Thank you, Camilo, and thank you the organizers for having us here. It's really been a pleasure and also very productive and because of that I changed the title. So, um, sorry for the change. This is all joint work with Tristan and Nader who are both here. So there will be some overlap with Camilo's talk tomorrow, but I hope not too much and you should really view of these talks as complimentary of each other, I hope. So we're going to deal first of all with the Euler equation. Incompressible, density one, etc. So this is 3D Euler for the velocity field U, P is the pressure. Although we're going to talk about turbulence and turbulence is generated at the boundaries, the statistical theories of turbulence deals with the behavior away from the boundaries. So for all practical purposes, we might as well consider a domain without boundaries and for throughout the stock, the domain is T3. And, okay, the conservation law that we have for 3D Euler is the energy, the kinetic energy, and oh, by the way, you can put force if you want, if you want. And if you can compute the rate of change of energy in 3D Euler, you get d by dt of E, well, is equal to minus U grad U inner product U minus grad P dot U. And if you want to include the force, you can. And because of incompressibility, meaning the vector field is divergence free, you can move this out. You get the divergence here. Here you also get a divergence. And if you're allowed to do this integration by parts, you get either the energy balance or if there's no force, energy conservation. So this is if U is smooth. So the on-sagri conjecture deals with how smooth you have to be in order to do this computation. And, okay, it's not very complicated. We are dealing with weak solutions. So weak solutions just means you're solving this in D prime of time cross the torus. So in particular, you can test with smooth objects. And in particular, you can test with Sj squared of U. Sj is the projection liter with paleon frequencies less than 2 to the j. And if you do that, then you can do d by dt of the Sj U L2 squared, 1 half is equal to what is called the energy flux. The flux through the boundary of this Fourier shell, and that will be equal after integration by parts to U tensor U gradient Sj squared U plus the pressure if you want. But the pressure disappears between because U is divergence free. Sj squared U is divergence free as well. So this disappears. And if you want, you can put this maybe for a lot of the talk, we'll ignore the force. So the question is, when is this zero? Well, you can put one of the Sj's here. And you can also subtract this. Why am I allowed to subtract this? Because this is zero.  So I have no problems in showing that this object is zero. And this is called, oh, plus the force. And this is called the energy flux denoted in some papers by capital pi sub j. So this is the energy flux through the shell 2 to the j. So now you may ask, what is the bound on this? And I'm not doing this in very chronological order. But it was proven in a paper of Konstantin Cheskidov, Friedlander and Schwittkoi in 2008, following an earlier idea of Konstantin E and Titi from 94, eink, 94 at the same time, Duchenne-Roubert. But what they proved is that pi j, and by the way, this has nothing to do with Euler. This bound I'm showing here is for any vector field, right? It's less than summation over i, bigger than one, 2 to the minus 2 thirds j minus i, 2 to the i, and here's the litterwood-Paley piece, delta i u l3 cubed.  Just the usual. So now it's very clear. When does this go to zero? So you can ask, when does this, as j goes to infinity? That would correspond to energy conservation, right? The flux vanishing. And well, the answer is here. This, you recognize, of course, is the best of space, b 1 3rd 3 l3 in time. And if you just have a supremum over the litterwood-Paley pieces to be bounded, right? If you just have this, then you will get that the limit is finite. You will not get that it's zero. To actually get that it's zero, you need that this litterwood-Paley pieces actually go to zero. And this is the result, the main result of, just kill of Constantine alphabetically replaced. Yeah, sorry, I cannot spell. Just kill of Constantine Friedlanders-Schwittkoi. This is the result that if the weak solution of Euler belongs to this, then energy conservation. But this is really the reason why, because of this flux estimate. And the flux is really the important thing, because it's really what connects Onsager theory to Kolmogorov theory. That's the only reason I introduced it. Now, of course, this is the last in the sequence of papers. And I have mentioned Eing, 94, Dushon and Robert, Constantine E and Titi, who have slightly weaker assumptions. In particular, in this paper of Constantine E and Titi, you can do 1 3rd plus epsilon 3 infinity, right? So what is the Onsager conjecture about? It's basically about the sharpness of this bound. Is it true that for weak solutions, for which the little pearly piece is 1 3rd 3, don't go to 0? So for instance, 1 3rd 3 infinity. Is it true that energy is conserved or not? So we could state, this is definitely not how Onsager in 1949 stated the Onsager conjecture. And I think Camillo will discuss this subtle difference. So we can rephrase the Onsager 49 conjecture as it has a rigidity part. And the rigidity part is exactly this. If the weak solutions are as smooth or smoother, then they conserve energy. That's rigidity. So smoother, whatever that means, than L3 in time, best of 1 3rd 3 C0 implies rigidity. Just to be clear, it does not imply uniqueness. It implies energy conservation. And B, the opposite of rigidity is that that given space less smooth than that, x, which is whatever less smooth means, L3 D1 3rd 3 infinity, there exists weak solutions of 3D Euler in x, which do not conserve energy. So that can be stated as the Onsager conjecture. And there has been in the past years tremendous progress on this question. It is currently still open, but there has been tremendous object, especially by the program started by the Lelysynsek Eheidi. So I will mention that in a bit, but before I do that, let me erase Euler from the board and write Navier Stokes and say a few words about connections to the Kolmogorov 1941 theory of turbulence. So Navier Stokes in 3D, I should not have erased the equation. It's very important that we put in the force now as opposed to Euler. This is a theory about turbulence, about the long time behavior of solutions to this equation. And of course, if you remove energy, you have to put energy back. So think of F as a stationary in time object, stationary and smooth. It's supported at, let's say, supported in frequency for all we care about in B1, whatever. It's at low, medium scales. Okay, so if you check what the energy of Navier Stokes, what happens to it, d by dt of one half u nu L2 squared, well, you can actually do the same computation. So I actually should do Sj here, but let me not do it. You will get also the limit as j goes to infinity of pi j nu.   Energy is definitely going down, thank you. Now as opposed to Euler, you can show for Navier Stokes that this goes to zero under much weaker assumptions. You cannot actually do it for Euler A half, it's not enough. You need a bit better than Euler A half. And I think Schimbrot has the best result. I really don't want to insist on this, but I think the best result is this, LpLq with one over p plus one over q less than a half and q bigger than four. This is due to Schimbrot in the 70s. So I think this is the weakest result, saying that this is zero. So in particular, if you wait for this solution to become statistically steady or at least to reach a place on the attractor, because you're putting it force, which is very smooth, the solution will inherit smoothness from the force. In particular, it will obey this, and therefore this is not here in Navier Stokes. So if you're looking for a solution which has lived forever, whatever statistically steady means, then in Navier Stokes you have d by dt of energy, energy dissipation rate, and of course what you put in. So what is the Kolmogorov theory about? So the Kolmogorov, the main assumption of this religion, is that if you call e nu to be the energy dissipation rate due to the dissipation due to the aplosion, will you take an average against what? Well, if you knew you had a unique ergodic invariant measure and a strong law of large numbers, it would be just the average against the unique ergodic invariant measure. If additionally you would have a strong law of large numbers, you could up posteriorly deduce that this is just a long-time average, let's say. So whatever this means, this is some average encoding statistical stationarity. If you want to think of long-time averages, that's okay. And the main assumption of Kolmogorov is that the lim inf nu goes to 0 of e nu is some number epsilon, which is strictly positive. So this is one of the fundamental assumptions of the Kolmogorov theory of homogeneous isotropic turbulence. Homogeneous meaning it happens everywhere. Isotropic meaning it happens in the same way in all directions. So in particular you should keep this in mind. It's not some kind of sparse thing. It's really filling everything up. Okay, so the question is, is epsilon related to Onsager? The answer is of course yes. And the relation is that, so you can show, you can do a formal argument on statistically stationary solutions, is that if you would have a statistically steady solution of Navier-Stokes and you could prove, you convert this weekly in L2 to a statistically stationary steady solution of Euler, then you could prove that actually epsilon, well, these are a lot of ifs. So the word prove with a lot of ifs. I'm not sure what the value of that is, but this should be exactly infinity. Thank you. So this is one of the main connections between the two theories. And I predict, assumed by Kolmogorov, is exactly what makes this not conserve energy. And again, this is derived typically in physics textbooks under the assumption of statistical stationarity and so on. Okay, are there other connections? And the answer is yes, of course, much finer in some sense, or as fine. And these have to do with structure functions. And this is of course related to the stock more than anything. So what are structure functions in this Kolmogorov theory? They're just, you take a vector L, you look at increments of the velocity field by L. If you want, in a lot of books, let's say in Frisch's book, you only see the longitudinal structure function when you look at things parallel to the vector field L. This is a scalar function. You can raise this to the power P, you can integrate this, and you can take a statistically stationary average, and you can call this number SP, whatever, new, off. So these are the structure functions, the P structure function of Kolmogorov. And what is observed experimentally, as far as I understand with tremendous error bars, so I will not spend too much on time on this, you observe that this is proportional exactly to minus 4 over 5. I hope I didn't screw this up. What does this mean uniformly as new goes to zero? 4L inverse in the inertial regime. L inverse is a frequency scale, and L inverse has to live between the frequency where you put in energy and the frequency where dissipation kills everything. So this is supposed to hold in this regime, and this is called the Kolmogorov four-fifths wall. So now, why did I put this on the board? You should look at this, and you should look at this space, and you should say, ah, it smells a lot like the same thing. You have power three, and you have to divide an L into three pieces. So if you divide by L, you exactly have a velocity increment with exactly L to the one-third raised to the power three, L to the one-third raised to the power three, and this is bounded, does not go to zero, becomes like epsilon. So this is a very, very deep connection between these two. I will not discuss the number four over five at all. Proving this, proving the on-segure conjecture are in some sense the same thing. Except, the on-segure conjecture is about Euler, and this is a statement about Navier-Stokes in the vanishing viscosity limit. So from that point of view, currently we have no hope on this, we have some hope on that. Now, in the theorem I will mention, the joint work with Tristan and Adair, we are actually not able to prove this. We are able to prove something about S2. So again, the Kolmogorov theory predicts that this should behave like a constant, epsilon to the two-thirds, L to the two-thirds. As far as I understand, experimentally there are deviations from this law. This is called intermittency. This is super robust. This is as far as I understand, not as robust. And this is, of course, exactly the same. If you translate this finite difference thing to Fourier, which you of course can, it's exactly to saying that the energy flux to the wave number k, which is exactly d by dt of the projection on less than k, UL2 square, sorry, behaves like epsilon to the two-thirds k to the minus five-thirds. So the famous five-thirds spectrum that we hear in all the physics books is the same as the second order structure function. And this is related to what we're going to talk about, because this has spatial integrability 2, not 3. So with this introduction in mind, let me state the theorem. Ah, I cannot state the theorem yet. No, sorry. I should tell you a little bit what is known. So there have been a lot of progress on part b of the on-sector conjecture. If I just tell you the theorem in absence of the context, it will not make any sense. Camillo, I'm sure, will review this and give a lot more details than I am, so I'm going to go on purpose a bit fast on this. There have been works of Schaeffer, 93, I think, Schniermann, 2000, where they prove that there are weak solutions of 2D Euler in the space L2, which do not conserve energy. There are many of them. Transfer the C from Schniermann to Schaeffer, actually. Thank you. And these constructions are in some sense wild. They seem out of the blue. And then the first time this was put in a rigorous framework, and this is the framework of convex integration, is in the work of Delelis. I should make sure not to spell this one, but this one is hard. Delelis is in Saqehidi in 2009, where they use convex integration to construct weak solutions of 2D Euler, which are bounded and do not conserve energy. And the beauty of this was not just, okay, you improve x, but finally put this into a context related to the Nash isometric embedding problem and so on. It doesn't seem out of the blue anymore. Now there is a serious obstruction at L infinity to go past it. There's a very serious obstruction because of high, high interactions. And Delelis and Saqehidi, a couple years later, were able to successfully use special solutions of 3D Euler, special stationary solutions called the Trami flows, to push this to C1 tenth minus in spacetime. And for a while this was the quote on record, and since this work there have been three more significant improvements. By using more structure of the equation, in particular using a slightly more carefully material derivatives, at the same time almost, I set and Buckmaster Delelis and Saqehidi, 13. So again, by keeping more of the transport structure of the equation, they have improved this to C1 fifth minus in spacetime. And a lot of, it seemed like this required a completely new idea to go forward, and indeed it does require a new idea. The Onsacre conjecture, the Kolmogolov spectra, are not about Euler spaces, they're about L3, L3 integrability. And with this realization, Buckmaster, in his PhD thesis, when is the thesis, one also at the same time, he constructed solutions which are C1 fifth, with the additional property that almost everywhere in time, they're C1 third. And almost every time. Now, this almost every time is not so nice, it's not an integrability, but this group of authors, so I will just draw an arrow, have improved this to L1 in time, C1 third minus. And to date, this is the world record, quote unquote. To achieve this, they had to basically explore the full power of time intermittency, if you want to make a connection with turbulence, and I'm sure Camilo will mention some of these works at least. So, what is the main theorem that I'm going to talk about? I'm going to put it on this board. And again, this is joint work with Tristan and Nader. There exist weak solutions of 3D Euler, which live in the space C in time, H1 third minus in X. By H, I mean the sobolef, the L2-based sobolef space, which do not conserve energy. Okay, so a couple of remarks are in order. First, this is precisely up to epsilon, of course. There's a minus here. This is exactly the Guamangorov spectrum. It's exactly this. Except it's for Euler, it's not for Navier-Stokes, right? It's a major difference. So, Guamangorov spectrum. The second and rather more curious remark is that if you compare this result with this one, and you freely use the power of interpolation, so again, so Buckmaster, Delelis, and Zekeli Hidi prove that one third of a derivative belongs to L infinity in L1 in time, L infinity in space. That's basically what they prove. What we prove is that the same thing, but we have L infinity in time and L2 in space. Now please interpolate these spaces. You get exactly R3 and R3. These are not just some random spaces. So if you just do interpolation, because it does not mean that there's any intersection between the two subsets of solutions, that's why I put a dotted line just to be clear. The two subsets, the infinitely many, uncountably many solutions constructed there and the uncountably many solutions that we construct may have the empty set in common, and they very likely do. So this dotted line will be, I guess, the scope of another paper. And then we can put the on-sector conjecture to rest. So I will try to sketch a little bit the proof in the remaining 20 minutes. So I need to erase some boards. I will not be able to fully compare our proof with the previous proofs because it would really take many hours to describe the previous proofs. But there's the advantage, of course, that Camilo will mention some of these previous proofs before. So I will freely point out some differences, which may seem a bit random at first. Okay, so the proof is based on the same idea of using a convex integration scheme. So what is the main idea of a convex integration scheme? You approximate the solution of 3D oiler with some vector field, whatever. You make a guess. So you make a guess vq. And you're making also an error, rq. So this is your guess. This is your error. And these will not solve oiler. They will instead solve the oiler Reynolds system. Ah, I don't want to write the pressure. So this is the array projector on divergence free functions. So you're not solving oiler. Solving oiler would mean putting 0 here. You're not able to do that because you made a guess. It's a random guess. So your goal now, the goal of the game, you know you have some information on this. And in particular, you know that this lives at certain frequency in Fourier. So this lives at frequency lambda q. Are you projecting both for the divergence or? I guess there's no need, right? Maybe I should have written the pressure. Because I think these errors will propagate. So I should write pq in the triple. And these approximate solutions that you have made leave us a certain frequency lambda q, which you should think of as a huge number to the q. They're not. They're actually super exponential. But for the purposes of a talk, I think it's fair to say we can think of them as being exponentially growing and q will grow to infinity. So what is your goal? You want to add a perturbation to this approximation. So you want to define vq plus 1 is vq plus wq plus 1 a perturbation in such a way that adding this object makes this error that you have made smaller. You're adding a correction at the higher frequency and you will therefore obtain a new solution, an approximate solution like that, which lives at the higher frequency. You'll see why. But hopefully this in a certain norm is much less than that is a certain norm. If you can continue to do this in the limit, you will have zero in that norm. Now if in this process you also manage to prove that the sequence of vqs remains bounded, then you will have a theorem. Otherwise, I mean you may correct it and then the vqs you make them so large that they don't converge to anything. So you have to do two things, decrease the Reynolds stress error and make sure you're not adding two wild things. So just for the purposes of notation, let's say beta is a number less than a third, beta is exactly whatever this minus means. It's the regularity that we get. And let's say the error we have made in L1, okay, you should notice something. This is an L2 and I'm all of a sudden writing L1. The Reynolds stress is quadratic in the velocity field. It's like v10 or v. So in some sense, you have to put it in L1. You cannot afford any other space. And let's say the error we have made has a certain size. And because of the regularity beta we want, let's say it has size lambda q to the minus 2 beta, which instead of writing this object, we're going to write delta q, plus 1. Okay, so our goal is that this one has size lambda q, delta q plus 2, which is smaller, right? Because this is a negative power, this increased, so this is smaller. So this is our, be our goal. That's how you can close the induction. By the way, I'm skipping a ton of details. Now, what is the equation obeyed once you have made this? And we can write this. So here you have the old Reynolds stress.  And then you have two more errors. Once when this material derivative will fall on your perturbation and once when your perturbation is stretching the previous vector field. Okay? So following the notation introduced by Camillo and Laszlo, we call this the oscillation error. You should think of this as high, high interactions in frequency. Nothing more, just high, high. This we call the transport error for obvious reasons. And this is the Nash error for reasons that, in the asymmetric embedding problem, you have a very similar term. So now the goal is to rewrite this whole right-hand side of the equation as divergence rq plus 1 and estimate the size. And hopefully it is of size delta q plus 2. So let me say that lambda q is approximately lambda zero q. So in the dyadp, so in distance 1, or how wide of a range you're allowed to be. So vq, because it is a sum of many pieces, lives in a ball of this size. Each of the corrections live in a shell, in a little bit palier. That's a subtle thing I will mention it in the proof. It shows up at the end. You don't think of the whole dyadp. You may have to cut it. Yes, you may have to cut it. In all these previous constructions in fact it's isolated spheres. It has no thickness. In this construction in fact one of the main ideas is that we have thickness. Which is basically related to saturating burst and inequalities, intermittency and so on. And it's not literally I think the dyadic shell. I mean wq plus 1 has a small piece which is actually even leaving with smaller frequencies. So it's much smaller. It's subtle basically, how you choose exactly. Yeah, it's a very subtle issue. We actually make it supported because we hit it with a projector. Yeah, we hit it on top with a projector. Just so that we can do it. You should write down the q plus 1. Where? Ah, here? Thank you, thank you. Okay. So, you should have already noticed something. If wq plus 1, tensor wq plus 1 has any hope of cancelling rq. Which in turn has that size, delta q plus 1. And you better have wq plus 1 in an infinity l2 is of size. Ah, I don't want to make this precise. Delta q plus 1 through one half. Okay? Because otherwise it cannot possibly cancel it. And it lives at the next frequency. Lives at lambda q plus 1 as Alex pointed out. In a shell in fact, whose thickness is rather subtle. Okay, so now I want to just point out very quickly one thing. Ah, very quickly. Because of this term, you can actually cannot go with beta more than a third. Because of the Nash error. And you can very quickly see this. You can just compute how big divergence inverse, whatever that means, of wq plus 1 dot gradient vq. Right? The size of this, to get the size of this you need to invert the divergence. Right? In l infinity l1, you can check what the sizes are. And you get lambda q over lambda q plus 1. This lambda q is because you have a derivative falling on the low frequency guy. This lambda q plus 1 inverse. It's because you're inverting a negative one operator of something which lives at the higher frequency. And you have the sizes of the objects. Which are just delta q plus 1 to the one half, delta q to the one half. Okay, and you can now check that this will be of the desired size. Okay? So it's a computation. You can very quickly do it if you assume geometric. It takes a while if it's super exponential. By a while I mean a couple minutes. Okay, so now what is the main idea? I didn't mention anything about the proof. So in the remaining 10 minutes, all these ideas, by the way, are in all the previous constructions. I didn't say anything new so far. So now I will say something's new. Why the previous constructions are getting stuck at that regularity, or at least one of the reasons, is that the transport error is very high. Okay. You're making high frequency perturbations. So the material derivative can hit these high frequency perturbations.  I mean, you get one of those exponents. So what you do is then, instead of having stationary Beltrami flows, you let them flow with the velocity field of the previous, with vq. Okay, when you do that, certain property that you used, that's why you started with Beltrami flows, namely a certain linear algebra cancellation, gets destroyed because your oscillations are not exactly Beltrami. So you can let the transport equation run for some short time. How long time that is, you need to put in a time cut-off. When your material derivative hits that time cut-off, you make an error, which at first seems to have killed your scheme. Because it's as big of a size as what you started with, you have no improvement, you think you're done. You can play some games, and that's how you get here, by interpolating. But basically, this is the main enemy. So, what do we do, in fact? How do our corrections actually look like? So again, I will not write Liré projectors, and I will not write Littwood-Paley projectors, but they're here. Basically, this is a sum over frequencies of some amplitudes and some object Wc, which are built high-frequency oscillations. I will tell you exactly what those are. I will not tell you much about what they are. What is this index set? You're looking at the sphere of radius lambda q plus 1. This is an actual sphere. And in the previous constructions, this index set were a couple points. Always including the diametrally opposite one. A couple points. This is basically, I'm exaggerating, some coefficients, lambda q plus 1, c dot x, composed with the flow of the field. So this would be at time zero, but of course you let this run. And what happened in the previous constructions, when Wc interacted with W minus c, the phase cancels, you produce something low-frequency, and that low-frequency object is exactly what cancels this lower-frequency object. Okay, when you do that, that's this interaction. How about this interaction? This interaction is high-high and spits out high at the same frequency, lambda q plus 1. And that one you win because you're inverting the divergence. So here I want to emphasize it spits out something at very high frequency, lambda q plus 1. And you win by inverting the divergence. And this is why the oscillation error actually had a bit of room, apparently. Not once you optimize with the transport, of course. So what is our construction instead? Our vector, our object's W, let me start by doing the north and the south pole. There's finitely many of them. Think of them as rotated. They have, instead of a point in frequency, a vector, they have a Dirichlet kernel. So basically here at the north pole, at the south pole and at every other point, dots that were in the previous construction, we put a Dirichlet kernel. What do I mean by that? We really take this vector, c bar, which is of size lambda q plus 1. And we add to it an object which is really a 2D Dirichlet kernel. It has certain features, this Dirichlet kernel. It has certain separation between the two elements, the next consecutive elements. And it has certain number of elements. Okay, so separation and number of elements. Let's call this number R just so that we fix that thing. It changes in q. Sorry? It will depend on q. So the higher the q, the more objects you put in. Okay, what happened? What is the advantage of this? What is the norm of this new object that we are constructing? I guess it depends on c. C bar, which is the center. Well, like any Dirichlet kernel, when you compute the L1 norm, you lose a log in L2 square root of the number of elements. In L infinity is the number of elements. So what we're doing, keep in mind, we want this relation and we already had this relation for the coefficients. Right? Already this is of the correct size. So now if I do herder inequality and I make this of any size more than one, I will screw my correction. Doesn't exactly work like that. You can use almost orthogonality. This is much higher frequency than this. So when you estimate the L2 norm of this, it's like the product of the L2 norms, in fact.  That's the correct rescaling if you want to maintain something like this. This almost orthogonality really comes if you have sufficient separation in your frequencies there. If you don't have enough separation, this will get screwed immediately. Okay. L2 norm 1, well, by rescaling, the L1 norm is now small. RQ, whatever. It's a tiny number. Before, when you just had one element, it was order 1. So now all of a sudden, when you're computing the transport error, and with this I think I have to very, very quickly finish, when you're now computing the transport error, these cannot fall on these Dirichlet-Beltrami waves because we flow them with the vector field of the previous correction. They can only fall on the corrections. Sorry? There's an extra V. Yes, there's an extra V. Thank you. Thank you. So this is... We have a room full of editors. Yes, yes. This is this, right? Right. It cannot fall here because it's transported. So it has to fall there. So now when you compute the L1 norm, this is what you have to compute, the L1 norm of the transport error. Again, you need to use almost orthogonality. This lives at much higher frequency if you have enough separation than this one. So when you compute the L1 norm, you get the L1 norm of this times the L1 norm of that. This will give you exactly the same things in the previous constructions, except you have now an RQ inverse now. You have improved your main enemy with exactly this object. Now, of course, this improvement does not come for free. Let me write... Let me blow up this Dirichlet kernel. Okay, so these guys help you when this interacts with exactly the south pole. You're good. You know, when this interacts with that, you're good because they always spit low. Okay, so one more minute. They spit low. But now, what happens if... There's no color. Okay, what happens if this one doesn't interact with the south pole but with this one? How high is this interaction? The answer is not very high. It's that high. So now what you... It seems like you're stuck, right? The error you made from high-high interactions is actually not high. It's not sufficiently low either. It's in some intermediate range. And this seems to screw you, and then it doesn't if you now include more Reynolds stress corrections. You come with another velocity field and you correct those interactions which are of medium size. Those which are of high size you can always control by inverting the divergence. And this is where the thickness of the shell comes in. The thickness of the shell is how high do you have to push the errors so that you win from the divergence? Okay, you optimize in that. And now we basically have a chain of Reynolds stress errors in every single step. Every single one correcting the medium errors you have made from these high-high interactions which don't spit high. The advantage is in every single step of this chain you have slightly decreased the amplitude and your separation is slightly increasing also. So you have fewer and fewer enemies. And we can prove that after finitely many steps independent of Q you stop and you achieve the desired bounds. And I'll finish here. Which one of those parameters depends on T? On T? Everybody. W is what I showed you is only the initial data. These are flowed by the velocity field. These have to depend on XT because they're correcting a Reynolds stress which itself depends on XT. Any chance of running two iterations at the same time? So to get one solution that you put in T? The two iterations at the same time maybe not. But to get to T maybe yes. But not by running them at the same time. It seems that this temporal intermittency that you guys built doesn't really gel with this spatial intermittency that we built. And there's some idea we don't know yet if it gets you all the way. It gets you a bit. But we don't know yet how far.


Felly, y cychwyn i'r blaen a'r ad faithau yn ei deilio y bydd frydym yn fwy o'r hawdd. Felly, oherwydd, arwt i'r ad faithiau yn dlunio ar ddweud y cychwyn, mae'r ad faithiau ynificio ar gyfer ei ddechrau i gael ziwb ac yn绕. Felly, yma, mae'n ffordd os yw'r ad faithiau sydd ar gyfer y cychwyn i'r gwneud yn ddechrau i chi ar y cychwyn i'r ad faithiau ac yn yr ad faithiau sydd yw eu cwestiwch cyfrannu ac ydych chi'n gyfrannu'n acotifolion. Yn ystod y Ddod, arfer y cyfrannu cyfrannu sy'n cyfrannu i'r taxid o'r darlogau i ni wedi arbennig y targedau. Roedd y targedau rhai y gallwn ei fanolodol iawn ac na'r targedau arbennig y targedau yn y fynd eich neu bandannau arbennig arbennig Ie, rydyn ni'n gweld yn ddod i'r wneud i'r unedd iawn o'r ffordd. Rydyn ni'n ddod i'r holl yn rwynt i'r holl i'r holl yn yma. Yn ymgyrch chi'n gwaith yw'r holl yn ymgyrch chi'n gwybwysiadau a'u gydagol ar gyfer cyfaint gweithio. Yn ymgyrch chi'n gweithio gyda'u gweithio ar gyfer cyfaint gweithio a'u gweithio i'r holl yn ymgyrch chi'n gweithio. ar y cyd-fair. Roedd nid o'r ffordd, oherwydd roedd yn ei peth yn seffaith bwysig â'r bethau gyda'r halfasucurol. Ond, oherwydd fel y cyd-fair, y clywbeth yn hynny'n cerddwyd mawr o bach o bach, cyd-fair byddwyr yn cael eu'r cyllidhaeth karadwadau ar y cy yrwe. Felly, mae un entscheidiau yn rhan ymdoul, mae'r 170 arfer yn bach o'r cyd-fair. Rwyfyrr ar hynny'n mwy arall o fod yn helyshno i chi roedden, mae'r cyd-fair, gyda siaradid ar y cwblu pan georddoedd cyntaf, gallwn hefyd yn cael ei dyn nhw. Mae'r cyfrifosydd cyfrifosid cyfrifosid cyfwiydd ar ei dyn nhw yw ymlaen i'r adyn nhw, mewn gwneud cyfrifosid y marmarau, y rymian ydw i'r sgall, ydw i'r adeg ond yn allan iawn nesaf. Mae'r archiolwg yn cynnig gyda cyddig o'i cyfrifosid tîm iawn o'r negedeg yng Nghymru, I arfer yw'r un ffordd yn gennym o'ch byw. Rwyf yn dechrau gynnig o'r newid i yw serion cookwyr a'r idea o ran ein newid. Ond oes i wneud byw oedd 상au i gwirionedd tynnu a'r holl hwnnw o'r cyfnod iawn i ymwneud yma. A usesi'r ffordd yw'r cyfeithio yma, oherwydd carefio gan Eulaf i'r gwirionedd Cymru o'r Llywodraeth Mae'r gwirgau sydd yn eu cael ei wneud, yn ystafell ac ein llwst. Ond, ond pan pobwynt, mae'r buddynhau monumentiau yn ôl yma yn maen nhw'r holl bonne i wyddweithio. Felly, mae'r holl gawr i'n ôl am y hollewyr a'r hollewyr yn yr oedd. Ond yna, mae'n mhlion o'r hollewyr mewn teimlo yn ddim yn yr oedau i hollewyr. A oedda hi fe nhw'n hollewyr a'i cyfannu hynny o'r gwirgau sydd y byd? Mae'r brodwyr yw'r hanfodd ymlaen i'r Constant NGO. Mae'r brodwyr yn ystafell. Rhywbeth dda i'r brifysgfa Ychynig. Ydw i'r pob yw'r cram о'r Cyfrifysgfa Fyrdd y cwrwp yn ystafell. Rydw i'r brodwyr eich bair o cwrwp, oka'r gwbeithio gael oherwydd Ac oedd yn bai'r brodwyr gwiaith. Rydw i'r cwrwp yn gyiar Rylen oherwydd ymlaen, ac mae'r holl gympr metre oherwydd ac oedd yr haes i'r cyfrifysgfa These are just the mega-projects. So, Sulu Culew is a good example of what happens when the state wants to basically make cash out of an adored district which doesn't necessarily conform to the identity of Turkishness. Plans to redevelopment of Sulu Culew were revealed in late 2005 and the areas demolished through 2009 to 2010. 571 predominantly Romani houses were destroyed, displacing over 5,000 residents. The area stood as the home of the oldest central Romani community for over 1,000 years and was defined by UNESCO as 1,000 years of Romani cultural heritage. The Sulu Culew's residents were offered strange deals concerning their moving and only their own property basically were they allowed to then buy a new property but not on the same level and the new houses retailed over 10 times per square meter than the residents paid. In doing so, the Aberystwyth regeneration not only gained loads of cash but also encompassed the Romani culture into formal avenues of the state. Often these lower levels minorities have just been left to be impoverished by themselves but now in these areas they've been forced out and this goes on. This is not only Sulu Culew, this is Tala Basha, Orkmidon and other kind of areas that are in the very centre of the city. So it is formalisation of the heritage narratives which currently breathe such discontent amongst these communities especially places like the Roma who have kind of a very vibrant and not necessarily Turkish inclusion. This is not the first time that the city has undergone radical change and just in the actual construction of the Republic in the first place the city was well the capital was moved from Istanbul to Ankara for the precise reason of actually moving away from what was perceived to be old and not progressive. Istanbul still became a very centre of industry and violence essentially so whereas the city journey off from the period existed as 56% was in Turkish and that then grew to 99% due to pogroms and expulsions of Greeks, Armenians and many minorities basically being subsumed into Turkishness. This came through kind of economic stripping and kind of social also like lack of inclusion. Places that the government kind of at that period during that kind of period of redevelopment actually grew. Places like Gezi Park came to kind of be a symbol of this new city and actually redefine relationships amongst these communities and the other communities so what you ended up having was kind of a synthesis between this old city and the new city and these old communities and the new communities and you know kind of a mix let's say but many of the kind of older generations still mourned the soul of the city as it's dilapidated and still do today. If you go around in the city you can often speak to people who say that you know they don't necessarily know the heritage of the area that they were but you can speak to their next-door neighbour or a friend who is Greek or Armenian and they would be able to tell you the truth behind this so this kind of synthesis exists these two or many many many more narratives that are overlapping and interlinking in the city and causing these kind of strange micro narratives let's say but conversely the actual archaeology of the city is as Edwin exactly says an ideology and so in making plans for taxim making plans for Gezi Park to have a barracks to reinstate the old Ottoman barracks which was actually also the first football stadium um he's directly going against uh not only well kind of weirdly going against the archaeology but also for his idea of what the heritage should be i the autumnisation and the stuff which directly contrasts with the the idea of Turkishness but simultaneously doesn't actually link back up with the kind of inclusion of minorities in society although it originally started as intended to be like that um places like Gezi Park have become new lifeblood for the city a new kind of area of liberalisation and new mixed synthesis between these things so you have LGBTs and other minority communities also existing in this new republican space and so being targeted again by Edwin um he's also trying to get rid of this new progressive kind of minority inclusivity of things um which resulted in the Gezi process which were in 2013 and I think that maybe quite a lot of people know about those um so they kind of grew out of a lot of different narratives a lot of different directions including suppression of minorities and general public space where people could practice non not necessarily the same um behaviour the different parts of cultural heritage for Sulu Fili that's turned into a music venue where the kind of Romani music and even contemporary versions of Romani music including hip-hop created dominated slaves and became very well known as um as a form of intangible cultural heritage which was being paraded around in spite of the uh the huge amounts of government attacks and pressure and of course the loss of their actual ideological homeland which was Sulu Fili which was then at that point already destroyed and they had been scattered some 30 kilometres actually most of the families from who had been scattered and not moving back to poor residents but still fractured and so the Sulu Fili continued in the outline and other districts which became a very interesting thing onto itself um so the public appreciation of space really hit high in Gezi and re-evaluated what the space was supposed to be and and people realised that the space wasn't necessarily the same for everybody um dialogues which didn't usually exist or didn't typically exist between kind of larger scale ideas of republicans and nationalism minorities lgbt's football forums all started talking and appreciating each other's um kind of position in in the city um and it kind of devolved after the protests ended to this last quote and which is wherever you are chance to prepare a banner and take a picture of it tie a cloth around a tree take the parts organize a forum talk with one another sing our songs geziad everywhere you are so people were aware that this can occur through the materiality of the city and actually fundamentally it is the materiality of the city which is going to make a difference so um and so this evolved onto the streets and graffiti such as this such as this is our meaning graffiti in kutlush um showing and kind of appreciating other parts of Armenian culture um so and by reproducing their own material memory they placed themselves firmly in the landscape combating the larger scale kind of destruction of the city and reconstruction so they're producing their own narratives their own material narratives in the city um the eco-based desire to uproot disperse cohesive suba subaltern groups is stronger than ever though because of this and because of the movements of geziad and the aftermath um with stuff like this um they target are now targeting spaces which couldn't perform in this kind of multivocal way um and so by smacking a massive great mosque right on taxim um and tearing down the api me and the after cultural um mechizzi and opera household by a kind of a renewal cultural heritage during the cat um he's essentially placing and completely overriding um what was essentially and is the spiritual home of of this kind of latter unity um and so through targeting the space and you know and he is essentially and then they are essentially changing fundamentally the way in which um these cultures these kind of smaller narratives are kind of appearing um in Tsiglukule um a Roma guy said our houses have courtyards these courtyards have many functions we used to wash out dead there we have fruit trees mulberry cherry and plum trees we start animals chickens birds horses donkeys we live in apartments now but living in apartments is not our way of life we do not have this culture um and this isn't just in Tsiglukule this is occurring on a wider scale um and without kind of acknowledging that these micro narratives are incredibly important actually for the production of smaller kind of forms of heritage or less grand heritage we always think about stuff like and the Iosofia and we think that's happening in the Fumos but the city is constituent and always has been constituent of many different kind of parts of the world and parts of the nation and without recognizing that these things occur we're going to lose them and that would be a terrible shame and so in order to protect globally significant horror historic fights we have to be awareness of local ones and as the government turns away from kind of westernization way from ideas of kind of formalization of heritage products and and and the west is a guiding light basically and that could also transfer quite easily into the rejection of things like UNESCO in the same way that the government is more or less rejecting the union and and in doing so the transience of the kind of underlying material in the city it produces a threat because without the people there actually on the screen protecting that protecting these values protecting the actual their own personal heritage it's going to go and there's nothing that we're going to be able to do to stop that anyway so I hope that that wasn't too fast for you and if you have any questions please ask me at the end


Now that we have talked about the concept of application layer, let us see how it actually works. Hello and welcome to week 3's video on port numbers. So we know port numbers are used to distinguish communication for different applications. For example, the websites use port number 80 so your browser knows that this packet is being sent to a server and it has come from a server and when your browser is sending out the request it attaches its own port number. So now we have seen these things in packet tracer so what we thought it would be a good idea to actually see them on via shark and what we'll do is let's open via shark and let's see how on real time traffic we can distinguish traffic for different applications. Now we'll use the Wi-Fi adapter that is used in this card so we see a lot of traffic that is going around, you have different protocols going around here. So in the meanwhile what we can try is try opening say some website. Let's for example just open our old dummy website which is this. Now since we have generated some traffic we can go back and see what's there on via shark. So what we'll try to do is we'll try to filter out traffic. Now we know the protocol that the website is using is HTTP so let's filter out this and let's try to see what is the port number here. So what we see here is the destination port number is 80 and source port number is 53712. So what it means here is that this packet is intended to go to a web server via the protocol HTTP. So when the this packet comes back then the source the destination port would be this and the source port will be this that is when it comes back comes back from the server. So let's try to see some other HTTP request packets and see if we can see. So these are all requests that are going to the web server. Now you see this one here the source port is 80 and the destination port is 53715. So this means this packet has come from a web server and it has come to the application which was using the port number 53715. So this way port numbers are used to distinguish different communications that belong to different applications and even if it's a same web browser and it has multiple tabs here so all the tabs would use different port numbers. So let's try and see if we can find any other protocol that the laptop is communicating through. So let's try DNS which is domain name service and yes we have some packets from DNS. So again we see here the source port number is 53 and the destination port number is something like this is a response packet and let's try to see here. So we have the source port as 53 and destination here and here again the reply has come. So here what we see is the source port is 53 and destination port is 55987 and here what we see is for one of the replies which was 51807 the reply has come back which is with destination port 53. So this is another example of how different protocols use port numbers and so these are some standard port numbers 53 is a standard port number for DNS requests. Similarly 80 is a standard port number for web browser based HTTP requests and let's see another protocol. So let me show you something called ARP. Now the interesting thing here is ARP is a protocol which actually doesn't work on the application and higher layers it basically just works till the internet layer. So all you see here is the IP addresses and the MAC address. Basically what ARP does is it gets the corresponding MAC address for IP address or vice versa. So we call one as ARP the other as reverse ARP or reverse address resolution protocol. So all the protocols that work on the application layer have these port numbers. So this is a brief demonstration of port numbers. There is one more crucial concept with port numbers called port address translation. For port address translation what we would want you to do is go through the article on port address translation and try to see what it means. Once you have read the article go to the next video and we will talk more about port address translation. Thank you.