So I guess I'm introducing myself. I'm Lloyd Williams, I'm one of the second year residents, and I'm gonna try to go through this relatively quickly so that there's time for discussion. This patient was not presented, seems there was some confusion. We were unable to even get an adequate exam in the pediatric clinic, so it wasn't felt like it would be possible to bring him in and for any of you to get an adequate exam with so many people. He was previously presented by one of the medical students, but since then, more information has been acquired, and also a lot of people were away at the academy, so Dr. Greese is hoping that people can offer some suggestions. This is a four-year-old male who presents with, who presented very light sensitive and his left eye had a red eye, a lot of eye rubbing, squinting, perhaps some clumsiness and some changes in performance. He'd had an eye exam greater than a year ago, where the findings that I'm gonna show you were not seen. He has a past medical history of hydrocephalus and a past ocular history of congenital iris and retinal coloboma with microaphthalmia in the left eye, otherwise an unremarkable history. We were unable to get a good visual acuity. He doesn't cooperate with the exam. He has some evidence of a cleft lip hyperteleurism and a small eye on the left, so we scheduled an EUA. On the EUA, his pressures were normal, and a vascularized mass was found in the anterior chamber with a hyphema. There was no view of the iris or lens and no view of the posterior chamber. And at that time, our differential diagnosis included retinoblastoma, persistent fetobasculature, medjiloepithelioma, uvidus, toxocara. From two weeks later, a repeat EUA was done. The hyphema and anterior chamber mass were still present and there was still no view of the fundus in the left eye. The exam of the right eye was basically normal. And the patient was referred to the shield's at will's eye. Laboratory tests were also done. Toxocara was negative. ESR was one. Chem 8 and CBC were roughly normal except for a low hematocrit. HSV IgG was negative. I'm not sure what the significance of a positive IgM in this particular patient is. So they were evaluated at will's eye by Jerry and Carol Shields who did not think that the patient had retinoblastoma or medjiloepithelioma and that their thought was that this was most likely a lateral persistent fetobasculature and they suggested serial ultrasounds and enucleation if the eye became painful. During one of the EUAs, we also did an ERG which was flat. So it's our thought is that this eye has very little visual potential. Sorry about the typo here. If it is medjiloepithelioma, the Shields thought that it would be possible to follow it without risk of extraocular extension. And so their differential diagnosis, although they thought medjiloepithelioma epithelioma was less likely, their differential diagnosis included persistent fetobasculature and medjiloepithelioma. They thought that this eye was likely headed for glaucoma, had a poor visual prognosis, that a fine needle or open corneal biopsy might be possible. And they thought if this were done in enucleation it would also be a reasonable follow within roughly a week of biopsy. We did another EUA now five weeks after presentation, after the patient returned from Wales and found it still a normal IOP with a new sheet of yellow-white tissue across the pupil. The mass was still there. The hyphema or mass maybe enlarging is difficult to tell and I'll show you some pictures in just a little bit. Whether or not this is increased size of the mass or a consolidation of the hyphema, still no view of the iris or lens or posterior pole. I'll show you also ultrasound at MRI, but a brief summary of them is that there's no unequivocal calcium. The eye is in fact small with a flat AC and a hypervascular AC. Posteriorly there may be a persistent hyloid artery. So here's a, I'll just go through four pictures which are pretty similar of the anterior chamber. This is from November 10th. This is November 8th. I'll go through them in this order. So this is October 25th. You can see this mass. This is, these vessels are deep to the cornea in the anterior chamber, not on the cornea. This is, I'm sorry, I don't have the right order here. This is the first picture, 10, 8. This is the second picture, 10, 25. So that's the difference there. And this is the third picture, 11, 10. That's correct. It's anterior to the iris, posterior to the cornea. In particular, it's difficult to see whether or not this one is in fact growing or if you're just seeing more of it because the hyphema is changing. This one appears to be growing as well, although that could also be consolidation of the hyphema just showing you more of the mass that was already present. And so then this is the most recent whereas you can see that this is, in fact, more prominent than it was in this image. So the ultrasound, these are from Dr. Harry. This arrow represents the lens and this represents a peripheral vitreous membrane and a tuft visible on the optic disc. Flat AC, elongated ciliary processes. A vitreous membrane with fibrous material on the optic disc. And thickened cornea in the area, perhaps a fibrovascular membrane. This is a CT scan, which doesn't show any clear calcium here in the left eye, which is smaller. This is a T1 non-contrast image where you see thickening here or a membrane.  You see the mass here. And here's a T1 fat set post-contrast. Does anybody wanna see any of those images again? That's been done and that was pretty flat.


Veliko bilo. Počutke, prvno, s tešem organizacijem. Čelo se, da se počutim. Prvo, da je nekako vse vse vse vredno, ali za vsega bila, bilo očaj. Neko, zelo prijezdi. Od palju minutnih vsega. Počutim počutim počutim. Tukaj, da sem počutila 3D, ali baham, da sem zelo počutim, da bi se izvajila tektiv. V nekaj težiši je zelo. To je očaj skupniko z Fabrizio Pranšonem So, what's the plan? Well, I just start very, very simple situation about the Cochitheorie. So I want to say very quickly how to solve the Cochitheorie locally and global in time, and once we have a global solution, you want to understand the properties for large times. In particular we are interested in the issues of the growth of the solvable unknowns to deduce some upper bound, the polynomial bound that Jullio was mentioning before, in v svoj svoj svoj zelo. Zelo smo nekaj stručen, nekaj je nekaj ator, nekaj nekaj periodist, je tudi tudi 3D in 2D kompaktive, nekaj nekaj nekaj in nekaj nekaj nekaj klasične strategije v Burgeni, da se počuče tudi z polinoma glasboj, kaj je način vzelo modificovati energij. Vzelo smo, da v 2D More general results, in in 3D I think that you get some new result that as far as you can see you cannot get by using the bourguent technique. At least we were not able to do that. And then if there is time, but maybe not, I will say how this technique can give even better result in one day. Again about this issue of the sides of the growth. Now, what is the problem? It is the same, my Giuliolla, a few minutes ago. And here we assume, in general, for simplicity I assume a pure power non-linearity, but since now I tell you that this technique that we use allows you to deal also with more general potential. On the contrary it is not completely clear to me that the bourguent technology can work also for rougher non-linear potential. But first of all, please let's work on the pure power case. We are in this setting, so MD will be a d-dimensional compact manifold. Laplacian G is the Laplacian Beltrami operator. S is the regularity of the initial data. And at least you hope that for an initial data in Hs you propagate the regularity Hs for every times. And you want to understand these growth of Hs. But let's go step by step. And the special case that we will be interested in is the following. So 2D, essentially M2 is a general compact manifold. And P is any integer non-linearity. So no restriction to the cubic case. And in 3D we can deal with the cubic case. And we can say something in the sub-cubic case. But without any assumption on the topology of the manifold. So this is the usual. We learned what is the focusing between this morning and the day. And so as you can see, I assume here the sine plus, which is the same, is the opposite of the other one. So we get a positive energy. So what is sure is that the H1 non, if the solution exists, cannot blow up, not even at infinity. So no problem about the growth of H1. And let me spend a few words about the local and global Cauchy theory. Because if you want to start to analyze the long-time behavior, let's start by the local, by the global existence of one solution. So what is the status of the art, as far as I know? So in 2D, give me any 2D compact manifold. Give me any non-linearity, no restriction, no necessarily cubic. Then the Cauchy problem is locally well-posed in H1. So once you are locally well-posed in H1, for free, you globalize just by the conservation. What is the other result that you can prove? Well, in 3D, give me any 3D compact manifold. Give me any initial data in H1 plus epsilon. Then for cubic, then you can solve the cubic NLS in this regularity. And you see already in 3D the globalization is not free. Because you can solve locally in time in H1 plus epsilon. So this is essentially what I said before. So in 2D the globalization is free because you are at the level of the energy. In 3D the globalization argument is much more involved. It is due to Bjerke Gerard and Zvetnikov. OK, but I will come back to this approach later on. But just let me say a few words. So what is already this first point here? The second point here is not free. You need to work, especially on generic compact manifold, because the free result is the following. If you are in 2D, what is really easy and trivial is to solve locally in time in H1 plus epsilon. So already the globalization in 2D, if you just rely on the sobolef embedding, is not free. And in 3D, just by using this sobolef embedding, you have to work in H3 over 2 plus epsilon. In the previous slide they said H1 plus epsilon, H1. Why you can do that? Well, because instead of just using sobolef embedding, we have to use the Strikard's estimate. So you have to put in the game the dispersion in a more serious way, not just by L2 estimate plus sobolef embedding. And I also want to mention one result that was not mentioned today at least, which is the, as far as you know, the first global existence result for NLS in 2D, which is the result by Brésil Saint-Galouet, who said the following, they are able to globalize in 2D for cubic NLS any solution with initial date in H1 plus epsilon. That you cannot do that just by conservation law, but you have to add this extra sobolef embedding. You see, this is a sobolef embedding. In 2D you want to control L infinity in principle with H1, but you cannot. So you need to add some high frequency contribution, but this high frequency contribution can be hidden in the log. And this is something that helps to globalize. And you globalize, and moreover, you get also a bound, which is double exponential. So H1 plus epsilon cannot blow up in finite time, and moreover, you have the double exponential bound. How you can improve the situation? Where to improve the situation, you have to play with the dispersion. So you start to look, as Juliola said before, at some smoothing effect on the free waves. So you look at free waves, you take an initial data phi, you take the free evolution, and you start to look at some regularity, which is not necessarily L infinity, L2, but you start to play with Lpq, with the pq that move under this condition, which is a typical for Strikard's estimate. What you see is that, in general, if you are on RD, you can control this quantity by L2. If you are on a generic compact manifold of dimension D, you have to lose derivatives, and the loss is of the order 1 over p, where p is the regularity in time. Of course, in this situation, that Juliola was mentioned before, where m2 is T2, you can do much better. But this result is true in general, for every compact manifold. So this is the main difference between this thing and the one on T2. And this result, as far as I know, was proved in the same time, I don't know the details of this thing, but between Bjorki, Gerard and Svetkov, in the same time by Stafilani and Tatar. So we want to take advantage of this to solve our Cauchy problem at least local in time at low regularity. More precisely, in h1 in 2D, h1 plus epsilon in 3D. And, OK, let me go further. And let me explain very quickly why this Strikart's estimate helps to solve at least in 2D with initial data in h1, not anymore h1 plus epsilon. Well, you see, the game is very easy. The slide before implies this Strikart's estimate. You see, I lose 1 over r derivative if I take lr regularity, and this number here exactly the one for which r and this number here is Strikart's admissible. This estimate is proved, and thanks to a sobolef embedding, I go in l infinity. How many derivatives I need, I need 1 minus 1 over r derivative, and this 1 minus r derivative gives me h1, and thanks to this estimate, you see, I can control l infinity just by h1, that I cannot do for the time independent sobolef embedding, and that, thanks to dispersion, along three waves. If you combine this estimate with this free, essentially estimate on the dual meltdown, you get global existence in h1 and global solution because of the conservation. This is the 2D situation, so the globalization comes for free. Now, which are the questions we are interested in? Well, as I told you, we have an initial data in h1 and you have a global solution in h1. You can also prove without big efforts that the hs regularities preserved along with the evolution. So if you start from hs, which is very big, for every time you remain hs, but what is not so clear is whether or not this hs norm remains bounded or not. If there are solutions that whose hs norm grows or not, and if it grows, how fast you can grow. The second, even more basic question is there at least one solution that grows when t goes to plus infinity at least for other orders. The answer to this question, at least on 2D compact manifold is still open, but there are some results in other situations, like cylindrical symmetry on the manifold and so on. We have several fundamental paper on this topic, starting from Burgen, here it is a list of names, not necessarily working with Schrodinger equation, but the same question you can ask for generic dynamical systems. Here it is a list of names and I don't want to stay too much on this slide, but let me come now to the first remark already done by Giuliolla. If you start to ask yourself the growth of sobolev norms in R2 it is impossible simply because the nonlinear solution scatter. Since they scatter they behave for large times like free waves and for free waves Hs is bounded, is constant so at least you cannot run. So M2 is not so interesting so in some sense what is more interesting is the compact case. And another remark is that ok, once you have a reasonable cosci teori in H1, which is the case thanks to the estimate by Giuliolla and Tataru in the Birkjard and Zvetkov, then for free you get unexponential growth on the Hs norm. Why? Simply because in general just by tame estimate you can show that the Hs norm after an increment tau that depends just on the size of initial data you can control these Hs with a constant time this. You iterate this bound and for free you get exponential. Which is better than the double exponential in Galway for example. And moreover this is Yes. Not to growth bound. At the most the growth can be exponential. An upper bound for the growth. But can we do something better or not? Well there is the already mentioned work by Bougain who worked in T2 cubic nonlinearity and was able to show that in fact you have polynomial growth. And for instance I don't say that this is a big issue but for instance if you look at the same question for higher order nonlinearity I don't know any written results in the literature telling you that you have polynomial growth. But the technique by Bougain in this context should work and what is the basic idea of Bougain the basic idea of Bougain is to prove that this is a nonlinearity. It is based on some smoothing estimate in excess B spaces but the key estimate is this one. You see? Hs minus at an increment tau which depends just on the size of phi minus the initial data t with one constant but the power here has to be below 2. If you are below 2 then just by an elementary argument to introduce the quantity alpha n of time tau n then you write down this estimate implies this thing here this inequality here and just by simple iteration you forget this polynomial bound. N to 1 over epsilon means polynomial bound. But how Bougain could prove this kind of bound here? It is a fundamental that the constant c here is equal to 1. If you get another constant the best you can do is exponential but here you can get a big constant less exponents. This is the key. The idea of Bougain is very simple. We have to measure the difference in Hs at two different time. T1 and T2. What to do? We start with the fundamental theorem of calculus. This is the entity of this. You compute this time delivery but this is real part of this quantity. And now what it does? It replace ut by the equation. When you replace ut by the equation ut bar that just by simple self-adjointness argument is imaginary, so it disappears and you end up just with nonlinear part replaced in ut bar. Imaginary part. And here for simplicity they call 2s1 equal s. Well s is the regularity of the sobolev known that you want to understand. And now it is the key point. If you work a bit you see that this Laplacian 2s1 is Laplacian s1 to the square so you can put s1 derivative on the nonlinearity. Now when Laplacian s1 falls here you get zero because you have imaginary parts. So the most complicated thing is to deal when the Laplacian falls on this term, more specifically not in the bar and this is the most dangerous term that you have to control. But at the end of the story you start to control Hs you still get in this quantity Laplacian square. So it is not so clear how you can go down to 2 minus epsilon on the growth. But the magic thing is that here you have the bar. And this is what the bourgain spaces like a lot. The bar may give some miracle. So you have to play with these XSB spaces you have to spend some regularity in time given by B to gain something in space and you can go down with the power of Laplacian s1. What is the definition of X? So X, I'm not really an expert of this, but I can try. So you have to write this. One possibility is to take the solution the this can be adapted to general dispersive equation.  But acting with the free evolution. And the so you have B is in some sense the regularity in time, s is the regularity in x and thanks to this definition you can play a lot to push some regularity from B to s. This is the miracle that allows us in some sense to get a smaller power than 2 in x. And this is the game of bourguine. Now what we can say what is our result? Well the result that we can prove is the following. Ok, what is the the assumption? We assume that we can control l4 for free waves with s0 epsilon loss s0 can be chosen to be equal to epsilon if m2 is t2 but in the worst case we can prove in these two papers here that if s0 is 1 fourth then it is always satisfied. So it is never empty this assumption at the level of free waves. And our theorem is the following. Consider nls on 2d with any non-linearity not necessarily the cubic one no extra assumption on m2 then you have this polynomial loss. You can also this s0 one responsible of the loss of derivative in the free stricards estimate.  And this we can prove without seeing any difference between cubic, higher order non-linearity and so on and so forth and can be adapted as I told you not necessarily to the pure power case. It is just a question in some sense of integration by path as we shall see. It is a different strategy compared with the one by bourguine. And you should also say that the previous result at least in the cubic case was obtained by zonk and they also have to say that we never used the XSB spaces at least here. Probably by combining this technique with the XSB we can still improve. This is something that we are thinking about. But for the moment let me go really with the simplest thing that you can do. And what is the key point? The key point is that we don't use we introduce some modified energies. I should say that the user of modified energies is not something in you in the literature it has been used at least in those papers but even in other papers. But what we did is to combine first of all every problem as its own modified energy. Second thing we combined this with dispersion which is something not to don't be found in some sense. And how to proceed? We proceed like this. We introduced this energy which looks a bit strange I agree with you but I will do some comments in a few minutes. The leading term is the H2 norm square but we subtract some quantities which are there with proper coefficient. Because when we compute the entity of this modified energy, see what happens. We have the first time just one derivative so no second derivative on the contrary, Burgen had Laplacian to the square. We have just one derivative one derivative. Here we have one derivative and here we have dTU but dTU if you replace the equation gives you Laplacian U but to the power 1 not to the power 2 as Burgen did before but to the equation caused by this time derivative you gain one power of Laplacian and here it is where you gain the polynomial bounds. And just to give yes so, what to do? Well, notice that this energy is constructed in such a way that you have some algebraic cancellation where you compute time derivative but essentially the leading term to the square. Because in the second term you have the Laplacian U which appears in a linear way and here it does not even appear. So essentially at e2 if you control e2 essentially you are controlling h2. And so if you integrate this relation here what you get? You get that e2 but essentially h2 to the square minus h2 to the square right hand side here. And the most delicate term to be controlled at the following one you have this term and here you just use elder nothing more than that. You put this dtu in l2 you put u in l infinity because you are 1d you are 2d and in 2d essentially you can control l infinity by h1 plus epsilon and you just need to put to grad u square in l2 and this is w14 to the square. So at the end of the story what you have to control is essentially this l4 in time w14 in x and this you can do just by the dispersion you do that and you conclude the proof that does not depend on the structure of the nonlinearity, does not depend on the fact that you have a pure power and so on and so forth. This is the 2d that we did for an exercise to understand how to deal with the 3d case and just to give a very rough idea of where this energy comes from where you see the beginning is exactly the computation of bourguine bourguine said, ok let's compute dndt of h2 square so you have to compute dndt of this plus a nu l2 square this first identity is the same as bourguine also here you replace by the equation and you end up with this the point is that bourguine said ok I am enough like this so is the following this time dt we put outside you see and if you put dndt outside that can be absorbed in the energy of course you have to move dt also on the nonlinearity and you keep doing this game and at the end of the story you construct step by step the modified energy this is how this modification appears but after this cancellation the bad term by bourguine is disappeared and the remaining term involves just Laplacian nu to power 1 no more Laplacian nu to power square which was the dangerous term and of course we don't exploit too much the algebraic structure of the reminder for the reminder we just put absolute value and we put other but probably playing more with the reminder we can also improve what we do and another important thing is that this computation does not depends on if you are in the flat case or not even if apparently there are hidden some integration by parts just because this fundamental identity that is of course true in the flat case is true without any modification on every compact reminder does not depends on the curvature so the tool that you need is essentially hidden here this is how you deal with the piece of the energy but let me go further so let me come now to what we can do in the 3D case so in the 2D case every 2D compact manifold forever 2D compact manifold ever in no linearity we have growth polynomial growth what we want to do in 3D where in 3D on a genetic compact manifold first of all let's ask ourselves for which no linearity we have a global dynamic first of all and the unique case that I know in sub cubic no linearity so if you give me something which is a cubic plus epsilon and you don't assume nothing on the geometry of the manifold I don't even know if there is a global solution but in this situation you have a global solution and the point is how to guarantee that you can globalize the solution because you see assume that we start in 3D we are in 3D we take initial data H2 the local existence result comes for free just by sobolef embedding because 2 is above 3 over 2 question, can we globalize or not? which you cannot do that for free and as far as you know the first answer to these questions is by Bjerke Geradensecki and they give you very quickly their strategy in this way you can see where you gain if you use the modified energies the strategy is very simple put down the integral equation satisfied by U and you apply Minkowski inequality so the L infinity H2 norm of the solution is controlled by H2 of the initial data just by the Duhamel formula plus L1 H2 of the non-linear term and the non-linear term is cubic now let's go to the second step H2 of this cubic term can be controlled by H2 at power guard and of course you have to put this term in L infinity and you get L2 in time L infinity in x to the power 2 question how to control L2 L infinity because you cannot rely anymore on a sobolef embedding if you rely on sobolef embedding you lose other power of H2 and you cannot do that so here it comes the idea of Bjerke Geradensecki to use the dispersion so the Strikert's estimate was mentioned before and the target is the following the target is to show that L2 L infinity to the square can be controlled by this log you see if I can put here the log then I imagine here a log I am exactly in the position to apply a kind of gronval lemma that gives for free also a double exponential bound now how to control for the non-linear solution L2 L infinity you have to use the so called Strikert's estimate this version just a few comments about the Strikert's that you need to use well in general when you write down Strikert's estimate what you have to do well you have a non-linear solution in general you need also a forcing term because you have to work with the dual mill operator with initial condition equal phi and in general what you expect you expect to control some norm of v by using what by using some hs norm of the initial data which is what you expect if you don't have forcing term plus another norm that involves just f this is not the case of this family of Strikert's estimate for two reasons the first reason is that you have the project of pn and this is something that is not a serious problem because the project commutes with the equation so if you localize the forcing term and the initial data you have no problem but the key point is that here you don't take some norms of the initial datum phi you have to exploit the solution himself for f you have a norm but for the here you don't have just phi you have the evolution of course if f is equal to zero it's the same thing because h1 half is preserved along the evolution this is essentially the h1 half norm of the initial data but if f is not zero you don't control anymore in the standard way by Strikert's this is the key remark and why you have to do that because if you play like in a non clever way you lose derivative on the forcing term and it kills you so in this way you don't lose derivative on the forcing term but the price that you pay is that here you have the whole solution not just the initial data and this is technically this comes from what well it comes from the fact that when you write down the Strikert's estimate on a compact manifold when the frequency grows the times in which you have the Strikert's they come with semi classical times so what to do is to start for semi classical times and then you sum up if you sum up and then you apply the so called Kriste Kiselyev argument you lose two derivatives and you are killed in this way by this cutoff in time you are safe because you don't lose nothing here which is fundamental ok and now how do they proceed they say ok now imagine that in our case of course f is youtube because we want to apply this estimate no linear solution and what you do well you sum up, you take those estimatives since you want to control 12 infinity and there you control 126 you have to lose one half derivative you lose one half derivative one half derivative here becomes one derivative here and there you lose one half derivative and then they sum up because this is an estimate true for localized solution but you have to sum up on the frequencies so you have a sum of this term here plus a sum of those term here over capital N and you just use Minkowski on the left hand side this is where you lose a bit because use Minkowski and when you sum up you have to lose something but let me come a bit later on this point and essentially what is the estimate we arrive at this estimate here and now let's check that this is not a problem this is not a problem why well because you see in W1 half 6 over 5 I want to remove the sum but to remove the sum is free if I can lose derivative I can put one derivative here and then remove the sum and when I remove the sum you see what I can do I put a gradient in L2 and U squared in L3 and I get h1 and 6 squared which are conserved by the Hamiltonian so the unique term which is dangerous to sum up is the term coming from here when you sum up the L2 in time h1 in x of the projected pieces of the solution and how do they solve the problem well you split this sum in high frequency and low frequency part where n0 is a frequency that you will select later on you see this sum you split below n0 above n0 when you are above n0 you can gain 1 over nier if you lose h2 this is for free and when you work at low frequency you just do koši švarts you take a square root of log of n0 which is the amount of the dyadic number below n0 and then you sum up the square you continue this estimate and at the end you see where the log comes from just because you have to select in appropriate way the frequency n0 but what is the problem is that you have to sum up without the square if you could sum up with the square then you lose nothing and this is what we can do with the modified energies we avoid the usage of L infinity and the mean koši you can use just square of quantity so you can use all the power of the harmonic analysis with the squared function and now the conclusion at least for them is easy because now I have controlled L2 in time L2 in time L infinity by this log you replace there but since you have this one times the log you have double explanation now what we want to prove for 3D in which sense we can improve we can improve in the following sense we have two kind of results the first one is for cubic the same nonlinearity treated by Bjerge, Gerard and Svetikov and what we say is that in the cubic case global existence is already proved by themselves but we will prove the global existence with a better bound we just take explanation which is what you expect in some sense but you cannot prove with their strategy and the other thing that we can prove is that if you take something slightly below cubic so in the sub cubic case then you get also polynomial but why in this statement I put HM with M generic where I put H2 because if P is not enough regular then it is not so clear that higher order sublif norm higher regularity is preserved but of course in our situation we can regularize a bit of nonlinearity and still the proof is true and here you can really take HM with every M but since I wanted to focus just in the pure power case then the unique thing that we can do is the to work with H2 but still you get a polynomial growth on a genetic compact 3D manifold which as far as you know was not proved elsewhere and I don't see how the Burgen technology can work because here it is already a problem in H1 now this is something and just let me tell you why by using the modified energy we can go from double exponential so the modified energy that we use are exactly the same I have introduced before in 2D and you can use also in higher dimensions if you want so this is the E2 that should play the role of H2 but it produces in the computation of the derivative such a cancellation that cancel the worst term this is the key idea on the contrary but Patrick with his collaborator they start by putting absolute value 2ML formula absolute value and Minkowski here you don't use this you start by computing d dt of E2 and if you compute this you get this term which is not complicated the more complicated is this one because you have dTU that if you replace the equation is a Laplacianu but still to the power 1 here you have Laplacianu to the power 2 so you have a lot of room to try to play the ground volume and how to control this term in 3D well it's really very simple because simply you see you put dTU in L2 this is essentially H2 then you put this grad U square you put in L6 and you get T2W16 to the square and you remain with this extra term U that you put in L6 this is just another inequality this L infinity L6 is under control just by because you can control H1 so you end up with this quantity ok so the quantity that before for them was L2 L infinity now becomes L2W16 ok and the W16 you can deal without summing up the square not by taking Winkowski and you don't lose nothing when you sum up in some sense why let me give some details so the starting point is the same stricards estimate as before still localized at the frequency pN but before we started with the L we took one half derivative if here we take one full derivative ok you take this and what to do now I don't sum up this quantity sum up the square if you sum up the square on the N I sum up also the square here and I sum up also the square here now you see the nice thing is that here you can use planche RL to sum up by using high frequency low frequency this time that before was our enemy and now is our friend because it is for free L2 in time H3 over 2 square but here I have to sum up the squares but it is not a problem because you see when I want to sum up this W16 project I take the solution I project on the with the L2P I take a sequence and sum up the square this is L6 in time L6 in x L2 in time discrete L2 in N but since 2 is smaller than 6 I can reverse the sum ability I can put smaller 2 in time and this is nothing else that L6 of the squared function but L6 of the squared function by harmonic analysis is equivalent to L6 and so you conclude that you can really end up with this estimate the same as before by removing the project that I could do if I work with an infinity this is the the key point and once you have this now to control L2 W16 is essentially easy because you see I have L2 H3 over 2 and yeah this F becomes the non-linearity and here I just use Gallardo Nuremberg inequality I put H2 to the one half H1 to the one half this is conserved this is H2 to the one half when I take the square I take H1 but I add another H2 before so you take H2 square controlled with H2 square which is a sharp case to apply exactly grombal getting the exponential growth no need to lose more on the exponential and why in the subcubic you gain polynomial growth well it is simple because here I can work with 3 over 2 minus epsilon and if I apply the Gallardo Nuremberg inequality I take on H2 one half minus epsilon so I control H2 square with H2 to 2 minus epsilon which is what you need for the polynomial so this is nothing else than that so simple idea is to combine modify the energy with the correct stricards estimate to be careful when you sum up the square and the nice thing is that you have the correct monotonicity between the exponent that allows you to interchange L6 and the smaller 2 and then you apply just harmonic analysis result on the squared functions and you get this exponential value just one word I said everything about H2 how to deal with H1 million for example the point is that if you start to compute the modified energy associated with the power k of the Laplacian it starts to be a bit nasty the situation but what is the idea well instead of considering power of Laplacian consider power of the time derivative what is the advantage is that the time derivative they are straight so they commute very nicely when you apply Leibnitz rule and so on and so forth no need to take care of the Laplacian to to commute to the Laplacian with other terms and essentially once you control this then you have controlled this hk to sum of 2 so the correct modified energy at higher level order is this one you have to compute d d t which is not really nice it is the result but essentially you can isolate 2 or 3 bad terms and other terms can be controlled without dispersion essentially at this point you just apply older process I don't exclude that by combining now with the Bulgarian spaces you can get even better things and this is the situation I just I want to say that the unique point where you use differential geometry is this identity but you don't even use differential geometry because there is no curvature here it is just an algebraic identity that you can compute chart by chart and that's it and I just wanted to say one last slide to say how this technique we think can be used to improve further in 1d ok if you look at the result we have proved for instance in 2d of course if you are on the 1d tors I can consider every solution in the 1d tors as a solution in the 2d tors by a trivial extension so the same bound we get in t2 should work for t but it want to be a bit better if you work to our result in t2 what we did well we get here the growth t what I say is that at least for h2 we are able to prove that you get square root of t and we have some computation showing that this is true in hm for general m but you can gain order you can gain something on t and what is the new tools in 1d that we don't know how to use in 2d well as usual introduce the thing that the key point is to introduce this quantity here the capital N the j and the capital T that satisfy this couple of transport equation ok why this because all the terms that you see when you write down the derivative of the modified energy can be written in terms of j t and n and to have this nice solution this nice relation absolute for instance how does it help you see what you get when you compute the second derivative you get this term and this term for instance one trivial relation between this term here and this term here is this one you see I rewrite I like this I put dTn then you have n dx u2 that can be written in terms of j and then like this like j2 plus dx n2 then this dTn by the equation is minus dx j so je minus dx j j2 and the terms coming from dx n2 is exactly 2 so you have a non trivial relation between the first term and the second term if I find another relation you have done and to find another relation is to exploit now here I don't use the t I use the t because the t involves this dx u2 I get another identity starting from the t another system you have a system satisfied by 1 and 2 that gives you something nice that you can control because you see here the nice thing is that at the end you have a relation telling you that this difference can be controlled by at the integral of a term that involves four derivative shared on every term and once you have four derivative shared on four terms you need to get the square root of t function of course to do that for h1 million is not I mean we have some ideas but this is the way how this modified energy can be exploited in the 1D to get even better balance and I think that I thank you very much and I stop here in the case of 3D compact manifolds well of course without any other assumption for the moment there is three cuts inequalities we have but there are examples like of course teorai but also spheres or gel soul manifolds for which in fact it is possible to solve up to the quintic and in that case it is certainly possible using the bogyan spaces to get polynomial here but using your using my technique I didn't try but for sure once you have for instance if you look at the work by Svetkov, Er and Atar 1 t3 they can reach the quintic and the pausa there in your next could deal the large data case by using these UV spaces I think that you can get polynomial bounds by using my technique I didn't try but probably I didn't try yes, yes, probably just by an elder and the would think here is that that you need is not in h1 it's in h2 and all the tools of low regularity are needed just for the globalization so probably all the technicality needed when you ask for h1 are not needed it's unclear for me the technique of bogyan if it works to get the exponential bound in a generic 3D compact manifold that's what I'm saying probably but it's not completely clear unless you don't have a vector structure that allows you to use the bogyan spaces like t3 spheres I agree with you the modified energy do you have any intuition of what does it have the physical meaning for me it has just algebraic meaning just I don't see really but you see the construction is that you have to do a step more compare it with bogyan just this dndt you don't replace by the equation but you extract another term of the energy and you gain one power of the Laplacian for free without any effort no, no, absolutely no, this is the point we don't use bogyan resting all the refuel all the plus tricards so just the linear tricards


The title of today's webinar is The Impact of the Economic Crisis Caused by COVID-19 on poverty and inequality. You're still learning about the massive economic crisis generated by COVID-19. And one of the most pressing policy concerns is the implication of COVID-19 on poverty and inequality in the global south. The reasons expect that within-country inequality has increased. As most vulnerable workers, especially in informal employment, have been badly affected by the pandemic and government lockdown policies. It is less clear what might have happened to across-country inequality. May also increase as East Asia starts recovering faster than other developing regions. It's also the case that richer countries have shown some of the sharpest income decrease. However, much of the discussion that we have had on poverty and inequality has been speculative. And the evidence-based implication of the pandemic on inequality remains weak as of today. So in today's webinar, we have privileged to have one of the leading thinkers on global inequality, as is the likely impact of the pandemic on national and global inequality. You're also fortunate to have a distinguished discussant for the policy world to share insights in the policy implications of impact of COVID-19 on inequality and what the international development community can do in this respect. Now let me introduce the speaker and the discussant. Speaker is Professor Phanser Mourgenon, the Emeritus Professor of Economics at the Paris School of Economics. He is the director of the Paris School of Economics on 2007-2013. Before that, he was chief economist and senior vice president of the World Bank. Phanser is one of the seminal thinkers in developing economics, and has made many important contributions on research and poverty and inequality. He also has a long association with WIDER. He was a member of WIDER's advisory board from 2001-2008. It's a pleasure to invite Phanser back to our WIDER unit. Now let me introduce the discussant, Mattua Roig. Mattua is the chief of the emerging issues and trans-section of the Division of Inclusive Social Development of the UN Department of Economic and Social Affairs. Mattua has over 20 years experience in conducting policy research and analysis, leading to inequality, employment, international migration, and the link to development. So it's a pleasure also to have Mattua here as a discussant. Now I'd like to invite Phanser to present his keynote. Right over to you. Okay, thank you very much, Kunal. Thank you very much to WIDER and to you for inviting me to this webinar. I should say, I would say normally it's a pleasure to be back to WIDER and I'm really simply sorry that I am on my chair in my apartment in Paris rather than in a conference room in Helsinki. But this is our life is these days. Okay, so the topic for this presentation is about COVID-19 and the impact of that crisis we are going through for the last 12 months on inequality. And the way in which this crisis may have increased or exacerbate existing inequalities or created new inequality. Now, this presentation has been entitled some first reflections, first basically because it is certainly too early to say very much on the impact of the crisis on inequality. And we lack the data necessary to make any precise study of that impact. And hopefully in the coming year, we will have more information and it will be possible to get to more definitive conclusions about this decision. Instead of referring to a specific topic like global inequality or poverty, I entitled this presentation some reflections basically because I would like to look at various dimensions of inequality. Inequality is not only income inequality, it is not only consumption of wealth inequality.  It is all this at the same time. I simply wanted to look at those various dimensions sequentially in order to get a full picture of the relationship between the crisis and inequality. So let me move on with the presentation. So, COVID-19 has been presented very often these days as both showcasing and exacerbating inequality. For various reasons. First, it is expected to hit harder people who are living in precarious conditions. And despite the fact that the policy measures may have dampened this effect. Second, the lockdown is likely to have affected more low income people with irregular jobs, very often informal workers, especially in the south, but also in the north. And this is contributing necessarily to inequality. At the global level, the global economic crisis, which has been the consequence of the pandemic, has increased global poverty without any doubt. And presumably, but we'll see that a little later may have contributed to more global inequality. And finally, the human capital loss in low income countries among precarious households like the effect of poverty, the effect of unemployment, the interruption in schooling will affect future inequality and this is something that we would like to also take into account. So to early to estimate all these effects, but one can speculate to what what they could be. And in particular, I think it is important to make clear the various dimensions we want to look at through which we want to look at the inequality and must say that I could not resist the making this a critique to the last report by Oxfam on the progress and and inequality. Because it is a typical, despite the fact that this report is providing very interesting information on inequality and the pandemic. There is also a very argument which I be, which I find very spacious, which is basically to say look, a lot of inequality because the 10 largest billionaires have been able to recover their fortune in nine months, when it will take a few years for to recover their initial level of income. But of course there is a complete confusion here between the price on stock markets and the fact that the day the months of pandemic struck stock markets crushed in more or less everywhere in the world. And then they went back to some level in some countries at the initial level in some other countries are not yet there. So, this is a complete confidence has nothing to do with the standard of living of the people with the consumption expenditure of the people. So, we should be really careful in when looking at inequality issues to make sure that we're talking about something which is well defined, and to have rigorous arguments. So my presentation will be about this very quickly on each of these issues in equality of casualties, what we can say, or an equal is the cost of the lockdown and the impact of compensating policies. What happened at the global level means global inequality. And finally, if you remarks on the likely legacy effects of the lockdown and the possible reversal in the downward global inequality trend that was observed over the last 20 years or so. And then the inequality of casualties across countries. The case is extremely clear. You have here map of the incidence of the pandemic. We know that it is very often very likely underestimated in many low income countries in particular in Sub-Saharan Africa. And if we're consistently estimated, it is, there is no doubt that the crisis has struck much more severely advanced countries than developing countries in general. And one of you there is definitely not an increase in inequality in the world and on the contrary. So, this is one aspect which needs to be to be emphasized. It is not clear that we cannot say anything about the future. There is this you already today saying that the pandemic is starting to hit seriously in Sub-Saharan Africa, which has been more or less isolated, which had not not been too much hit until until now. Within country, we have something which is quite different in the sense that we have little information on the socioeconomic status of the COVID victims, but what we can do is the same kind of geographical correlation. And if we look at the incidence of the pandemic in a country or in the city, we can try to look at the correlation between the socioeconomic status of areas in our sub districts in a urban area and to see whether there is correlation. On this picture here, what you have is the incidence of the pandemic, which are the red circles and the color of the various sub districts, sub world units in London area in UK, which are dark for low socioeconomic status as described by the proportion of so-called unsecured occupations, temporary contracts into jobs, self-employed people, or we would find exactly the same kind of correlation with poverty of children. And you see that there is a clear correlation. The virus hit much more strongly those parts of the London area with a low socioeconomic status. The priori is the same analysis could be done at the global level, but as we have seen, it would give exactly the opposite result, although some people do not completely agree with this statement. The previous correlation must be taken with care because we have to take into account what was the initial level of mortality and to see whether the virus increased mortality in a disproportionate way in some areas or for some socio-economic groups than for others. And this study by a Belgian economist, the Costa and colleagues, is showing that things are definitely not that easy, not that simple. On this chart, what you have at the top is the excess mortality, therefore, presumably due to the pandemic, for men and then women on the right, 65 and more. And this is by the size of the income of those people, 65 and more, in Belgium. Many of them, of course, will be retired people, but then it will be decided in terms of the pension that they receive. And this gives me the idea about the socio-economic group they were belonging to at the time they were acting. And here on the top is that you have a declining trend, not trend, you have a declining shape. And the meaning that rich people or rich socio-economic status are less affected by the COVID than poor ones. And from that point of view, you could say, oh, this is inequality. If you look at the bottom chart, the Peace Corps is defined as the excess mortality by the existing mortality as observed in previous years, as observed before 2020. And you see that the downward-slooping shape that is on the top graph is disappearing. It is now becoming something more or less horizontal, which means that mortality due to the pandemic is more or less proportional to the mortality of people 65 and more that we observe regularly. So depending on whether we want to look at inequality as related to health and the correlation between casualties and income in absolute terms or in relative terms, we will conclude that COVID has increased or has maintained inequality or has maintained the level of inequality where it was. So this example is important because it shows that things, conclusions are not that easy to make. The next topic I want to look at is how an equal the cost of the lockdown has been and what has been the impact of compensating policies in the countries. Now, the lockdown policies have been applied universally in advanced as much as in developing countries. Coupled with teleworking where it was possible with partial employment compensation, then it was maybe compensating compensated in terms of other formal employees were not so much affected by this lockdown. And the lockdown was really affecting, as I said before, much more the informal workers who could not leave their area, their residence area in order to go to work elsewhere in the city or basically where the demand for the services simply had stopped. So in all countries, and again, this is also true, but because informal sector is less important in advanced countries, there's still more true in developing countries, informal sector has been badly affected. And because of that we may expect that indeed there was an increase in inequality because we know that in that sector, the income are lower. Now in addition, we could say that those sectors have been more affected directly by the pandemic. Now we know that in many countries, some partial compensation was provided through ad hoc public cash transfer programs. And this may have compensated or at least partly compensated the direct impact of the economic crisis due to the lockdown and the pandemic. So as I said, it's too early to estimate the impact of the crisis, but with some micro simulation exercises and some very limited surveys we may try to say something about this impact. On this chart here you have a very simple proxy for the impact of the lockdown on inequality. This is taken from study by Richard Blundell and colleagues at IFS. And basically what they did, they looked at what they call the shutdown sectors, the sectors which were most affected by the shutdown, where activity went down because of the shutdown. And they looked at the deciles, this is what you have in the bottom left design graph, they look at the design to which people working in those shutdown sectors belong. And what you can see on this chart is that you have much less people belonging to the top decides working in those shutdown sectors and presumably affected by the shutdown, then you have low income people, people in the first decide working in those shutdown sectors. So because of that, we may expect that the day we will have the data, and this will come soon because in a country like the UK or other advanced countries, you have Liverpool surveys which are taken more or less every quarter. So we should very soon start having to have some information on this. But this is a proxy or a sign that inequality may have increased because of that. You also have some people who did some micro simulation, so they look at a sample of households before the crisis pre pandemic, and they tried to figure out what could be the change in the distribution of income in that population, due to the pandemic, based on GDP forecast, based on information on an employment and employment. They did what is called now casting. Basically they have some partial information about changes which have taken place in the population, in particular in terms of employment, and they they try to apply to the micro data. So those changes which were observed are estimated at the aggregate level. And most of these studies found that the lockdown led to a strong increase in income inequality. But it was almost completely compensated by cash transfer policies in advanced countries, and not only advanced countries, even in some emerging countries. Now the time dimension is important because we should look at what is going on within the year rather than the full year which is the standard time dimension of all these analyses but I'm not insisting on this. On the other hand, it is clear that these estimations are very precise but they give some all of magnitude and they tell us something about what may have happened. The following charts are taken from a study on Latin America by Nora Lustig, and I think these are quite interesting. The solid line represents the direct impact, a simulated impact of the lockdown on the distribution. So these come percentile by percentile. So we see if we look at Argentina for example that the top percentiles have been less affected by the lockdown than the bottom percentile. This is what we're expecting when we know that the informal sector was more affected. And the dotted line would correspond to the impact of the crisis of the lockdown after compensation through the transfer policies in those countries. And you can see that in Argentina and Brazil for example, poverty has gone down, the impact on poverty has been reverted because of those policies may have been reverted I should say, because of course this is our simulation. And the inequality if we look at the case of Brazil has gone down rather than gone up. As a matter of fact when we look at the solid line, it is difficult to say because we see that both the very bottom part of the distribution and the top part of the distribution were less hit by less affected by the crisis than the others. So this I think is an interesting conclusion. Finally, if we look at a few adult panels survey which are available, we have something which is quite interesting. The come here survey has been a survey which for any other reasons started in January 2020. And then was taken in May again, and in September again. Same people, it is a panel survey. Five countries, 1600 individuals in every country, France, Germany, Italy, Spain, Sweden. And this is on average for all those. If we put the five samples together for the whole seven for the eight, eight thousand people, what we find is this evolution in equality increased in May, just after the lockdown or just at the middle of the lockdown in many countries. So definitely the lockdown had an inequality impact on the distribution, but by September. Okay, this is telling me that I should stop very, very shortly. But in September inequality had gone down, simply because compensating policies came out and made equalized the distribution at that time. I must say that I would be surprised by that, that the inequality went down, but probably will have to work more on this, but this basically say that the impact is not as simple as one could think. And also I want to mention a study on Bangladesh by same rain and the colleagues, they re interviewed people would be interviewed in by their research center in 2018. They only were able to get 5000 households or there was a nutrition of the initial panel but by 37%, but you had still more than 50% being people being interviewed. And what they found in that case is a surge in poverty, of course, this is what we know, but they observe a noticeable increase in inequality. More or less everybody was very much affected even the top decides, we are very much affected by the lockdown, less 30% drop in consumption expenditures, but the top was less affected, definitely than the bottom. And definitely there was there an increase in inequality, and this was taking into account of course the compensating policies. Now, a few words about because I'm in a hurry, let me go directly to what we can say about the COVID crisis and global income inequality. So I'll go directly to those charts, which I have done, I mean, based on calculation which I have done for this presentation. And that simply I looked at took some previous estimates of the evolution of the global inequality that I have worked on for many time had this book several years ago which was basically about about which was partly about this. And I tried to figure out what would be the impact of the crisis by using the GDP per capita forecast by the IMF. Some of them from January some other from last October and trying to see okay what has been the impact and what has been the evolution of inequality. And if we simply say that we take those figures of GDP per capita growth, we apply them to the data that we had for 2018, 19, etc. And then we see what is the resulting distribution of income within the world. This is taking into account both inequality between countries and inequality within country, except for the fact that because I didn't have any data to do that, I kept the inequality within country constant. And so there is something missing here. We are basically looking at only the impact of the between inequality change on the global inequality among old world cities and. But something is missing because we don't have the unique change in within country and equality yet. And what you can see in the circles is the evolution we see that the trend that the downward trend that we have in is essentially flattening in 2019, 2020. And if instead of comparing to the previous year, we compare to what would have been implications of continuing the growth trends observed in every country in the world from 29 to 2020. And then this is a dotted line and you see that the actual change in inequality is a different between what may what we may observe, which is the solid line. And what we could have observed if growth had remained the same in all countries. And we see there that definitely there's been an increase in inequality with respect to this counterfactual about the growth trend. And inequality basically remain constant, didn't increase remain constant. If we look at the comparison between 2019 and 2020. This is more or less in agreement with some results that Angus Deaton got doing the same kind of exercise lately, you find more or less the same thing. And that is measuring the impact of the crisis by looking at the difference between the forecast by the IMF of growth in 2020 as released in January to the forecast, which we have made a year ago before the pandemic. And then he said the difference in the forecast is telling us something about what has been the impact of the pandemic. And this is something very similar to what we have here. Okay, so let me conclude very quickly. And this is the last point. What is the likely legacy on the crisis. And the reason why I believe that maybe we're going toward the reversal of these downward global inequality trend that you have just seen on the previous chart. And the legacy effect include the consequences of a severe increase in poverty in many countries of the world and especially in the south, of course, the loss of human capital, due to the increase in poverty and the fact that there was some health problems the health of the population certainly has been affected by the increase in poverty, loss of human capital through the interruption of schools, huge increase in dropouts in many countries.  So we truly have dropouts and the impact of that would be felt for a very long time. And finally, the increase in the indebtedness of Sub-Saharan Africa. We knew this was already a big issue before the pandemic. Now those countries has had to borrow a little more during the pandemic during 2020, and the increases of the debt has been more or less 10% of GDP. And this means that one way or another, this debt is not sustainable. So we have to be done, but most likely some the growth will slow down in those countries because there will be some package some changes in policy that will be producing that chain. So all these factors will probably produce a slowdown in the growth of those countries. And because of that, it's maybe the case that for at least a couple of years, the global trend that we have seen for the equalizing of the global income might stop and we might go toward reversal and toward an increase in global inequality because the poorest country in the world will be affected for some time for some durability in the affection. I mean, in the way in which those countries will be affected and this is something which believe we must be prepared. And because of that, the main conclusion that I would get from all these we said that the help that should come from that help us to come from the global development community, especially in terms of the debt, something has to be done. It is not clear what what can be done and the situation is slightly more complicated than in the 90s in the late 90s. At the time of the hippie initiative with which reduce the debt of low income countries. So what the G7 or the G20 can do is something really quite important. We have to go beyond simply a stopping or a moratorium on the debt service. We need to really to get to reduce the debt, and this will take some time as we as we know. The other thing on which we need to have a community aid coming from the development community is of course about vaccination. And I will finish on this, as I said at the beginning, we don't know what will happen with the pandemic in particular in Southern Africa. It may be the case that and we have some signs today that the situation is worsening in several countries. Because of that, it is very urgent for vaccination to become available in those countries. And we know that today, those countries will have to would have to wait one year before really some significant vaccination program would start. So what could be done about this. This is something that was completely ignored until now. And I think that this should become a priority in the global dialogue about the development that will take place in the coming weeks, I think so. And I will stop here. I've been sorry about this. Thank you so much. It's a really interesting presentation. I mean there are several interesting points that I thought came across which will follow up in the Q&A. One was a spatial and temporal dimensions of the pandemic. That's very important. The second was I thought really interesting the U shaped effects of pandemic and cash transfers in Latin America, which is something somewhat content intuitive when question is, we see that in other parts of the developing world. Of course implications of global inequality. I'm going to now ask Marta Roy to make her comments on your presentation also thoughts on exactly questions of implications of the pandemic on global inequality. Marta, over to you. Thanks very much Kunal and thanks very much. It's a pleasure to be here. Thanks very much for the invitation. It's a pleasure and an honor. I'm giving myself a timer and I'm going to try to try to stick to 10 minutes or less. Let's see if I get to it. Let me share my screen. Okay, so on the policy implications, Francois has said that the pandemic exposes existing inequalities and can make them worse. One is what can we do about it. From our point of view here at the UN secretariat, we have presented this crisis as a tipping point as a fork in the road, a moment where the inequalities may be too large to be ignored. And something has to be done about it. So what are the implications for policy. And here I want to distinguish two different types of implications. I can't seem to have trouble moving my presentation. Can you see the next slide. Yes, it is fine. Go ahead. Perfect. I'm sorry. So within country inequality from where we stand, we, I would say, and this is my opinion, there's reason for hope. There seems to be more awareness among member states and more willingness to go for both policy options than there was in the past, say after the 2008 crisis. There is some for hope also in terms of the rescue packages that have been put in place, larger also than in 2008, and with a significant amount of discretionary spending on social protection as Francois has shown. Many of these are those rescue packages based on preliminary evidence may have been able to arrest the growing inequality trend. So reason for hope when it comes to addressing within country inequalities, at least in terms of the commitment that governments are showing inequality between countries that's more uncertain, because obviously countries are plagued with domestic problems and when countries have played with domestic problems they are less likely to show solidarity across their borders. There are some shows of solidarity that there are dead relief measures, very, very short, very focused in terms of letting countries making sure that countries are able to implement their emergency rescue packages but very, very short leave and very not very there are also efforts to coordinate the vaccine, you know, make sure that the vaccine is available to all countries but then again, you know, including COVAX for example, but then again what we are seeing as the there's almost a perfect correlation between GDP and the availability of vaccine and the speed of the vaccine roll out. So for now, there isn't, we don't see that that anything is changing in terms of addressing inequality between countries. When it comes to inequality within countries, there's increasing commitment but what happens after the rescue packages are withdrawn. Of course, rescue packages are short term measures, one of measures and there's a risk that once again they will be withdrawn too soon. There is increasing number of voices calling already for a move to fiscal austerity as soon as possible. What happens after these packages are withdrawn and here I want to just give a very, very basic message that we have repeating for many years but I think it's worth repeating is that policies matter. Well, too often the discourse on inequalities is rather fatalistic in the sense that growing inequalities are presented as an inexorable trend that results from technological innovation or global integration or in this case a pandemic, as if nothing could be done about them. It is clear that something can be done about it inequalities, the both levels and even trends of inequality are very different among countries, even within the same region, even at the same development level, even equally exposed to trade. So there's something one can do to arrest the growing trends of inequality. This is the first point I want to make. And policies matter for good and for bad. Within the last decades, for example, many policies have been actually counterproductive when it comes to arresting the increasing inequality trend, or have just done nothing to arrest it. Let me give you some examples. One is that redistribution taxes and transfers have failed to go to curve the trend towards growing inequality. This graph presents a very rough indicator of redistribution, which is the difference between the genicoefficient of market income shown in orange and genicoefficient of disposable income. Again, a very rough measure of distribution that are better analysis. Francois mentioned the commitment for equity initiative has better analysis, but in any case this rough indicator shows that it's not that policies have become less redistributive, or in some cases, yes, the tax system has become less progressive over the last decades, but it's just that they have not become increasingly redistributive as inequality in market income has grown. So that's an example of a policy that has not helped. There are others. Public services, for example, on average has increased in healthcare access to healthcare, the cost of education, the cost of housing as well and this of course affects families that are poorer. There's growing insecurity that's very important there's growing security and precariousness in the world of war changes in the world of war have actually pushed inequality upwards as well. More importantly, there's a growing disconnect between labor market policies in this in institutions, and the changes that are taking place in the world of work in the sense that we have increasing precariousness we have a growing number of people working on their non standard contracts meaning temporary contracts, zero hour contracts, including in the growing gig economy, we have persistent informality in developing countries. And yet we have labor market policies and institutions and even social protection systems that are still designed to for labor markets where most of the working population is in the formal sector and working on their standard contracts. For example, many social protection programs are still tied to a contract with specific employer, other social protection programs, social insurance programs specifically are tied to salary employment. Many are not portable between different jobs, many are not portable across countries so there's a disconnect here. There's a disconnect. And these disconnects, we argue, together with growing inequalities has very negative impacts, very obvious impacts, including on deaths from COVID-19 that could have been prevented but it has also negative political impacts. I'm just looking at the time. Do I still have a couple of minutes. Is that okay? Yes, okay, that's fine. So let me focus for a second on the political, on the negative political impacts of growing inequalities and the disconnect between policies and the reality of today's world of work and beyond. The important thing is declining trust in institutions, increasing this content among the population. Here I'm just showing the example of the United States where trust in government, people who trust governments highly, has declined from the late 1950s to 2017, this is when we have the last data from the Pew Research Center from 70% to 20% and this is also the case perhaps not as deeply but it's also the case in many other developed countries and trust remains low in developing countries. This is a consequence of inequalities but it also poses an obstacle to addressing inequalities because it disenfranchises part of the population, it creates this engagement with public mandates, it makes it therefore harder to implement public mandates to make those investments that would help addressing inequalities in the same in the first place so it creates a vicious cycle and it weakens the social contract. It is in this context that the Secretary General, in a speech that he gave last summer, talked of a new social contract for a new area. His speech was aspirational and then so he gave a list of policy outcomes of a possible new social contract. I'm not, I'm just gonna not going to go through them because I think we all know what these policies are. Some have to do with promoting equal opportunity access to quality education, healthcare and so on other have to do with ensuring equal rights, by putting prejudice and discrimination. There's also some space for redistribution. All these policies of course are linked to each other, promoting equal opportunity is important but one needs tax revenue in order to promote equal opportunity. The issue with policies to reduce inequalities not so much that we don't know what needs to be done, that there's not technical advice to know what has worked and what has not worked in different country context, or even that there's not enough capacity administrative capacity to implement some of these policies, is that there are important obstacles to implementing some of these policies and I think it's important to understand what the obstacles are, in order to be able to mobilize support for these policies. Very quickly, challenges to many of these policies one has been with us for a long time is that many redistributive efforts face opposition for by very powerful sectors of society, this is not new. Again, what's perhaps new is that we have the class coalitions of the past, the class coalitions that supported strong redistributive agendas are do no longer exist. And there are still other working class that supported big new deals in now developed countries doesn't exist and the middle class is shrinking. So we have a change in coalition right now. We have part of the working class that feels excluded that objectively is excluded but also feels excluded and that is important because part of this working class is providing support to fringe political movements that do not necessarily support redistribution. So again, this is not to end on pessimistic note but just to say that it's important to understand the obstacles with the obstacles we face that may weaken the social contract as well in order to think of what to do, and how to convince government and their constituencies that addressing inequality right now is important. In the first half time I was going to say that the secretary general also talk about a new global deal. The need to strengthen multilateralism is a COVID maybe a moment where we succeed in reinforcing global solidarity I was going to speak about tax cooperation and a reform of that architecture but maybe we can talk about this during the break. Thanks so much. And I'm sorry. Thanks so much. Yeah. Now I think what I think is so important what you said was that even before the pandemic hit. We were not doing very well on on sg 10 on inequality. Right. And the pandemic is actually brought into shop of focus our failures in addressing the quality, even prior to the pandemic. So I think that's a very important point and the question for a new social contract is obviously something that one is to think about. So if you have any questions in the q amp a that I can see. Let me ask the first three I can see here are questions from Carlos gratin, Ricardo Santos, and a very interesting question of Marcus excerpt.  So what I suggest if you can take this week, there is sort of questions all together and please be focused and be to the point. So Carlos, do you want to go ahead. Thank you very much. I was thinking about if you have any reflections on the impact of the pandemic on horizontal inequalities. I think in mostly in terms of age so the pandemic clearly has affected mostly the elderly, especially those living nursing homes, which at the same time could be aggravated by the failure policy failure, you know, managing the crisis in these institutions, but also thinking in terms of the impact on ethnic minorities in some countries for example in the US or immigrants temporary workers, etc. Thank you very much. I think you had two questions you want to go ahead. Yes, I'll try to be quick. Okay, thanks. So, first of all, thank you so much for the presentations and for so it's a pleasure to talk with you again from Mozambique. I would see a new presentation that like it's an excellent presentation I would see some possible caveats in particular on the possible muddle the waters on global inequality effects of COVID. So for one on the instance of incidents of COVID. So if poor countries are younger, i.e. a higher percentage of the population are of ages with lower COVID morbidity. COVID incidents may be lower than average Chetris part of us, but then the elderly in those poor countries may be relatively worse off than elderly in richer countries. And when we look at impacts of lockdown, again, if if poor countries are more rural, i.e. a higher percentage of population lives in rural areas, and rural areas are more sparsely populated and this is necessarily not the rule in all global countries. Then lockdown effects might have been mitigated again, all other things considered. But the urban residents in poorer countries that implemented lockdown may have suffered more and may have had lower access to social compensation measures than urban dwellers from richer countries. So what about vaccination? One could arguably suggest that countries with a younger and mostly rural population again is sparse may manage to normally overcome COVID with a lower rate of vaccination. But what can be the medium and long term effects of an effective effort to reach herd immunity in the global north, and the ineffective effect of reaching the same type of immunity in the global south. So this last question is probably more suited for epidemiologists than for us economists, but they would probably there would be probably an impact on social economic inequality nonetheless. Over. Thanks. Thanks Ricardo. Marcus, if you're online, do you want to ask a question? Yes, thank you very much for the enlightening presentations. Professor Bogeon, I have seen professors Dayton and Milano Beach, explaining that Chinese stronger growth rate inverted its impact on global inequality. Now China is rich enough to increase global inequality when it grows more than everyone else. Does your final result come from that too. Thanks Marcus. I'm going to have one more question coming in if that's okay for Francois and Marta, and then you can answer these questions. Giovanni Balancisi, do you want to ask your question I've admitted to you. Yes, can you hear me. Yes, we can go ahead. And thanks for the excellent presentation. My question just wanted to point to one of the issue on how much do you think that the evolution of functional inequality matters in determining the long term trends that were presented, especially within country, and then related to this, to what extent do you think that COVID could affect also the functional inequality, particularly if we consider the weakening bargaining power of workers and the strengthening bargaining power of certain lead firms, especially in the digital world that continue to grow and record increasing profits, despite the crisis. Thank you very much. Thank you Giovanni. Francois and Marta, do you want to start within Marta respond to the sale of some very interesting questions. Franco, would you want to start. Okay, thank you very much for the questions and thanks Marta for your, your comments I mean, listening to you and many ideas came to my mind but I'm afraid that we won't have time to go over all of them. Let me start first with the question that we just heard on Carlos about horizontal inequality. I'm afraid that we, again, I mean in terms of economic inequality. We don't know very much when we know about health, or let's say the mortality due to COVID we know that they're very well that there is a big difference in terms of ethnicity. The age of course, this is one of the characteristics of this pandemic is to affect older people, but for the impact of the lockdown it's not not completely clear. I believe that one thing is sure. It is a fact that in terms of age the issue is not so much older people versus the rest of the population. It is about younger people and the rest of the population. The people who have been who have been most affected in developed countries has been the young people precisely because they were somewhat part of the informal sector. All those people were working in order to survive, in order to pay for their studies. We're working in restaurants, bars, having this kind of informal part time job, suddenly because of the lockdown didn't have anything to rely on. And the situation for many of those people has been really dramatic. Moreover, the rescue package which went through the official channels of redistribution policies very often could not reach those people because they were not in those official channels. So for all these reasons, I would say that in terms of horizontal inequality, the young people have been the most most affected in many countries. And because ethnicity is also discriminating, I mean, because ethnic people or ethnic young people tend to be also in the informal sector, more or less in all countries, I would say that there is also an impact on this. On the question by Ricardo about the fact that we should take into account when referring to developing countries, the fact that the rural area, the rural sector is more important. That's true. Because of that, you have a more dispersed population because of this virus may not have been as the impact has not been as severe as it could have been. This is true. In terms of the lockdown, remember that in many countries, we saw a reverse migration process. And the reverse migration process has affected rural areas because look at what happened in India, for example. Very, very soon after the day decided about the lockdown, many people left the big cities to go back to the villages. But this means that in the villages you had many more people to share what was available in those villages. Those people have also been affected. But it was a consequence of the lockdown in the city. And this is a phenomenon which I believe has been quite important in a few countries, but which were not able to analyze very well. Moreover, because it's not sure that in the future will be able to do that, basically because the information on this temporary reverse migration might not be available. This is an important point to take into account. And what you told about the fact that in the future, things may not be that bad in those countries. Things may depend very, very much on the mutation from the fear of the virus. We know that some variants are much more aggressive than others. And in particular, these days, the concern about many countries in Sub-Saharan Africa is precisely the fact that there seems to be an increase in the aggressivity of the virus due to the fact that the variant is completely different. And if this is okay, then the problem may be very serious in the future and vaccination may become a very important need in the future. Remember also that the new variants seem to be less discriminatory with respect to age. So your point about less people are younger in developing countries, which is absolutely correct, may not be as important anymore in the future if the variants are attacking or are being, are hitting much more severely people in middle age at middle age. In about Marcos point about China, as a matter of fact, I am the one who suggested to Angus Deaton that he should look at China and probably because in initial version he said China has contributed to equalize the distribution. And I told him, look, the problem today is that China has been reaching a level which is above the median in the global distribution and because of that, China's growth is today contributing more to inequality than to equality, which it did for many decades before. So, but in with my data, I find the same thing as Deaton and Milanovic. Basically, what is interesting is the fact that Deaton, I've not seen what Branko Milanovic has done, but Deaton is using the, as a measure of inequality is a standard deviation of logarithm. When I'm using the standard deviation of logarithm, I find the same result as he did. If I'm using the genie coefficient or the tile coefficient or other, I don't find this, this change, this specific effect due to China. I would suggest that if we were to look at Lorentz curves of those distribution, most likely it cannot be said that the Lorentz curve is shifting upward or shifting inward. This means that the Lorentz curve is crossing or crossing each other and the different inequality measures will give different responses to the question of increasing or decreasing inequality. To tell you about the functional distribution, you're right that in the past, the evolution of functional distribution in many countries has been in favor of capital and this contributed to more inequality. You have to remember that there is a big difference between market income inequality, where the functional income distribution plays a direct role, and the inequality of disposable income, or the inequality of equalized disposable household income when we take into account the distribution system. When you look at these developed countries today, and it is more or less a kind of picture that Marta has shown, you see that the increase in inequality of income after all the distribution taxes and benefits has not been that big. And as a matter of fact, we know when we look at averages that the increase is due to a very reduced number of countries, in particular the US, but in most European countries, for example, the distribution of disposable, I equalize the possible income has practically not changed. So this issue of the functional distribution is important when we look at market income. And in other words, when we ignore the impact of distribution. Now at the global level. I'm not clear about this. And is it possible to say something about the COVID. I'm not sure about this. It is too early. Now this is the last remark I want to make and must say that listening to Marta this came to my mind. There is something which I completely ignored in my presentation and this is really a big problem so I apologize for that. When talking about the legacy of the crisis. So let's just look at what will be the exit of the crisis. What will be the exit of the lockdown situation in which we still have many, we still find many countries, or at least a few, or this kind of measures which have been taken. What will happen when in those countries, the rescue package will progressively be dismantled. This is where the issue will really start. This is when we will see problems in the labor market. This is when we will see unemployment going up. This is when we will see firms, which have been able to stay alive during the crisis because they are supported by the policies, state policies, suddenly those firms will go bankrupt. This is true both in developing as in developed countries. So, the big issue from my point of view will be beside them. Maybe it will be at the end of this year 2021 maybe it will be at the beginning of the 2022. Probably the big impact of the crisis will be at that moment and then my gut feeling and my almost my conviction is that then inequality will definitely be increasing within countries because it would be very surprising that the distribution of the policies which have been put in place during the COVID might be maintained as such for a very long time. I mean the cost, the cost is really enormous and at some stage there will be an issue with the indebtedness of those countries. So, this is something which is missing in my presentation. Again, I apologize for that because this is certainly very, very important point for what will happen within countries and what will happen at the global level. And I stop here. Francis, thank you so much. I think that's a really important point that we still don't know yet what the impact of the pandemic will be when the state starts withdrawing support that they have provided to workers. So that's something we'll have to see the next year or so. Marta, I wanted to ask you, do you want to respond particularly in the questions on horizontal inequality and also on weakening bargaining power workers? Horizontal inequality because I know that you and your division will be working on this issue for quite some time. And on the weakening bargaining power workers, I think this is linked to the question that you raised was the social contract. You want to go ahead? So on the horizontal inequalities, I don't know, I can say better than Francois if they are going to increase or not. But policy-wise, what matters is what the cost of these different impacts of COVID on, say, different racial groups, what the cost is. More in the informal sector because of low income because of lower access to better education, the digital divide and so on. Or are they more affected because of discrimination? Too often in the press there seems to be an assumption that any racial differences in the impact of the crisis are due to discrimination. It's not so sure. I haven't seen any studies that control for everything else for the socioeconomic status of these racial minorities before making assumptions. But the policy implications are very different depending on the root of these differences between ethnic groups, for example. Just one thing though, there was a question on vaccination and global integration and what happens if populations that are younger and more rural reach herd immunity without necessarily being vaccinated in the south. One issue that may be affected is obviously international migration.  Many people will not be allowed to migrate or even travel anymore. That's a good point. So that's something that may stay with us. The issue of a new social contract is complex, but that I think is precisely why the Secretary General put on the table the need for a new social contract precisely because the different stakeholders don't have the same power that they had 50 years ago anymore. Because workers don't have the same power, don't have a voice because they are not unionized and unions are not fit for today's challenges. That's what was on the table. How do we give these workers that have lost their voice again? And that may help also regain trust in government and so on. I don't have an answer, but I think it's in everyone's interest that people who have lost their voice be given a voice. And I'm not answering the question here. I think it would be a much longer discussion, but just some reflections. Thank you, Marta. Thank you so much. Thanks to Francois. We are already six minutes past the hour, so I'm going to bring this webinar to a close. We had some really great questions and very interesting presentations. Just want to remind those who are watching this webinar that the next webinar will be on the 23rd of March. And that will be looking at the effect of the pandemic on a specific set of workers, government workers in a particular country context Bangladesh. You can find further details about this webinar, the next webinar on our website. Thanks so much, Marta. Thanks so much, Francois. It was really interesting. And I think the question, I think the way to think about this now is that we need to get better data really to have a better sense of what has happened to inequality, both within country and across countries. That we may have to wait for a bit longer because that's not going to happen. The data will not arrive right away. But I think it's important to get a better sense of what's happening because at this point it's still not very obvious, especially across country inequality as you mentioned. So that's going to be really interesting to see exactly what we see from the data when we start getting it the next few months. Thank you so much. And thanks to everyone for listening in and thanks for the great questions. Bye-bye. Thanks so much. Bye-bye.