 Okay, thank you so much. So as the speakers before me, I would like to thank the organizers for putting together such a wonderful conference. So the goal of my talk today is to discuss a novel approach of how to extend frequencies of integrable PDs to spaces of functions of low regularity or spaces of distributions. But before I state the main results of, on this topic we have obtained, I would like to give an application to the well-posedness of MKDV2 and the modified KDV equation. And let me mention right away that all of what I talk about today is joint work with Jan Malnoll. Okay, so let me begin with the KDV2 equation. So we consider the KDV2 equation on the circles, period one. The KDV2 equation is an evolution equation, dispersive evolution equation of fifth order. It comes up in the long wave approximation of water wave equations. Of course, the coefficients chosen here written down here are of very specific nature and in the approximation, many more of these type of equations come up. But this specific choice of coefficients makes that this equation is actually an integrable PD. First it's a Hamiltonian PD and the specific choice of equations makes that it's actually in the KDV hierarchy. Therefore, it can be written in a Hamiltonian way. So DDTU is equal to DX, the L2 gradient of the Hamiltonian H2, that's the KDV2 Hamiltonian. It is given by this expression, so it involves the L2 norm of the second derivative of U. So H2 is well defined on the subolive space H2 and has some non-linear parts. DX is usually the Poisson structure introduced by Gartner. So I recall this is the second Hamiltonian in the KDV hierarchy. So the zeroes Hamiltonian usually referred to as H0 is the L2 norm and the first one is the famous KDV Hamiltonian. So the bell positeness result which comes from this analytic extension I was talking to at the beginning for the KDV equation two reads as follows. It's well posed on L2, despite the fact that it is fifth order and the Hamiltonian is of second order. So it's defined only in L2. So for the convenience, we restrict to the symplectic leaf of subolive spaces with average zeroes. So this is a cosimere of this Poisson bracket. The same results would be true on any of such leaves. Okay, and then also have to introduce many folds M as zero D. So these are the functions U in subolive space H with average zero, so that the L2 norm is constant. And because the L2 norm is the zeroes Hamiltonian in the KDV hierarchy, it is invariant under the KDV two flow. So this manifold is invariant under the KDV two flow. So the well-posedness results for the KDV two equation reads as follows. KDV two is globally in time, C zero well-posed in H as zero for any S bigger or equal to zero. It means that the solution map S continuously extends to, I mean, from more regular data where it is known that solutions exist, it extends continuously to a map of the following type. So initial data in H as zero goes to solution. These are continuous functions of the time interval minus TT for arbitrary T into H as zero. And this map is continuous. However, this map is nowhere locally uniformly continuous on H as zero. It means that if you give me any open non-empty set, U on H as zero, then if I restrict S to the set U, it's not uniformly continuous. In addition, if I take S bigger or equal to one half, and T is strictly positive, otherwise it consists just of one potential, the zero potential is uninteresting, then this map actually is uniformly continuous on this manifold. But on contrary, if S is in between zero and one half, then again, the solution map S is nowhere locally uniformly continuous, even on this manifold, even if I fix the L2 norm, it will be nowhere locally uniformly continuous. Okay, then the second group of results concerns ill-poseness. And to describe these results, I have to introduce a renormalized KDV equation, which I refer to as KDV2-sharp. So it is again a Hamiltonian equation, and I subtract this term here. This corresponds to the Hamiltonian H2-sharp, but I subtract 10 times H0 squared. So again, this is an integral PDE because this is the zero's Hamiltonian in the KDV-aharke. And therefore the difference, of course, is again integrable. So then the result on ill-poseness goes as follows. The modified, the renormalized KDV2 equation is globally in time, C0 well-posed on each subalpha space H as zero for any S strictly bigger than minus one. So for all distributional spaces H as zero, KDV2-sharp is still well-posed. Again, it means that the solution map S-sharp continuously extend from H as zero to the space of continuous function in the time interval minus dt to H as zero. As an immediate corollary, it says that KDV2 must be ill-posed below L2 due to blow-up of phase factor. So if I represent the Fourier series, the solution of KDV2-sharp in Fourier series, let's write it down like this. So this would be the N's coefficient of the solution at time t of KDV2-sharp with this initial data. Then these coefficients compare to the corresponding Fourier coefficients of the solution of the original KDV2 equation with the same initial data V as follows. The N's Fourier coefficient is obtained from the N's Fourier coefficient of the KDV2-sharp equation times this phase factor. And of course below L2, this phase factor blows up because H to the L2 non-gets infinite. Okay, so the results improve on earlier results on well-posedness on the circle. And there has not been so much results in the periodic case, but more recently, a lot of results on the line in particular by Gua-Kuan-Kuan and Kennegan-Pilo where they looked at solutions on the energy space contained in H2. Okay, so much about KDV2. Now corresponding results for the defocusing MKDV equation, we also have some results on the focusing one, but I don't wanna talk about right now. So the defocusing MKDV equation, everybody knows on the circle is given by this equation. So it differs from the KDV equation by the fact that instead of U, I have here U squared. Again, it's a Hamiltonian PDE. This is the symplectic gradient where DX again is a Poisson structure from Gartner and this is the L2 gradient. The MKDV Hamiltonian is given by this expression. It's DXU squared plus U to the fourth. So in KDV, you will have U to the three. Here you have U to the fourth. So here we have an ill-posentness result of the MKDV equation below L2 and this goes to state it again. I have to renormalize the MKDV equation. I call it MKDV-sharp. So it is obtained, MKDV-sharp is obtained from MKDV by replacing U squared by U squared minus the L2 norm. Again, this equation is Hamiltonian. H-sharp is its Hamiltonian. It's H minus six times H0 squared. And again, H0, the L2 norm, is conserved by the MKDV flow and therefore it will be easy to compare the two flows. And in order to state our ill-posentness results, I have also to introduce the Fourier-Lebesgue spaces, FLP, P between one and infinity. So this is a space of periodic distributions whose Fourier coefficients, the sequence of Fourier coefficients are in LP. So of course, due to Parseval's identity for P equals two, FL2 is just the usual L2 space, but for P stricter than bigger, this is a big space. It's a big space of distribution and it gets bigger and bigger as P goes to infinity. In particular for P bigger than two, this contains elements which are not even measures. Okay, so the result for the MKDV goes as follows. MKDV-sharp on FL, on Fourier-Lebesgue P is P between two and zero at the following properties. It's locally in time C0 well-posed, meaning that for any initial data V in LP that exists in neighborhood V and the time TV so that the solution map as sharp continuously extend in this way as before. In the case V is equal to zero, we can have a slightly stronger result, namely the equation is not only locally in time, a C0 well-posed, but actually globally in time well-posed on V0, so this would be true for any T bigger than zero. So again, as in the KDV two equation, we have an immediate corollary, saying that MKDV equation is ill-posed below L2 due to blow-up of the phase factor. So again, if I take the Fourier representation of a solution of MKDV-sharp and write it in this way with initial data V of X, then I can compare the Fourier coefficients of MKDV-sharp with the ones of MKDV in the following fashion. The N's Fourier coefficient of the corresponding MKDV solution is the one of the MKDV-sharp equation times this phase factor. So again, if I'm below L2, this phase factor gets infinite. So of course there has been many work on the MKDV, so maybe first a few comments. So we expect MKDV-sharp equation to be also globally in time zero and zero well-posed everywhere on FFP, but there is one thing we don't know to prove at the moment. And I also mentioned right away because I might not have time during this lecture to explain anything of this, that the key ingredient in the proof theorem two is to view the MKDV Hamiltonian as the corresponding Hamiltonian and in the NLS hierarchy. So it's the fourth Hamiltonian, so NLS is the third Hamiltonian in the NLS hierarchy and MKDV the fourth. So does this mean that your result is also valid for the complex version of the equations? Yeah, yes, very big yes, absolutely. Actually we do extend the frequencies to the system. We call it MKDV system because it's actually, of course we have to be in a neighborhood of the real case. It cannot be arbitrary far away because we probably have hyperbolic structures of the equation. So of course there has been many, many work on MKDV on the circle. Essentially it is well-posed for HS as big or equal to zero. So then there are also other works. So maybe I don't want to spend too much time on this but just mention maybe a little bit more in detail. One result by Luc Moline on ill-posedness of MKDV below L2. So he proves that MKDV is ill-posed in HS for S strictly bigger than zero in the following sense. The solution map is not continuous as a map from C0 infinity with the norm HS coming from the sub-left space with S strictly smaller than zero into C0 minus DT periodic distributions. So we can view corollary two as sharpening Luc's result saying that ill-posedness of Luc is really due to the fact that you have blow-up of phase factor. Okay, so let me give a short outline of what I want to do in this talk. So the method we will, we prove these results are the so-called non-linear Fourier transform or Birkhoff map or normal form transformation. So it makes, these are new coordinates which make that the equations when expressed in these coordinates are actually linear and the dynamics are described by frequencies omega n. So the crucial part in improving these well-posedness, ill-posedness results is by extending these frequencies in the corresponding spaces of functions of low-regularity or distributions. So as applications of these analysis of these extensions of these frequencies we have already mentioned or the well-posedness of the PDEs as stated but also that actually was the original motivation of why we started to do this is it applies to Hamiltonian perturbations of such PDEs. So it has important implications on the properties of the action to frequency map and also it allows to analytically extend the Hamiltonians and to discuss its convexity properties which are important for Nekoroshev type results for perturbations. So to keep the exposition simple I will explain the method actually for the proof for the KDV frequencies but even in this case actually we improve known results. At the end if I have time I will discuss very briefly the de-focusing and KDV equation to see how it connects with the NLS system. Okay, so to start I have to review quickly this non-linear Fourier transform or these normal coordinates for KDV. Actually these coordinates are good for the whole hierarchy for any Hamiltonian in the Poisson algebra generated by the KDV equation. So the setup is the scale of sub-alef spaces h, i, c stands for complex value these are periodic functions. We include s bigger or equal to minus one so some negative s's are included here. So as I said we have the Poisson bracket which was introduced by Gardner so these two functions f and g on this space is sufficiently regular L2 gradient have a Poisson bracket defined in this fashion. And note that the average is a cosimere meaning that for any functional if I put the average as the functional g then this is identically zero. So this cosimere leads to symplectic leafs and in the sequel I will only restrict to the zero leaf so average zero and again the z refers to the complex numbers. So then to state the results of normal coordinate cells I have to introduce sequence spaces so these are LPP sequences, complex values so this stays for c and they have weight s. So this is the weight and to the sp and zero stands for that the element of the sequence in the middle is actually zero has to do with the zero leaf. Okay and then we also introduce the real subspace of this consisting of all complex sequences which have the property like for Fourier sequences that z minus n is a complex conjugate of cn. And then to each of such a sequence we associate a sequence of actions for n bigger or equal to one they are obtained by multiplying zn with z minus n for all n bigger or equal to one. So just as an exercise because I will use this later we will consider sequences in LP spaces with weight s plus one half then the corresponding actions has double weight so this is two s plus one and p gets half because of the square. And as a special case if we are in the real subspace then this is a complex conjugate of this so i n is bigger or equal to zero and then the sequence of actions is actually in the positive quadrant of this LP two sequence space. Okay and then the integrability of the K-DV hierarchy can be formulated as follows. So as our base space we take h minus one zero so the real space of functions in sub less space with average zero in s equals minus one. We take then there exists a complex neighborhood of this space in this complex space and this complex sub less space and a map from this neighborhood into the corresponding sequence space associating to each potential the corresponding coordinates complex Birkhoff coordinates refer to as complex Birkhoff coordinates so that the following properties hold. When we restrict this map to the sub less space h s zero for any s bigger or equal to one then actually it's a real analytic diffeomorphism onto the sequence space there is a two missing here s plus one have two. Second fees canonical meaning that to pass on brackets of the coordinate functions or the standards one. Of course here we have complex coordinates so we don't have one but an i instead all other brackets vanish. And thirdly the most important property is that the Hamiltonians in the K-DV hierarchy are real analytic functions on the actions only. So it means that in H K-DV so this is the Hamiltonian is defined in H s so the corresponding sequence of actions is in weighted L one space with weight three so if my potential is positive then the actions are in the positive quadrant L three one and correspondingly if I have the K-DV two equation then I get this kind of real analytic map. And lastly actually fee of zero is zero on the differential of fee at q equal zero is essentially the Fourier transform up to this little weight which comes from the Poisson bracket. So as I said this map is usually referred to as pick-off map and the coordinates as complex pick-off coordinates. The principal property of fee is this three it means that it linearizes the PDEs in the K-DV hierarchy. So then we come to the frequencies the topic of my talk. So the frequencies of K-DV and K-DV two equations are given by, so we take the Hamiltonian and the ends frequent is obtained by the partial derivative of the Hamiltonian with respect to the nth action for any n bigger equal to one. And then the equation as I said linearizes in these coordinates and they read as follows. So the ends coordinate the time derivative of the ends coordinate is minus i this comes from the Poisson bracket omega n K-DV cn the omega n's K do not are invariance of the flow because the actions are invariant and hence the frequencies and hence these equations can be very easily integrated and the same for K-DV two. So key ingredients in the proof theorem one, theorem two are the extensions of these frequencies to spaces of functions of low regularity and distributions and in addition asymptotics of the frequencies as n goes to infinity. So the setup we choose to achieve this extension is as follows. So we write, so as I said I do this only for the K-DV frequencies just to save a little bit of time. So we write omega K-DV n as this is the frequency of the area equation sort of the linear part of the K-DV equation plus let's say a remainder which we call omega K-DV star n and of course it's enough to extend this omega K-DV star and in order to obtain asymptotics for omega n actually you want to extend not each frequency by itself but we want to extend it as a map. So we look at this as a sequence. So we have to determine from which space to which space this it's usually called the frequency map or the action to frequency map which in integrable systems and in perturbation theory plays an important role. Okay, so here is the main result. So for the K-DV star frequencies. So the omega K-DV star analytically extends as follows. So we can view the frequencies either on sub-left spaces or on the corresponding spaces of actions. So I wrote the two at the same place because this might be a little bit more difficult to decipher than just the sub-left space. So the extent to H minus one, zero. So on the level of actions this means we are in the positive quadrant of L1 sequences with weight minus one and it goes not quite to the same space for S equals minus one but a little bit larger space namely an LR space with weight minus one for any R strictly bigger than one. But if S is between in the open interval minus one, minus one half, then we go from the space L to S plus one one into the same space. And if S is bigger or equal to minus one half and then again we have somewhat a weaker space on this side, this has to do because this has asymptotic expansions in one over N and it appears at some point and you cannot improve. Okay, then we have asymptotics. So essentially the N's frequency or omega NKDV star is minus six IN. So this is the N's action variable plus an error term which can be estimated in this way. And this error term depends a little bit on the S. So I mean there are more precise estimates but just to keep it simple I stated in this way. So if S is between minus one and minus one three and this is the form of the error term we get and if S is bigger or equal to minus one three it actually doesn't improve. Again, due to asymptotics which you cannot change in one over N. Okay, certainly we have seen that if it's S between minus one and minus one half then omega KDV star is a map from the space to itself. And therefore I can look at its differential and we can prove that its differential is actually fragile. It's actually minus six identity plus compact. And then I mentioned also because for one application I need this that we have a special case. Actually we can also extend the frequencies to Fourier Lebesgue spaces but I just take one example. This would be Fourier Lebesgue S equals minus one half P equals four which leads to an action space of L2 plus. So also in this case, omega KDV star can extend as a real analytic map and the differential is flat on. Okay, so this is the essential result on the frequencies and I will discuss a little bit on the proof in the second half of my talk but first I want to give applications. So of course one application we have already discussed but I would like to go back a little bit. So first we can use these frequencies to analyze the PDE, the PDE is considered, the integral PDE is considered in Birkhoff coordinates. So it allows to analyze the dynamics of the angles in a very precise way. So we decompose any of such frequencies which come up in an integral PDE in the following fashion. A n plus B n plus omega n star. A n is a polynomial in n with constant coefficient. B n is a polynomial in n with flow invariant coefficients and then omega n star is of the form leading part plus remainder. So just to give an example, for the KDV frequencies the A n's would be the frequencies of the area equation. If I consider it not only on the zero leaf but on the whole sub-alive space actually I get a B n which is 12 times the average of q times n p and then omega n KDV star is minus 6 a n plus remainder. And similarly for the KDV2 equation this would be the frequency of the linearized equation at zero. B n would be this kind of expression and again omega n star would be linear and I n plus a remainder. So the B n's and the omega n's are possible cause for the PDE for not being uniformly C0 well-posed or not being locally uniformly C0 well-posed. The growth of these coefficients might create phase D coherence which will lead to these vicarious notions of well-posedness. And of course it will lead to imposedness in case of blow up. Okay then of course we can analyze this using these frequencies the PDE and the original sub-alive spaces. So if ST is a flow map in the sub-alive space Hs0 we denote by STv the corresponding flow map in the Birkhoff coordinates and then the flow map ST can be obtained from the one from the Birkhoff map by conjugation through this Birkhoff map due to the fact that V is in practice. So how can this be used? Let me illustrate this just for illustrative purposes for the KdV2 equation on Hs with S bigger or equal to 1. So one can prove that omega 2 star is actually a continuous map from the space of actions into L infinity which is uniformly continuous on bounded subsets. And so are the corresponding Birkhoff maps and its inverse. And because B and 2 is constant on the leaves with D equals constant the KdV2 uniformity is 0 well-posed on this sub-manifold MS0T. So by analyzing the frequencies in this way one gets these type of results. OK then just maybe as a remark let me mention from this analysis it suggests to think of a possible role of the critical sub-alive exponent Sc in the analysis of ill-posedness. So if I look at the Fourier if I write down the solution in Fourier space then I can write it in this way. So An of t is an amplitude exponential i omega t the phase factor and exponential i and 2 pi x. So a possible scenario for ill-posedness in Hs is and the possible role of the critical sub-alive exponent is if S is smaller than S0 we have ill-posedness due to the amplitude blow-up of the amplitudes. If we have already ill-posedness before Sc it might be due to the phases. So that's essentially true in the MKdV equation following our results I explained. OK so second application as I mentioned is the convexity of the KdV Hamiltonian. So to explain it I introduce so this is a KdV Hamiltonian and we re-normalize by subtracting a term which gets very large if I go into functions or spaces of distributions. Then because KdV Hamiltonian is defined on H1 therefore the corresponding spaces of actions is a positive quadrant of L1 sequences with weight 3 is real analytic. So there was a conjecture by Sergei saying that H star KdV analytical extends and is strictly concave on L2+. So the L2+, is the space of actions I was mentioning in Serum 3. That said I will use it later for applications. Here it is. Now in earlier results Big Bayef and Cooks improved that this Hamiltonian is actually concave. So they prove this for finite gap potentials and then by continuity show that it's concave once you know that you can extend it continuously. And then last year actually we proved together with collaborators that the H star KdV actually extends to L2+, and it's strictly concave near I equals zero so at the origin. And using our theorem 3 now with the property 4 actually the conjecture 3 is almost true in the following sense. So this renormalized KdV Hamiltonian is strictly concave on an open dense subset O of L2+, with the origin being an element of this open dense set. It means the following that if ever my action is in this open set then the Hessian of the KdV star, I mean this is the same actually as the Hessian of the original one because I just deducted the linear term. The Hessian of the renormalized KdV Hamiltonian is strictly concave in the sense that I can bound it from below by minus C, L2 norm of J, for any J in L2 where the constant C can be chosen locally uniformly around it. Yes, absolutely, thanks. Okay, so as I already mentioned, convexity property is relevant for stability results of the KdV type. Okay, so maybe there's also application to the action to frequency map, maybe I leave this in order to be able to explain a little bit. But Thomas, where it loses strict concavity, it is degenerate concavity, there are flats, is that how it loses it? So it's open dense, so we don't know. I mean, Sergei thinks it's everywhere but I don't see any reason why there should not be some zeros in between. Right, but that's the failure of strict concavity. Yeah, it's always concave, but strictly concave on this open dense subset. So for local perturbation theory, no local way is enough, of course. But, and the fact is we prove that the differential of the frequency map is fret home and that allows us to together with analyticity to have the open dense thing. Okay, so outline of theorem three. So I recall theorem three was the analytic extension of the frequencies. So I need first to introduce what are called the flow exponents. So on the real line, the corresponding notions would be probably the transmission coefficients. So these have been already introduced in the seminal paper by Peter Lex in the 75. So for Q in W, so W is this open neighborhood of H minus one zero. We look at the Schrodinger operator, so this would be a distribution. You can do spectral series still for this and the spectrum is discrete. The eigenvalues come in pair. Maybe I write them so if it is real. So this is the lowest and they are, if Q is real, then they are on the real line and they satisfy, they can be ordered in this way. So the pairs never coincide with another one but the two elements in the pair might coincide. Then we define the gaps. So this would be these intervals here and the gap lengths which would be the size of the length of these intervals. And then we have here another gap G zero which is to the left of L zero. Which is infinite. Okay, then I have to introduce the discriminant of the thing. So of course if Q is regular, the discriminant is given by the trace of the flow K matrix and one can show if Q is sufficiently regular, then actually it has a product, this product expansion involving the eigenvalues of the Schrodinger operator. And in case you're in low distribution, you use directly, you define the lambda by using this product expansion. Anyway, product expansion will be all over in the analysis of, in the estimates of all the quantities involved. And then we introduce a canonical root of delta squared minus four. This usually gives rise to the hyper elliptic surface associated to the spectral problem. And this is well defined on C minus the gaps. And for Q equals zero, just to give an example, it's sin times square root of lambda. Then the flow K exponents. So this are the, so we have the flow K multipliers would be the eigenvalues of the flow K matrix. And if you write them in exponential form, you get the flow K exponent. And one of them can be written in this way. So for any Q, again, even if flow K series doesn't make sense, because delta makes sense, you can define this in a perfect way. It's well defined, also it doesn't, even if there is a little singularity here, this is integrable. Okay, so these are a list of important properties of the flow K exponent. So f of lambda is analytic away from the gaps which are open. So if the two eigenvalues here coincide, actually this flow K, the f of lambda extends analytically into this double eigenvalue. And we have specific values at the periodic eigenvalues at lambda zero, by definition it's zero, because we start integrating at lambda zero plus. And here one can show you the fact that the potential is periodic, I get here minus i and pi. And then we introduce a version of this flow K exponent by normalizing it differently, fn of lambda, so that at lambda equals plus, lambda plus minus n, this is actually zero. Then one can prove that fn changes sign across the gaps. So here we have different signs, but there are limits as we approach the two sides of the gap, and hence the square of fn actually analytically extends in the neighborhood of gn. We have estimates, we have an explicit formula of the gradient, and the fn's are related to the actions by this contour integral. So gamma n would be a contour around the gaps. Okay, and then most importantly, there is an asymptotic expansion of f of lambda. So actually we use later on only f for of lambda, so I write down the expansion right for this. And of course we have a little problem if we do this for distributions, it's enough to do it for real finite gap potentials. Finite gap potential means that only finitely many of these eigenvalues are simple and the rest are all closed, and it means that they'll potentially see infinite. Okay, then one can show that there is an expansion as l goes to infinity of this form. So it's actually a meromorphic function, has a pole at infinity of order two. Similarly, so we see at the point of the residue, the Kdv Hamiltonian is involved. Similarly, for the Kdv two equation, we use instead of f to the four, f to the six, and this makes that the Kdv two Hamiltonian appears again at the place of the residue. So later on if you do contour integration, it's a way of picking up the Hamiltonian. Okay, now contour integral of four around simple eigenvalues because we know outside this is an even power, so we know it's analytic. In particular, it's analytic here, so because I have finitely many gaps open, I know that outside the big disk this is analytic and therefore I can use contour integration to pick up the residue here, and on the right hand side, I will have by contour deformation just integral of our contours on the pairs of eigenvalues. So S is a set of Ks where the eigenvalues lambda K minus lambda K plus are simple, meaning that gamma K is different from zero. Okay, so this gives us the corresponding formula for the Kdv two equation. And then the idea is to develop these expressions further in order to get the formula for the Kdv two equation. Now let me explain this a little bit in more detail. Now here is the crucial idea. So from Hamiltonian formalism, we know that when I express the equation action and angles, then the ends frequency is nothing else than the derivative in time with respect to the ends angle. And this is equal to the Poisson bracket of the HKdv Hamiltonian with theta n. So of course, in order to have the angles defined, I need that the ends gap is not collapsed. So this goes only for simple eigenvalues. Okay, so this allows us to get a formula for the omega for the ends Kdv frequency in the case where n corresponds to an open gap. So it is equal to this expression. I substitute our expression we got from the residue calculus for HKdv. This is the expression. By the property F5, I know how to compute the gradient. I insert it here in the Poisson bracket and I get this. Now we use a very important identity from our book, saying that the Poisson bracket of delta. Okay, so this is where Pcn has this product expansion. These are actually related to holomorphic differentials on the Riemann surface. So these are very well known quantities and play anyway a very important role. They come here a little bit out of the sky, but they are not. Okay, so here sigma and K are the zeros. These zeros are completely determined by this equation. So this is a normalization of the holomorphic differentials. And they can be easily shown in case the potential is real, they have to be within the gap for N different K, for K different N. Okay, so we insert and we get this expression here. And now we expand F of three as Fk minus ak by cubed and get formulas for, get in this way a formula for MKd, for the N's Kdv frequency. So for this to express this, we introduce the M's moment of the, of the Fkl's. And these M, now we use sort of important symmetry properties of these moments. And that is actually why we can succeed in extending to such low regularity. So first of all, for all M's, these moments are identically zero. And this has, so I don't wanna explain because I'm running out of time. So also if gamma K is equal to zero, then these moments for all M and N are zero, for this specific K. And finally, the zero's moment is nothing else than the integral of this holomorphic differentials over the gap scheme, gamma K, which is nothing else than Dirac delta times two pi. So using all this, we get the following formula for any real finite gap potential of average zero. We have this formula. Omega NKdv is two and pi cubed. So actually this comes from expanding, from this, if I expand this expression, I get this term and this actually gives precisely this. And correspondingly, the frequency of the Kdv to Hamiltonian, the N's, has a formula of the four omega Np to the fifth. This is a linear part plus this part, which we already came up with when we discussed frequencies plus omega N2 star, where omega Nster can be expressed of the moments of F and K of the moments two and four. Okay, so the strategy for proving theorem three is we use formulas of theorem six to analytically extend omega NKdv and omega N2 by analyzing in detail these moments by showing that this series, by showing that they individually, they analytically extend to H minus one, first of all. Second, by showing convergence of the series by deriving asymptotics of these quantities. And for this we use sort of quite involved estimates for product expansions of terms, which involve in these moments. Okay, so maybe I leave this and then just maybe one last little thing. So coming back to the NLS system and the MKdv equation. So I said at the very end, if I have time, I would briefly say something, how this can be, how MKdv comes up in the NLS system. So I recall the NLS system is a system of Hamiltonian PDs for phi plus phi minus in HSHS, complex valued, given by this expression. It's a Hamiltonian PD where the Hamiltonian is given essentially by the NLS Hamiltonian, but with these two components. And the Poisson bracket is the usual one with this factor minus i because I'm in the complex version. So if I restrict this system to a real invariant subspaces I get actually the defocusing NLS and the focusing NLS equation. So if I restrict to phi plus equals phi minus complex conjugate I get the defocusing NLS. And if I restrict to phi plus equals minus phi minus complex conjugate, then I get the focusing NLS. And this NLS system has the NLS hierarchy. It starts with H0, H1, this is the moment. And then H3 is the NLS. And as I said, H4 is related to the MKDV equation. We call it the MKDV system. So this would be the first equation, one of the two for this Hamiltonian. And then this Hamiltonian system has also invariant real subspaces, namely in the case where in addition to phi plus being the complex conjugate of phi minus, they are equal. So u is actually real. Then we obtain the defocusing MKDV equation. And if phi plus phi minus is real but they differ by a sign, then we obtain the focusing MKDV equation. So then the idea is to look at the Hamiltonian H4 we actually expand the Birkhoff normal form theory to Fourier Lebesgue spaces. FLP with P negative, with P between two and infinity to these spaces of distributions and then extend the frequencies of the MKDV system to these Fourier Lebesgue spaces. And then of course we have to make a little analysis because it's a subspace we are looking at for MKDV, how the Birkhoff coordinates restrict in this case of these real subspaces. So maybe I leave the details here. We stop. Another question? Yes, you showed us how to extend these frequencies to wider spaces for Fourier Lebesgue. Yes. FLP for P bigger than two. Yes. What about P smaller than two? This is easier. This is probably easier. Yes, because it's contained in L2 so that the series is essentially made. Yes, but you get additional. Yes, yes, you can. Yes, there was a little bit less of interest but can be done. Yes, for instance you can do everything on the venous space. Yes, absolutely, absolutely. Absolutely. Nice space. Yeah, I agree, I agree. More questions? There's recent interest in well-behaved space, quasi-periodic or even space almost periodic solutions. And that involves then, for KDV, other ones too but I just heard of the KDV. And that of course involves some important spectral theory of that. So you have to do the good reflectionless ones but then the time evolution seems to be, at least this sounds like this is machinery that would work for that. So this is a question, is that a possibility? I think so, yes, I think so. Meaning gaps are no longer ordered but doesn't really matter. I must admit I did not look into it but I think the machinery should be extendable to this case, yes, absolutely. So, but we were more thinking of by looking at the action to frequency map in infinite dimension so there we can also show that this, but this is in the periodic setup that the map is actually a local defilmophism. So that means potentially you can do KM theory with infinitely many, at the total of infinite dimension because you can control the infinitely many frequencies through the action, you can go back and forth. All right, if not, let's thank Thomas again. Thank you.