 So first of all I would like to thank the organizers and also I envy all of you that will stay for two weeks. Unfortunately I had to stay for a shorter time and since it's a summer school instead of presenting the most recent complicated calculation I have been doing I decide to give a little bit an overview of what we have been doing for a certain dispersive equation that you can see you can look at in terms of or as an infinite dimension Hamiltonian system. So I will be speaking 45 minutes this talk was presented for 50 so I will just go faster don't worry about it I will you know I will skip you things that are not really as important as the rest. So this is a little bit to the program that all the what I will be presenting today so I start with a very simple introduction you already have seen some of the most important equations in this summer school so the wave equation with Carlos the Navier stoke with my smoothie so today we at this now you will see the Schrodinger equation and then of course we have been working and we have been talking a lot about dispersive equation and with them the Strykart's estimates then I will move to well here you see weak turbulence but there are many different definitions of weak turbulence and this is a very very weak weak turbulence just wanted to tell you a little bit about that and then I'm going to look at the Schrodinger equation as an infinite dimension Hamiltonian system and as such people have been trying to prove for this infinite dimension Hamiltonian system a lot of the really cool and difficult problems results that come up for a finite dimension Hamiltonian system and one of the problems I will look at is the Gibbs measure its definition and its use and then the Gibbs measure will bring up a probabilistic approach to the questions of well postness local and global for the dispersive equation and with that randomization of initial data and then if I have time at the end I will mention another problem which is linked with with the infinite with the dispersive equation as the infinite dimension Hamiltonian system that's no squeezing theorem and there are recent results on that and I would like to conclude with some open problems okay so as I promised the star of this talk is the Schrodinger equation so basically this is what you should look at this is the equation i is of course the complex number and this is the nonlinearity my nonlinearity is p p is going to be greater than one and lambda is going to be either plus or minus one and that depends whether you're looking at focusing on the focusing problem and Carlos already mentioned the difference that you can see when you look at the Hamiltonian or the energy if you want what I wanted to you to pay attention at is the fact that most of my the questions that I will address are in the periodic case in other words my x is in the torus of dimension n now there is an important point that I want to make which is linked with the with this torus and I will stress more later we talk about toride that are rational or irrational and just let me say from the beginning I will go back to this later the rational torus is a torus which has periods such that when you take a ratio between them you always obtain a rational number and they are irrational if for example think about one of the periods to be an irrational number and there are other one rational that would be the case but I will anyway I'll go back to this in a second and this equation and this initial value problem it's relevant in many situations and they come obviously from physics I will give you a very short in a short moment a little bit of one idea but the way you look at them or the kind of problems you solve for them involve quite a lot of different subjects in mathematics so harmonic analysis obviously we like these are waves so we chop them up in small pieces so we recombine them somehow so harmonic analysis clearly if you write analysis we do a lot of things with Fourier transform analytic grammar theory when I will go to the I give you a little bit of the statement at least of the street cards estimates and I will tell you something place where you use that probability I'll be talking about measures and dynamical systems that's for example when you look at the question of weak turbulence and more now of course all you want to see here it's you feel like it's going to be there are easy questions and easy answers but that's just the tip of the iceberg and I just wanted to present the easy version of this deep problem so that you have at least an idea so like I said this problem comes from physics and one of the really interesting questions which I'm not going to address at all but is obtaining this guy as a limit is there as the number of particles go to infinity or the temperature goes to the absolute zero of a phenomenon called Bose Einstein condensate and there is a lot of work done and there are many many developments that happened the last 10 years Natasha Pavlovich who is here yeah she has been working on this limit I worked a little bit on this limit and it's interesting and it's not at all trivial to take the limit from the many particles that are governed by certain amiltonians and obtain as a limit in a certain sense these equations and I like a lot this picture that appeared in science in 1995 because actually in 1995 a couple of teams working on the on the phenomenological aspect to the Bose Einstein condensate so with experiments got a Nobel Prize so really what happens is all these guys think of them particles if you are let's say in higher temperature they are all independently doing their stuff but as you lower the temperature they kind of lose their own identity that they reassemble as a wave and basically this wave is the solution of the Schrodinger equation now when you model a certain phenomenon via an initial value problem the first thing you want to do is at least check that that system the initial value problem has a solution that's the list of your worries now you of course want the uniqueness as well when possible you want stability and you've seen in the Bose lectures this morning that these are fundamental questions and we have in some cases I mean the cases that we can treat pretty good understanding of what happens in small time or for small data but the long time dynamics is a very complicated problem and I think you got an idea from this morning already so the questions of stability and long time in particular long time stability and so on it's very hard but let's first start with the small problems and like that of existence the uniqueness of solutions and let me go back for a second to if you consider this problem here one way of solving it is by reducing it to an integral equation via the Duhamel principle and you saw already this Carlos who used the same word is that was for the wave this is for the Schrodinger but this is the linear solution of the problem and then here you have the non-homogeneous piece so it's again the operator acting now on the non-linearity that age that Carlos mentioned so as a first step when you want to find the solution for this integral equation you see clearly that if you define an operator like the right hand side the fixed point is a solution and in order to understand where in which space you want to do the fixed point then you start thinking well I'm thinking about this right hand side as a perturbation of the linear problem so let me prove as many estimates as possible the linear problem that will give me the intuition of what should be the norms of my space and then cook up that space with those as via those estimates and prove that this are a piece here is a perturbation and you usually can do that for you know small data lower or subcritical in a certain sense non-linearity small time and so on so that's the first step so because I say you want to think of this problem as the perturbation of the linear part let's look at the linear part so we look at the linear part and the simple way of solving this linear problem is just by taking Fourier transform in other words you think of v as the remember we are in the periodic case here and I'm giving this very simple minded presentation by thinking I'm in one dimension if you are in higher dimension slightly more complicated I'll get to that in a second so you just look at it by thinking that v is going to be periodic function it's going to be written via certain coefficient with respect to your l2 basis and what are these coefficients well these coefficients will satisfy this all the e for fixed k so for each fixed k your solution ak t is this guy here where u hat zero is exactly the Fourier coefficient k of the initial data and then you can reconstruct via the anti Fourier transform that's what you have so then you say well I actually have a completely explicit form of the linear problem so it's going to be very easy to find a bunch of estimates for it because it's completely explicit well that's not the case especially in higher dimension so let me tell you what it looks like when you are in higher dimension just by repeating the same argument you will end up with this expression here and here in the one-dimensional case you have the k square in the n-dimensional case you're going to get a gamma k which is this object which is positive the ki is a k1 up to kn at the component of my vectors in n-dimension and these ci's are going to be related to the periods okay or on whatever direction you go and now this is the point of having a torus which is rational or irrational if all these ci's are rational numbers then the torus is rational if there is one which is at least one which is irrational then it's an irrational torus unless you can divide by it but let's just pretend that we can they don't have common factors and now let's go to the estimates that you want to go to want to do for these solutions and these are the street cards estimates and let's see how the rationality or rationality of this torus plays a role now excuse me from the point of view of physics there shouldn't be any difference okay at least that's what we believe but from the point of view of doing mathematics with it there is a difference so these are the street cards estimates that Burgen listed for us this goes back to early 90s and he proved some of them using in fact some concept from analytic number theory and why just let me make this point why analytic number theory not just analysis well in the case in which instead of the torus of dimension n you are n the solution of the linear problem is an oscillatory integral and if you do integration by parts you can get nice decays estimates or you can use theorems that go back to restriction of the Fourier transform of a certain hyper surface that have a non-zero Gaussian curvature like the parabola which is attached to the Schrodinger equation so you can do a lot of things but in the case of periodic situations the oscillatory integrals now become oscillatory sums and they are much more complicated to address and they are treated in an analytic number theory so that's why it uses those methods now I don't want you to look at all these p's and q's and things like that what I really wanted to pay attention is the following so here I'm doing this I'm taking the linear evolution of a certain function and that function has Fourier transform which localize in a ball center and zero radius n so in Fourier transform you are in this ball located anywhere x0 it's just n0 sorry it's anywhere in my space of the frequency and the radius is n and then I'm estimating this lqt lqx norms finite in time you do not expect a global in time street cards in fact and then what you see here on the right hand side is the l2 norm why the l2 norm well the l2 norm it's very important for the Schrodinger is actually invariant it's one of the invariant norms for quantities for the Schrodinger equation so you want to measure via that and what you see here in red is how much you might lose in doing your estimate so let me actually give you a specific example so that you can fix the idea a little bit so I'm considering now the l4 estimate which is the one for t2 torus of dimension 2 and in the rational case which was considered by Bourguin early 90s think of c1 and c2 for example being one or natural number and if you write down what this l4 norm is you can write it as the l2 norm of sq0 time sqt0 you use planche rel you do the convolutions and at the end what he has to estimate is the number of lattice points that are on circles now think of c1 and c2 to be one so will be the circle radius r r and there is a theorem in analytic number theory that tells you how many of those numbers you have in terms of how large the circle is and this is what it does now if you replace the circle with an ellipse then the situation is not at all as clear as this and the reason why you cannot conclude this striccarts estimate in the case in which you have an ellipse which corresponds to the case in which you have different period particular rational well then you know you don't have such an estimate so you have much weaker results and that's where the situation stand standard for several years and there was other contribution that came in order to find out if you really have the same striccarts estimate for this rational situations but there were partial results now a couple years ago instead a paper appeared by burgen and demeter on the proof of the l2 decoupling conjecture i'm not going to write here what the decoupling conjecture is but as a consequence of that they show that actually the striccarts estimates in any dimension and basically in full generality except for this extra little bit or loss of derivative which i put here in blue are true now i just wanted to point it out that the proof of the l2 decoupling conjecture from which this is derived as a corollary it's purely from given from a harmonic analysis point of view in fact that depends beside the traditional harmonic analysis in some new addition to it which is the was there before too but in particular in particular it's very much used in this proof um incident geometry theory and larry goose had um a work that was related to the kakei conjecture but anyway that he had done by using this incident geometry and combined all of that um burgen and demeter were able to prove the decoupling conjecture as a consequence of that this and that proof sees no rationality irrationality everything is the same um so it's a completely different proof that the original one that the burgen gave and i wanted to say that um on this kind of approach there is all the work of wolf and burgen goose and cario segar and so on it's a very interesting branch of mathematics okay so now let me move on um so since this result of uh burgen and demeter we now know that we can address you know in um the the results that were proven before via the street car system only for the rational in terms of well posings now are also true for the rational torres now let me remind you a couple of conservation laws for the equation i already mentioned the altunorm here it is that's corresponded to the mass but it is also not a piece of it which is the hamiltonian and which for the schrodinger is the gradient square and then here is the nonlinearity and the nonlinear part and like i said if we are in the defocus in case then this lambda is one and so this quantity the hamiltonian is always positive we thought we are not in the situation that carlos needed well we are in the situation that carlos dismissed this morning at the beginning because we are going to be thinking about this in the periodic case this is still um still a lot of open question the def the focusing case in the periodic case is really hard okay so um let's put ourselves for now in the defocus case and let me uh for you like i said i'm going to present the simplest possible problems in this context so that you can understand what's going on so this is a theorem that goes back several years of burgen and you take so we are in dimension two so that's where the l4 norm of the strict arts is important we have a cubic nonlinearity this problem is l2 critical in the sense that if you do rescaling like carlos was doing today the l2 norm is left invariant and um he burgen proved using the strict arts estimates that this problem means global sorry it's locally well posed in hs for s strictly greater than zero but by using the hamiltonian it's also globally well posed for data you zero in h1 there is also some partial result of global like well poseness below strict a little bit below h1 by using the um i method but i will not talk about that but the point is that the point that i want to make now is that if you take initial data that are smooth enough and you are in the defocus in case you have a global solution okay so now once you have a global solution we also call a global flow then there are many questions you want to ask for example how does it look like this global solution and one of the uh question that i wanted to address here is the question of weak turbulence now for me um weak turbulence in this context only means that if you start with an initial data which is localized in frequency near zero let's say um do you know whether or not the um this this um localized solution localized initial data as t goes to infinity moves in high frequency so this we call forward cascade so you the this is something that uh is not intuitive uh a priori because the conservation of the l to norm and the conservation of the Hamiltonian it's a um constrained the solution a certain way um but whether you can have these frequencies or the bulk of the solution move from living near zero which was at time zero to live farther away it's a complicated question and like i say goes under the name of forward cascade and let me immediately say that you cannot have any of this forward cascade if you have scattering why because scattering means that as t goes to infinity the solution becomes basically linear and any linear solution has a um an hs norm which is here which remains bounded and if the hs norm remains bounded then the bulk of the solution in terms of frequency cannot be too far away in terms okay because otherwise it would be picked up from this norm since it has this weight here so if the u hat tk gets bigger then when you hit it with the weight k as t goes to infinity that would become bigger as well but if you are scattering that's basically linear any linear solution has the hs norm bounded hence that cannot happen another um possibly enemy for a weak um turbulence in this sense is a complete integrability like if you have a bunch of conservation laws and you you can recombine them in some way and this conservation law has a piece of it which is related to the hs norm then you can you are able to sort of control the whole hs norm as s becomes arbitrary positive number so scattering a complete integrability might um you know might be your enemies for the weak turbulence um so you have to worry about that but in the general case that i'm considering which is this periodic case we do not expect scattering and we put ourselves in the non non-complete integrability situation so for example dimension two cubic is not integrable so let me give you a couple of theorems this one are relatively old theorems by now in a sense but just as an example of what has been approved in the context of this weak turbulence in the way i mentioned so the first so there are two things that you would like to say you would like to say that this hs norm although they might grow they don't grow too much in terms of time so these are bound from above and then you also would like to say that you can exhibit at least a solution for which there is a growth and somehow this is bound from below so let me explain a little bit what this theorem says so the first one says take the solution of uh Morgan the one that i really told you exist globally in the defaulting case then since that the global result is obtained via an iteration the trivial bound is an exponential bound in t and you wanted to do better than that and in fact you can and you can prove that the bound cannot be stronger than at least t to the s so of course if s is one then you have a uniform bound that comes from the Hamiltonian but if s is greater than one this is what we know that it's polynomial we expect much better bound than that maybe log of t but that's what we have so far so this tells you that it cannot grow too much now the question is can you actually have a growth so one of the theorems that were proved a few years ago this is approved with my collaborator so take t2 which is rational in fact i'm not sure that this proof holds for the rational case i think one should check that so just to be safe let's think of this to be rational there is a construction that happens in terms of the set of frequencies so that needs to be redone somehow anyway let's think of it to be a rational torus let's take s which is greater than one and k which is a large positive quantity and sigma which is a very small positive quantity then we can construct a solution for our cubic two-dimensional periodic problem defocusing let's call a uxt such that at time zero was of this size for the hs norm so less than sigma but if you wait long enough it's going to be as big as this quantity k now this theorem is not enough in order to show that you actually have a growth in time no even logarithm growth um so this is much less than what we would like to have and after we prove this theorem there has been a lot of work related to this and much more improved to that so strictly related to this is the work of guardia collosion then there is for the quintic n l s there is a work of house approaching and there's some more recent work of guardia house purchasing and for a different kind of system there is a beautiful work of patrick gerard grullier and gerard and collaborators so but that's just to give you an idea of where we are there are lots of open questions here okay so let me move on um and think of now of um my schridinger equation as an infinite dimension Hamiltonian system i will show you uh sorry this is a maybe maybe you know ask it this because i think that i will not be able here we go so um let's now think of our schridinger equation as a periodic schridinger as an infinite dimension Hamiltonian system i will tell you in a moment how you do that but before going to the infinite dimension let's recall a few things that come for a finite dimension okay so this is the way we write uh or you see written um a finite dimension Hamiltonian system so we have two variable um q and p um there is the Hamiltonian h which depends on q and p and there is um the equation q dot equals partial derivative of h with respect to p and p dot equals minus partial derivative of h with respect to q um and then if you look at the derivative with respect to t of h when h is evaluated on the solution of the system you see with a simple calculation that actually is zero so in particular h itself is a conserved quantity now um i say we are looking finite dimension now so let's call it y to be q one to qk and then p one to pk and this is just as a vector so that's why i put the transport here and transpose and this is r to 2k and 2k equals z my dimension you can also rewrite in this um maybe more compact way which looks a lot more like a schridinger equation and where j is zero minus one minus one and zero okay so that's my finite dimension so what can we say about this system um actually we can say very easily a bunch of things so the first one is that via new wheels theorem that you see recalled here so let's recall it for a moment again so let a vector field f from rd to rd be divergence free that's very important then the map phi t which is really the flow map relative to this od now here i'm talking about the of course leaves the volume of the leg bag measure if you like the finite dimension leg bag measure or the volume of one invariant okay so what does it mean that leaves that invariant it says that if i take a set a and then i evolve it through phi of t then the measure of a and the measure of the evolution phi of t of a at any t is the same okay so now for a Hamiltonian system new wheels theorem is applicable because in particular you have that the this right hand side here for the Hamiltonian system is divergence free hence as a consequence any Hamiltonian flow leaves the leg bag measure invariant or the volume measure invariant that's a trivial it's very simple so now slowly let me walk to you through the Gibbs measure infinite dimension is also trivial to prove that the Gibbs measure is invariant so first of all what is the Gibbs measure well the Gibbs measure is the volume measure now this makes sense just like the volume of the parallel part if you like then this is exponential beta is just a positive quantity don't worry too much about that this is the Hamiltonian and this constant here is just to normalize so that the measure of mu of your space is one okay so now if we are looking at again the Hamiltonian flow then my claim is that that flow leaves also this mu invariant this is called the Gibbs measure why does it leave it invariant well it's pretty trivial this part it's finite me this we are talking about finite dimension this d is finite its invariant is the volume measure and that's invariant through the new wheels theorem this part is also invariant because h we just know that it's a conserved quantity so as long as I stay along the Hamiltonian flow that h stays invariant so all together is invariant okay so this is very simple but what happens when I try to do something like this in infinite dimension Hamiltonian system so let me give you an example of one infinite dimension Hamiltonian system that comes from a Schrodinger now in in this context I wanted to give you a very simple example in fact I'm going to give you the one in one dimension and I will tell you a little bit more when you are in higher dimension what happens so we start with let's say this equation I'm looking here at the quintic nonlinearity I could have taken the cubic as well but the result that I wanted to mention which is which is relevant and important is a organ it makes sense also in the quintic case so I want to present that you have an initial data u0x and x now is in t and here t is just a circle so there's no problem about ration in ration there is no meaning of that there is just one circle one period but when you go to higher dimension that becomes an issue I just wanted to announce this the Hamiltonian is like I mentioned before is the gradient and then there is one third of this is the I don't know if this this constant is correct but it's a positive constant I'm looking at the one six thank you yeah I thought that was one six okay the point is there's a positive here so this is a positive Hamiltonian all right so then this problem here can be written as an infinite dimension Hamiltonian system in the following sense take you and look at it in terms of real coefficients it's going to be a real part and an imaginary part so ak is my real part bk is an imaginary part again I rewrite this problem by exactly like the infinite like the Hamiltonian equation that I gave you before except that now the k are infinitely many k belongs to z so sorry this I wrote the z n by my case in this particular case n is equal one but in general you can write in any dimension of course so it's an infinite dimension of this vector ak bk is an infinite dimension vector so you have to be careful for example to understand what you mean with the Gibbs measure so remember before in the finite dimension case the Gibbs measure will see the exponential minus the Hamiltonian and then there was the volume part in there so you would like to make sense of something like this well exponential minus beta the Hamiltonian this you can imagine that makes sense because this is just a conserved quantity but what doesn't make any sense whatsoever is this guy here which is an infinite volume okay so really as it is we cannot understand what it means and that's why I put it in quotation marks but label its rows and spear proved that you can in fact make sense of this measure as long as you are you consider the measure in a certain space and in fact you can make sense of this thinking of you to be a measure in a let's say hs tau so the so-called space but s has to be strictly less than a half if you are a half bigger you cannot this doesn't make any sense so the regularity of the space where you work if you want to define the Gibbs measure is very low and it gets worse and worse as you go higher in dimension so that's what I wanted to say okay so how do you make sense how did label its rows and spear and zitkov actually to make sense of this measure well let's step back a little bit and let's look again at the Hamiltonian so here is the Hamiltonian you can also add the L2 norm I said that's conserved as well so let's consider this whole piece here which is invariant this was the original Hamiltonian positive quantity and then I also added the L2 norm which was invariant and now let's think of it as two different pieces the lean the part that comes from the linear part of the problem in a sense so is the gradient and the L2 norm right here so that's the what I call the linear part of the Hamiltonian and then there is the other part I see Frank I put six there now so kind of new but anyway so this is the non-linear part right there is a minus here and this part well we didn't really do much we didn't make much more sense of this stuff because still there is the volume piece which is an infinite dimension so we have to take care of that but we can recognize something that we understand this part here this is the Gaussian measure and there is a lot of literature on that if you're interested there are books and books written on it and the Gaussian measure makes sense so what I put still in quotation mark here makes sense I will in the next slides I'll tell you how you do that so this part makes sense but you still have to take care of this and what you have to do is to show that this piece is in L1 with respect to the Gaussian measure and so you can think of it as a random nicodem derivative so in a very short one-slice argument I told you how you can make sense of this Gibbs measure in practice it requires a lot of calculations and I'm not going to go into that so since I mentioned the Gaussian measure let me just briefly tell you how you deal with it so the first thing you do is as a a lot of things in this infinite dimension Hamiltonian system you just truncate you truncate in finite main in boxes of frequencies of size n you use a lot of finite dimension stuff that happens there that you know how it works and then you have to take the limit and taking the limit is usually the hard part of course but once you truncate so you take frequencies are less than n then this makes sense this is the volumes the volume measure in a finite dimension setting now in r and r I don't know how many dimension anyway finite many dimension here this makes sense so this finite dimension row n makes sense that you can also prove that if you well let me just leave it at that I want to go back to the infinite system and another thing that you can say is that well there is a measure so there is probability underlining background to this measure and the way you can think about that is the following so you can take the map that goes from little omega to this quantity so now little omega belongs in a probability space big omega p p is the probability and induce map from the big omega to the functions that i'm considering is made like this you go from little omega to g k omega of square root of one plus k square this is a Fourier coefficients coefficient in fact i want to give the name of v hat k omega and this guy here and you can reconstruct it your function via the inverse Fourier transform in the following way so phi omega x is going to be this Fourier coefficient e to the kx and in other words it's also written like that okay so these are very special functions because as you see the Fourier coefficient is very special in fact it's made by one over square root of this guy times g k omega g k omega is a random variable you have to put assumptions what kind of random variable but the point is that you require them to be independent and the independence of these random variables are going to help you quite a lot when you have to do multilinear estimates and again i'm not going to go into details but the way you should think of in these problems is that your initial datas are going to be of this type so they are generic in a sense that this omega varies in here but otherwise they're pretty specific data okay okay by using that this particular data Burgen proved that this problem this is the quintic nls one dimension is globally well posed almost surely in a space hs where s is has to be below one half because you're going to use the measure and not just that the Gibbs measure which i told you briefly how you define it before is invariant with respect to the whole flow of given by this problem now a couple remark that i want to make is that when you you can do this also in the focusing case which is i will say even more interesting but in that case you have to impose a restriction on the l2 norm in order to make sense of the measure itself and that restriction is that l2 has to be smaller than a certain constant not an epsilon but some definite constant in there and then another remark that i want to make is the following the important contribution of Burgen in this direction which you can already see in one dimension but really it's a much more much stronger in a two-dimensional case is that well you can prove well posings global well posings at levels in which you do not have a conservation law so for example at one half minus epsilon so the the case in which that i show you at the beginning like the global well posing that i show you before Burgen was using the Hamiltonian the conservation of the Hamiltonian here instead what you use is the invariance of the measure and even more impressive is the fact that you can prove this like i say even the focusing case in which you basically show that what happens in the focusing or the focus is according once you sift through with this measure is the same so the the result is exactly just a one half minus for those particular initial data that i mentioned before and it's in order to prove some kind of propagation irregularities not as trivial as when you prove well posings in a deterministic way so you have to do a little bit of work and just i'm gonna close here let's see because i see that okay never mind let's go that i don't have time to do it but this whole probabilistic approach on dispersive equation i didn't write here the names all the people have been working on that but has opened up quite a lot of different directions in particular situations in which there are a contra example for deterministic multi-linear estimates that we cannot be that tells us that we cannot really try to hope for deterministic results or well poseness and also a lot of different approach for the problem itself in terms of showing questions of that related more to probability so let me just conclude with here we go some open problems so understanding the street cards estimates in a more direct way the fact that at the moment for a rational to write they are based on the proof of the decoupling conjecture it's a very restrictive way of thinking about it a more direct proof will be a really welcome then improve theorem or weak turbulence here i just mentioned one recent result i found very interesting before honey and german this is the understanding at godic structure associated with the problem which is linked to this Gibbs measure that i mentioned then finding theorems that allows you to prove go from local to global without having to use this gives measure too much because they are very rough when you are in higher dimension then use probabilistic approach to study property of discrete version of nls so we have been looking at continuous versions here discrete like with finite definitions to find a difference and so on this there is interesting work by chatter jker patrick and chatter j and then i didn't mention i didn't have time to talk about the um no squeezing theorem but that's another direction in which there are um some recent there is some recent work um and if you want i can talk to you more about that thank you do we have questions you're set sigma for the for the well possible like do you have any idea of what it is like well it's yeah it's so the data i already wrote for you um yeah the data the initial data looks like this so it's a g and omega so let me sum it so in this particular case of the of the the problem that i had there and this is my phi omega x so that's how they look like where these are set again sigma is only zoos and yep it sets like that elements like this and then what changes is the omega and the omega is going to be part of let me call the omega hilda which and then the point is that the probability this measure is basically is the same as the one for the invariant measure of sets like this let me call it a so then that's how they look like so they are very specific and i didn't mention i mean this is when you use invariance measure but there are other results in which you randomize the initial data and many more different kind of data than that but then you cannot use the measure you had to use other things to go from local to global thank you