 Well, thank you very much, Frank. Thanks to you all for coming and thanks very much to the organizers for inviting me here. It's a great place and it's been a great program so far. So what I was hoping to do in these three presentations is to tell you about a story about the cubic NLS on product spaces. So I hope you will like what I'm going to say, but I'll also try to use this as an excuse to wander a bit around and present some site questions related to this. So I want to talk about a very simple equation, the cubic nonlinear Schrodinger equation. And so here this is manifestly a nonlinear dispersive equation so there will be no need to really do any derivation. The point is to try to be able to do some explicit computations in these simple settings. And now maybe just a few explanations about the bold. So many people have worked on this and so those are people that I should at some point mention today. I wrote their names so as not to do it all the time. And maybe the one thing that I should say is that most of what I'm going to talk about is for those things that were initiated by Bourguin so I won't really repeat his name all the time but he really started all of this. And to some extent this is also true about the second line, the item that for the defocusing case really started a lot of the things that I'm going to talk about. Now so I want to consider this equation in the defocusing case and I will look at mostly a function which takes values in a product of R3 namely R cross T2 with complex values. So my goal for today is first to tell you a little bit about this. Second to try to motivate why I think this is something interesting to talk about, why the defocusing, etc. And third to give you an outline of what I hope to present in those three lectures so that after that you can decide whether you think it's worth your time or not. I'm going to try to stick to more basic things and maybe prove some things that are somehow folklore but well I hope this, you can find something useful in that. So first if you comment about the cubic NLS. So this is a simple dispersive equation, it's the model dispersive equation and to some extent the results that you get from that are fairly general because this is not an equation that is going to appear naturally in physical settings so it's not something that you derive from first principle, it's to some extent something that you derive from second principle in the sense that it's going to appear in many different dispersive settings as controlling solutions that have some form of asymptotic behavior, some form of concentration one way or another. So this is an envelope equation in many different cities. And so understanding this solution, the hope is that if you understand it well enough then you can really say something about some form of singular behavior for solutions in many different cases. So this is, so for example, some equation to water waves that under some long wave approximation they will be well described by the cubic NLS. This is also true for solution to the Euler Maxwell case because I essentially derive the Zacherov system from this. This is also true for equation in optics. But in fact it's also true even for some other model dispersive equation that would also qualify as fairly simple models. And so for example, tau, kilibb, qon, xiao, and vsan. And in fact, that's also really showed that you can't really control too well the KDV equation, the Quintic KDV equation, unless you really understand well enough the Quintic NLS in this case. And for the cubic NLS, it's also related to, well, understanding this equation in R2 is really something necessary if you want to understand well solutions to the cubic Klein-Gordon equation in R2. And this was work of, so this was exploited first by Nakanishi and then by Kilibb, Stowell, and Vsan. So besides the fact that it's a model equation, there is some use to study it. On the other hand, it's not too general in the sense that there are some special features here that really severely restrict the degree of generality of this equation. So the first is that it is de-focusing. Well, this changes a lot. It's the only thing I'm going to talk about mostly because, well, things would be even richer if you were to look at the focusing variant of this. And for the moment, that is just too much for me. It's also a semi-linear equation. And so, well, this means that you really can understand very well the local theory. And then to which extent can you extrapolate to really what a wave or a Maxwell that is a lot less clear for the moment. But something which is maybe also not too appreciated is that it's a scalar equation. And the moment you go to systems, in fact, the set of things that can happen becomes really a lot richer. So maybe here, I'll give you some examples of other dispersive equations. That would not satisfy this. So there's a curve system. And another thing which is very specific to this equation, which explains the fact that you're going to find it everywhere, is that it has a very simple dispersion relation. And in this case, a convex dispersion relation. And in fact, this really means that a lot of pathology that you would expect are not going to happen. For example, the stationary phase, you can really use the stationary phase fully, et cetera. And there are many cases where you don't have a convex dispersion relation. So one example that was already mentioned was the capillary gravity water wave equation. Another would be the Euler-Puesson equation for the ion. And so it's good to keep those limitations in mind when one wants to extrapolate results that you get from the cubic nonlinear Schrodinger equation. Now one thing, one last thing. So another hope that I had in these lectures was to try to draw some connections to previous lectures because this is an easier case to most of what we have seen where maybe you can see some of the aspects. And so in this case, if you remember the talk by Professor Kenig, one thing that was very important was to understand this profile decomposition. And the idea is that the cubic analysis should enter somehow in profiles for those equations. But the moment you go to quasi-linear equations, it's a lot less clear exactly what that means and what help that could give you. But on the other hand, understanding if this helps or not would be something that I think would really allow us to make a lot of progress in the study of those equations. The main problem being that if Schrodinger was ideal, then you don't really have a good perturbation theory and then what good is anomalous solution is less clear. All right, so now I would like to motivate a little bit more this equation on those spaces. So the fact that I'm going to look at the defocusing case, well that's because it's hard enough and if you were to look at the problem in the focusing case, at least if you have a sufficiently big mass, then maybe you would have even more things that would happen. Now why R cross T2? And so maybe here this is some moment for some commercial for some questions that I think are very interesting. So there are two main reasons and one is that there is something that a question about the interaction between geometry and dispersive equation, dispersive flows. And so, well dispersive equation that exists regardless of the domain where they are posed. On the other hand you would imagine that depending on the domain where you consider them, things should change. And so, but the question is what changes and on which geometric information is it dependent? And so the first question was when do you have a global flow? Ah yes, this is one thing that I had forgotten when I mentioned the defocusing case. So what you would expect in the defocusing case is that all solutions should be global whenever you can define a reasonable flow. And if you cannot define a reasonable flow, then presumably all kind of bad things should happen like there should be a solution that just cannot be extend past initial time, et cetera. So in this case we would expect a global flow. But of course this is based on the intuition of what happens in RD. And now if you go from RD to say a compact surface, then a lot of things could possibly go wrong. And especially because the only mechanism for things to get better over time is dispersion. And now how much dispersion do you have on a compact manifold, for example, is a question. Now however, to this question, the answer seems to be a little disappointing in the sense that there seem to be no geometric obstruction to having a global flow for Schwarz data, for smooth Schwarz data, for let's say, well H1 data. And this has been verified in all the cases when we can have a good understanding of the linear flow. And so verifying simple cases in 3D. And this including in the cases when you have a compact domain. So first it was verified, well for the cubic equation I think it's essentially done in the work of Berger and Zetkov. But if you can increase the nonlinearity to Quintik, so and then there was a lot of people that worked on it going to spaces with more dispersion than the Euclidean case, looking at cases inside the domain or outside the domain. But for a, but if you want to go to even the critical case, so if you want to look at the Quintik equation in dimension three, then there was a big advance by Hertha, Tarou and Zetkov that showed essentially this, first for small data, then it was extended for large data. And a lot of people have worked on it. So Strunk has worked on it, Xuecheng Wang has worked on it. We worked on it with Alexionescu. And here I can mention one thing which is that this is essentially open in dimension four. Even though it's not so clear why it should be so. So it was, so the arguments that we know allow us to handle at least one case, the case of the torus of dimension four. But even the next model case where you have, you can write down explicitly at the solution, say the sphere S four. This is really, I don't know if you can even have a local solution there. And then you can, so open on S four. And this seems to be a very hard problem, but then you can try to see whether there could be something in between. So what about, say, S two plus T two, or S two plus S two, all things like that. So maybe this is more tractable, for example. And another thing that I wanted to say about the question of global flow for the dispersive equation is that this is surprising that the geometry does not really influence the fact whether or not you have a global flowing for energy data. However, this is not to say that the local theory should be trivial because if you really try to minimize the regularity, then things start to, interesting things start to happen again. Still, the local well-positiveness in low regularity remains mysterious. And this, even in the simplest, or in the model cases, so if you really want to push it all the way down, so this is true even in T two, and here I should say that now there are people that try to understand that. And for example, I forgot to write them, but Faou, Germain, and Hani. And even in the other model case that you would want the case of the sphere. And so there was a lot of work by Bilgear and Zetkov to understand that case where something happens and a recent work by Takaoka. But okay, if you don't care too much to minimize the regularity, then it seems that you always have a local flow. Then the question becomes, what happens to your solution over time? How does this work? So what about the asymptotic behavior? And here, in fact, you essentially know that the geometry is going to play an important role, and you can verify it in very simple cases, but we just essentially don't know anything, except for the fact that if you're better than the Euclidean space from the point of view of dispersion, then you're better than the, I mean, then you're as good as the Euclidean space from the point of view of the description of the asymptotic behavior of solution. So this is, I would say, where we are trying to understand and in fact, just understand some special cases. And this is why I think this space, our capacity is particularly interesting. Because you know that on the Euclidean space, hardly things are essentially as simple or things are reasonably well understood. And now what happens if you start to take questions of RD? So, and the idea being, of course, that in this case you retain all the nice Fourier analysis, and so you can hope to understand, at least fairly well, the linear solutions. Now on questions of RD, so there are a family of them, and our capacity to place is at a spatial place. So in dimension one you have two, but already when you go to dimension two, you have more, and then you can arrange them in this way, and then you can arrange them in several groups. So first, there is the case where you don't expect anything to become better for large time. Those are the case when the domain is compact, and then it's really a very difficult question to know what happens to solutions asymptotically, and essentially we are at the level of just trying to find some interesting explicit solutions. And in fact, so here you can see the talk by Professor Poczesi. And I should also say one last thing about those references that, so something that is going to be important is the resonance system, and there is again a huge literature on the resonance system, and I didn't write any name there because I believe Professor Poczesi is going to give you more references. Now after that you can order those spaces with respect to the decay of the linear solution, imagining that at least if you have scattering this is how your solution should behave for large time. And so if you do that, then you see that you have essentially two groups. This group, where things are at least as good as R2, to some extent you can say that this is because you have one over T decay for free solutions, but so what happens is that in this case, in some sense the fact that you're on RD cross T, or if you want to go below doesn't seem to change that much. So in this case you have nice asymptotic behavior as in RD, and this is cheating a little bit because we're only considering analytic nonlinearity and you only have so many analytic nonlinearity. So in particular you have scattering for cubic and higher. Now here, well at least you can look at the first member of this family to know what happens and you see that you have scattering for 20 and higher 20 nonlinearity and you should expect modified scattering for cubic, and this is really what I'm going to talk about. Okay, and finally there is one last group in this description, it is this group where if we consider the cubic nonlinear Schrodinger equation, in fact something very special happens that we will see up here several times and it is the fact that in this case you have a completely integrable system. You have a completely integrable system at least this is definitely true on R, definitely true on T. I don't know for sure that it is true on R cross T but to a large extent it behaves in the same way and you have perturbations of that that are completely integrable. Yes, so essentially this is the pictures of questions of RD and then you see that the space that we're considering is really the first guy that doesn't fall into the category of completely integrable system for which you have other kind of tools where you expect something different than scattering to happen. And this is essentially what we're going to see. Okay, so now well we're going to look at several things about the cubic nonlinear Schrodinger there so let me try to give you an outline of what I would like to see in the three presentations. So first I'm going to start with some generalities about the linear equations which if you've really studied the Schrodinger equation are not going to be so new to you but well at least I would like you to have seen them in order to understand the rest of the talk. After that we're going to look at something more specific to this equation and this is always something else that I didn't say about the de-focusing case that in this case you expect solutions to be global and so then the game starts to be more to try to understand what happens asymptotically and people understand this as trying to track down the motion of frequencies of the amplitude of your solution on the spectrum of your, so along the evolution and so because at least if you were able to show that those frequencies they are not going to push their L2 norm to high frequencies those solutions are not going to push their L2 norm to high frequencies then you would automatically have a global solution. So this is what you need to get global solutions and this is going to be one of the theme that is going to guide us and here I want to start with a first, one of the first tools to track this down it's the I method and in this case we can really present it in maybe not in full detail but at least in some detail to see it in action and essentially it's something that says that the first big packet of frequency is not going to move too much for large time and as a result we're going to get a first result which is that, well okay let me write it down and then talk a little bit about it. Here there is something that's over 6s minus five. Anyway so something that should saturate at one and then go to infinity at five over six. So by carefully tracking down the amplitude of your solution in at least in the Fourier spectrum we can really beat the conservation laws and show that even for solutions where we really don't have any global in time control we can extend them to global solutions in our space and the price to pay is that the unknown grows slowly. Now if let's first remark that if I could choose S equal to one all of this would be a triviality because the solution is locally well posed we'll see that in H one half and just conservation of the energy would really give me that for free and I would have on top of that an energy that remains uniformly bounded over time. However here we can push it a little below and hope that by doing that something interesting happens. Yes and we see this I just want to make one last comment that here we'll get some exponent five over six that happens to be what you could do on R three. So even if you had a lot more dispersion unless you start using more of its estimate and a long time strict arts estimate and things like that which you would not have available here and now I should say that if you were to look at the case of R three then you can get almost to the scale invariant case S one half and this is a result of that sum. Another remark that I can say is that with our method the fact that you treat the fully compact case of all that you have this R component is not going to play any role so but let me stick to my space for concreteness and okay. And now one last remark is that you could be you could find it a little uneasy that if S was equal to one then we would have this but with a uniform bound if S is smaller than one we still get something that kind of looks like it but we can only let it grow and now the second result is that this is in fact unavoidable and the second result that I would like to talk about is the fact that in fact whenever so let's fix S bigger than one half there exists a solution a global solution that is going to start small in H S and go to infinity in H S over large time. Now, yeah. S equal one is not a problem. Now, yes, so this is obviously not true because there is one assumption that is missing and that you have to remove the conservation law. However, I thought this was something interesting because well first of all it tells you that this kind of growth you have to you can work a little bit on it because presumably you could grow a lot more slowly than that but some growth is unavoidable if you want something that will control all of the solution even if you want something that will control all of the small solutions. Second, observe that this is true for every S bigger than one half different from one. So really S equal one is something very special which really I would not necessarily have expected. And finally, I would like to also compare it to the result of Coran Tatao or I guess should be Pan Duzan who look at the same equation but in a different space they look at it there and they all think that this would not work and in fact what, so this, the cubic analysis completely integrable there and as a result you at least know that you have a conserved quantity that roughly corresponds to the H K norm for every integer K and if I can rephrase it correctly so you have a countable family of norm but you can really link them with a one parameter family of norm that corresponds to every H S I think S bigger than minus one half but certainly S bigger than zero that such that for every value of S you have a conserved quantity that will control the H S norm. So of course this would not work and this I thought is also interesting because in our equation we have in fact two conservation laws. We have the L2 norm which corresponds to S equals zero we have the energy which corresponds to S equals one that is conserved. Now based on this you might have hoped that there would be at least a curve that would connect those two conservation law such that you would have a uniform control under the solution and yet it doesn't work. So yeah anyway this is probably the main punch line of what I'm going to present but that would be a little sad because I think this result is in fact a consequence of what I would say is a more interesting result about that describes the asymptotic behavior of all small solutions to the cubic NLS in our space. So let me just save the equation and then I have to go back to my outline but there is one thing that I want to say before. So in fact this is a consequence of a theorem and now maybe to say it in a more elegant way let me stick to S bigger than one because in fact this is a much easier case and if you go to a smaller than one then you have some form of conditional scattering and well we'll understand why later. So fix, well no in fact yeah okay. Fix S bigger than one. There exists a norm X such that if initial data is sufficiently small so epsilon is just going to denote some universal small number then there exists a unique global solution C of R X. So we have a global solution that is not so interesting but what the important point being that we can describe its asymptotic behavior, a solution G, some other equation that I'm going to talk about later such that U of T so you can compare your solution with something which is essentially a linear solution because the main dependence on time is linearity here but it's not exactly a linear solution so it's a modified scattering statement because there is a secondary evolution there that you have to put in the envelope to really be able to match the two. So if I may make another abusive connection I think this is a little similar to what Professor Masmoudi showed in a simpler case where you have to first extract the main dynamics, the free flow but then you also have to extract a secondary dynamic before you can start estimating the errors. I don't know if you would agree. And maybe before I tell you what the G satisfies let me tell you that this is really some complete scattering statement in the sense that the converse is also true. So conversely, G solution of the present system with such gently small initial data there exists a solution of NLS such that so U was distinct. So this says that the asymptotic behavior of those two solutions are the same. It's not really a fully satisfactory scattering statement because you assume something about the initial data and you only recover some other, I mean you recover more but you recover convergence in a week or more. So it's not as nice as it would be saved for if you were to talk about usual scattering results. And now there is one last thing that I have to state is what is this resonance system. So it's some evolution on a much slower time and it is this evolution. So here I have to start, wow okay. So and here this is the Fourier transform of my function. My functions are functions on r cross t2 so there are two kind of frequencies, one that are integer valued, so the point in z2 and one that are continuous and so the c is going to be in r and k belongs to z2 and they are really the, so this is the full Fourier transform in hopefully denotations are sufficiently evident. Now what is the system? Well so here the first thing that you can notice is that c only enters as a parameter so it's more like a system of equation one for every c and so the initial data is going to depend on, I mean the only dependence in c is through the initial data. Now if you freeze c then what you get is precisely, well okay maybe not precisely but up to a trivial change the Birkhoff Hamiltonian system that Professor Procesy talked about that you see up here and so as a result this shows that every interesting behavior that you can observe in the, for this Birkhoff flow, well for this resonant flow, let me call it this way, you're going to be able to observe it as something behavior of a solution to the nls and now this is interesting because it turns out that this has a lot of interesting solutions. For example there are solutions that oscillates infinitely many times, there are mass between two modes or even more than two modes and so for example this would defeat any, this would say that this is the solution doesn't by any stretch become too simple because infinitely in time it's going to oscillate all its energy from one mode from one Fourier mode to another Fourier mode. But well maybe something that is essentially a consequence of another result by Kolander-Kilts-Tafilani-Takaoka and Tao is that for this system you can produce solutions whose normal flow to infinity, you have to be just a little careful if you want to do it in this context but this is where those special cases come from because you can observe them for the resonant system therefore you can observe them for the Schrodinger equation. Yeah and one last comment about this, I thought this was interesting also because usually you study the resonant system because it tells you something about solution on the torus but on the torus it is a little unfortunate that it only tells you the description of your solution for long but finite time and as a result it was never clear to me what results, what solutions you get, they tell you that some solution to Schrodinger behave in an interesting way for a long but finite time but I mean does the solution settle afterwards? And at least it says that maybe not on T2 but on Accra-T2 then you will observe this phenomenon for all time. All right and yeah so now if I finish the outline so what I want to do three is to present a proof of modified scattering which is, do I still have it? Yeah so you can probably think that the description of NLS becomes easier and easier as you go up in these packets and so I'm going to hopefully prove in full details the modified scattering here and then tell you in the last part how to modify it to get the modified scattering there we will see that to guess this result is not so difficult to really prove it fully rigorously becomes surprisingly difficult on Accra-T2 especially if you want to go below S equal one. So all right so this should be roughly the plan of this presentation. Okay so maybe now is a good time for questions about the general idea. After that we are going to jump into some basic facts but more technical things. Yeah. So that is an example you observe oscillating oscillations so you have a sequence of time so it goes to infinity. That's right. So did you, did it falls down? Probably. So what I would imagine is that you can make it also go remain bounded along a sequence. We'll see what this is. We'll see how to get that. But I mean I haven't really done all the computations so that's at least what I would expect. All right so. In general what is the strategy to prove that the resonance system actually approximates the solution of the endless? Oh so it depends a lot on your space. So presumably it is not even true on the torus and what we're going to see is that if you have a little bit of dispersion it is true and what will essentially follow from what we're going to see is that if you have too much of dispersion then it's only the dynamic for finite time of the resonance system that is relevant and the dynamic for finite time is really nothing. So. This is partly what we're going to try to see at least in this case. All right and maybe still, so let's see where am I. So yeah maybe I can say that a few names. So the people that have worked the most I think in this picture are Zaheer Hany and myself and Nikolai Zetkov and Nikolai Vichylya in various combination. The people that have worked on modified scattering I think the first main result starting with Daphne and Joe then there was something by Ozawa that I think is really isolated the main correction or the PD started the PD proofs. There was a complete proof by Hayashin Namkin what I'm going to present is something strongly inspired by work of Katow and Pusateri. And then so Professor Zotaru mentioned that they came up with a new proof. Well I'll let you compare the two. What I'm going to present is probably something which is less elegant than either of those two proofs but something that I personally like it a little better because it's really doing, trying to find a dent in the armor and then turning the crank to chip away a little bit at the difficulty. And now I think the next case is going to be when we prove L4 strict-out estimates. Okay, oh and I forgot one thing which is this is really just my personal though fancy about this equation but I tell you that we look at this space and why because this is at least a case when we hope to understand the asymptotic behavior of the solution and this is something non-trivial at least it has some closed geodesics it has some interesting topology, et cetera. Now I see at least one other reason why you would want to look at the analysis on those spaces and this is from my understanding of what interests some applied people and if I understand well the idea is not so much that you care too much about periodic solutions. What you really want is to model a situation when the density of energy is roughly one. So in other words you can say the energy of your solution increases let's say in this case like the radius squared. Now in full generality it's how to make sense of the non-linear equation or even maybe the linear equation in this sense is something fairly difficult and so then you come up with some functional setting to do that and so if you assume periodicity in both directions then you can restrict to one cell and things are good. Now this would correspond in this picture to cases when the energy increases linearly with the radius and the interesting fact is that in this case well you can ask that still you can get a nice theory let's say even in 2D when so you model this by functions which are periodic in one direction and let's say you take your initial data in H2 of this so to have very smooth data but you could also imagine taking your initial data periodic in the other direction and then you could probably also flow in initial data which are essentially compactly supported anyway. Now on this space or for an initial data that is in the sum of these spaces there is no real problem to define the linear flow just because you have a superposition principle. Now can you really define a non-linear flow? And I mean maybe because this is an algebra it has all kind of nice properties you multiply two guys then you are likely to fall in the nicer one so but so this is something which is a question that I think deserves to be studied more can you find good functional setting to get the Schrodinger equation on R2 but not either with initial data compactly supported or exactly periodic but say almost periodic et cetera. So if you have nothing to do when you take the LRB then this is something that I think would be well worth trying to think a bit about. Oh and well, okay, let me use this ball. Okay so now actually how am I doing on time? About what, 10 more minutes? I don't know, one hour? Okay so at least we can do and maybe even get to a strict estimate. So all right so what is the basic things that are going to be important for us is that the cubic NLS equation is a Hamiltonian equation with Hamiltonian that you have seen now and in general there is only one more conservation so that I know of and this is the mass. Now as we have seen on RT then the situation is very special you have infinity many but in full generality those are the only one, on RD you start to have more you have the momentum, the angular momentum if you can make sense of it. But I'll mostly focus on this and the fact that you start with these things which are conserved for any reasonable solution which are smooth enough means that it's often customary to study the equation at least to begin with in H1 which I will call the energy space from now on because this is what the energy is defined. What else should I say? So the equation on RD and this is a little important because then the momentum for example is conserved on the torus and for example if you restrict to only four modes this allows you to get a completely integrable system in the recent system so there are some use to those and we'll see them later. But what I wanted to say is that there are two symmetries which will be of importance to us. One is the scaling and that says that for any if you start with a solution for any lambda this would give you a new solution and this leaves HSC invariant for SC which is d over two minus one and the only thing that is going to be relevant for us is d equal three so one half and essentially the idea is that this allows you to control what happens if you start to concentrate your solution at a point and at this stage the geometry is gone and so if you have impositiveness below this space which you essentially have all the time then that means that the geometry is not going to help you it's not going to do anything you will not really be able to define a nice flow below this regularity. So this is going to come here as an enemy oh yes and there are two things that I have forgotten once is that we can get all the way to the scaling and presumably if you were to take as strictly smaller than one half you would expect this to still be true but in fact you would expect a much worse behavior you would expect that there is an initial data which is small which you just cannot continue as a solution in this space so up to maybe the equality for one half this really tells you that things are bad and the other comment that I wanted to do was that when we're going to look at the I method this is going to be a setting where you can start seeing what Professor Vishigia talked about in his lecture on trying to control high order norms by doing the modified energy. Okay and so the second symmetry is the Galilean invariance and it says that whenever you start with a vector in RD then you can create a new solution whose Fourier transform, whose Fourier support is translated by this vector and this solution is given explicitly and it's given by that and this gives you a first way to see the somehow finite speed of propagation in the sense that if your solution was added Fourier transform very concentrated around zero then when you do that it's Fourier transform is going to be concentrated close to the frequency V and then the solution whatever it was doing is going to translate I mean the main effect of the dynamic is to translate at this fast speed. Now the main thing to understand the modified scattering statement is the Fraunhofer formula which is something that is essentially extracting the first term in the stationary phase formula but because we have a nice equation we can do it explicitly and so I think this is instructive to do once. So this is 1.2 in the Fraunhofer formula and so in RD just by looking at the Fourier transform of a Gaussian you can find an explicit propagator for the Schrodinger flow and it is given by this and so as a result you can explicitly solve the linear solution with initial data U0 and it's going to be given by a convolution with this and so you get 1 over 4 pi I t to the V over 2 integral U to the minus I x minus y squared over 2t unit of y in t1 It should be x over 40 in the exponent Oh, really? Yeah, yeah Okay, my apologies So then I'm probably going to have a one half factor it is not going to be quite right but hopefully we can track it down together. So once you see that then what you want to do is to expand the square and so if we do that well the first thing that we're going to get is only in terms of x so it doesn't depend on the integration so you can pull it out and you get I x squared over 40 and then you have the rest and so start with e to the minus no, e to the I x over 2 plus y e to the minus I y squared over 40 in 0 plus y dy Alright, so now the next thing that you can observe is that if your initial data was compactly supported then y is never going to be something too big and so this is really going to be essentially one Now if this were one you just get the Fourier transform So what you can do is to separate it out and you would get u hat and I guess in this case and so there is a 20 of minus x over 20 and then a plus integral e to the I x dot y e to the minus I x over 20 what is it? y squared over 40 minus 1 unit of y dy and now the front of her formula is saying that u is essentially equal to that and why is that? Well you can see it very simply because just by the mean value theorem this is essentially like f of y squared over t minus f of 0 and now for general function y could be large but let's assume that you started with compact support then this is about 1 and you have gained a factor 1 over t and in particular it's going to be something much smaller and so what we're going to use several times the front of her formula is making this approximation that you or which is a free evolution of you to clarify things you are making t going to 0 no t is very large yeah, I assume too t so is t just conformable? probably right so you can also expand a free propagator as a Fourier transform and some modulation and some dilation so and for us what we're going to say is that letting the solution evolve through the linear flow for time t is roughly the same as applying this now this is interesting in that it already tells you what how dispersive equations behave so the first thing that you can observe is that the main term whenever you want to take derivatives is going to come from when the derivative hits this phase here it's not going to come from there this is really what you absolutely have to get right and yeah, so that is well, so the main term comes from when the derivative hits there otherwise you're better by some 1 over t factor the second thing to observe so it's going to be important so to isolate the oscillating phase in front of your evolution the second thing that you can observe is that our solution is this other way to see the finite speed of propagation in the sense that if u0 of x is supported close to some frequency x0 over h so I'll use this to denote a smooth function essentially 1 on a ball of radius 1 and then 0 outside of a ball of radius 2 and then h is a small parameter and so if your initial data is really supported around c0 then your solution is only going to be supported when x over t is more or less c0 and so all of this is for large time so in this case x would be minus 2t c0 and now finally the last term to be understood in this formula is this thing that gives you the decay but in fact this one you can essentially guess for free because if you know that the L2 norm has to remain invariant when you're transforming the coordinate this way then this is essentially the Jacobian that comes out so to make the L2 norm invariant ok so this is things that we can get easily about linear solutions and they will be more important for us when we look at modified scattering because then at least to guess what the modification is it's useful to first do it by replacing the free evolution by the front of our formula you get a simple a simple formula and then all of the tasks is to make that rigorous and another way to say this is that in general you would have an oscillating integral you extract the first term and then it's all a matter of trying to say that the next term is not so relevant it's almost over I will give you five more minutes tomorrow because that's great I'll just say so tomorrow we're going to prove L4 strict arts estimate and start to see how you can use them to get a local solution and then to push that into the i method thank you