort of, by linear estimates and so on. So, why is it complicated? Because for the way we have the same thing. Yeah, that's a very good point. So in fact- Because it's just, we use sobolef and we mix sobolef and three cards. So this is more of a more delicate. Yeah, so, yeah, right, right, right, exactly. So you see the fee equation, if you expand has a term, which is a times derivative of fee. And okay, you're right. So this actually statement is a complete triviality for the semi-linear, for the power type nonlinearity. Because in the, there, what you can say is that in the nonlinearity, you can use three cards to say that you have, you have very good off diagonal decay and if you compose into little poly pieces, right? And the point is that this translates to smallness of some Bessel floor, yeah, and that's it. But then here, you realize that because of the presence of the derivative and the wave equation only gains one derivative, there's no exponential decay between the low frequency part of this and the high frequency part of this. So this is actually the key difficulty that one has to face, yeah. Then the second question. Okay, this is, I guess, the natural extension of the formula I have with the idea for the critical wave equation. Yeah, yeah. Okay. So you don't go back as in the proof I have with Carlos to zero outside when you have a real monotonicity formula when you are done, in some sense. You don't have to worry about the problem. Right, right, but then we, right, but then that's when you construct a minimal counter-example, right, right. So, right, I mean, I guess the point is this is more a more direct approach. So we are now assuming that we have a minimal counter-example. We're assuming that give us any counter-example. But then, okay, so the thing I don't understand is we lose approximation of current inside and not outside. Outside is not easy. Outside is, I mean, outside is actually, I mean, because I find it's propagated out and flux decay outside is very easy. So, yes, you are, you control all these terms. So we start, okay. Okay, and the way I can discuss with, we continue and...