 As far as I can say, I enjoyed new things, and it was a great pleasure to see him more or less regularly, year after year, and talk to him. And I hope he will not be objecting my taking a subject which is very far from this conference. The reason I took it is that there is a recent progress there, and I'm excited about it. I'll write you along there. It's a strange story. Turbulence was the first major progress in turbulence in 1940s by Kolmogorov and his students actually let influence, at least I would say, the development of critical phenomena of theory of phase transitions. What is surprising, what's amusing, surprising is that theory of phase transitions made a huge progress. We calculate critical behavior with high accuracy now. At the same time, and what is more, we have a lot of deep ideas governing critical phenomena. At the same time, the progress in turbulence was not that great. We still have simple questions which we cannot answer. And my guess is that, although you can say that people didn't try, there are plenty of work, both experimental, numerical, theoretical, but still the state of the subject is much, much more poor than critical phenomena. The reason for that, I think, is that in critical phenomena, we have the two dimensionalizing model. An exactly solvable model which is highly non-trivial and which actually influenced the development of the subject, enormously influenced development of the subject. At the same time, there is no such model in the theory of turbulence. And what I will discuss in this talk is, well, some models which hopefully replace the icing model of critical phenomena. And also I want to discuss some new concepts which slowly arise in this field. First, there is a common phenomenon which exists in both critical phenomena and turbulence, and this is universality. Universality means that when you change a little parameters like interaction between spins or boundary condition of the fluid, small enough perturbations are irrelevant. They do not critical behavior remains the same. It's very easy to understand for anyone who has just a little experience in field theory. If we look at the Dyson's equations, then it contains bare green function and physical green function. Sigma here is a self-energy part. And it was a very important conjecture by Potashinsky and Pokrovsky that maybe in the critical region, the bare part of green function and other quantities, they are smaller, much smaller than the real physical part. And so all information about the bare green function, which means interactions, information about interactions, about the small details. If you perturb a little bit this bare green function, you will not change the critical function, the critical behavior. It's, by the way, I will add in parenthesis that it is very similar to the definition of topological invariance. In the topological case, you change geometry just a little bit and the value of topological invariant doesn't change. So it is, there is some similarity between these two points, these two concepts, which may be worth taking in mind. And that's the origin of the bootstrap. Bootstrap is the idea that critical fluctuations are composed out of themselves. And, well, I don't have time to go into more details. The solution of the bootstrap relation basically means that you have some complicated equations for correlation functions, which do not contain any parameters, do not contain anything. As I said, they just say that lines representing critical, lines in these diagrams representing critical fluctuations are composed of themselves. Running ahead of myself, in hydrodynamics, we also have similar situations, similar thing as similar to conformable to bootstrap condition is the fact that when you have a combination of vortices, each vortex moves in the fields created by all others. So there is some grand self-consistency of vortex motion. And I believe that similar language is appropriate in the cases of critical bootstrap and other things. But I have to move further. So basically, just before I move further, just to mention that bootstrap relations can be expressed as kind of Jacobi identity for critical fluctuations. Jacobi identity is the basic relation in new groups. And so I see some analogy here. Anyway, let me move. The one-dimensional model, which my point is that I will try to consider one-dimensional simplified, let's say, one-dimensional simplified model of turbulence and see whether I can get some general concepts out of it. The simple and commonly accepted model is that you have the Navier-Stokes equation. The Navier-Stokes equation here is one-dimensional Navier-Stokes equation without pressure and with external force F. It's commonly accepted that you can generate turbulence with the force being Gaussian or random force, which will not matter at the end. Which will not matter, really. The concrete force will be relevant. We will find some universal exponents, which because of universality do not depend on any details. So the whole motion in this model is the following. You have velocity profile if viscosity tends to zero. And if you set viscosity to zero, you will generate zig-zags and it will be physically meaningless. But in fact, the answer is that you have smooth velocity profile and it generates shock waves. And shock waves are generated by the mass-box rule that areas are equal before and after. So what can be learned to what extent this model can be solved? I don't have time to explain it, but we can derive the master equation for the correlation function of this type. You have velocity slumped sub-parameters and you have the correlation function of this type. And there is a quite remarkable fact that you can write down the equation, the master equation for this quantity. And that will be our starting point. Again, there are many subtleties which I cannot touch because of the lack of time. Especially there is a subtle appearance of Galilean symmetry in these equations. I actually want to stress that these equations, just trust me, I can't have time to explain it. They are extremely similar to the equations of statistical mechanics. This is the chain of equations, statistical mechanics, which allows one to derive the Boltzmann equation. And morally what I'm doing here is very similar transition. And I will give you a little later some concrete solutions of these equations, which will give us highly non-Gaussian distribution of velocity. There is some region in which velocity distribution is Gaussian, but what is very exciting is that there is non-Gaussian region also. Another lesson which we can draw from this model is the relation between turbulence, dissipation and axial anomalies of quantum field theory. I shall explain this briefly. There was a great discovery by Adler-Bell and Chekiv that if you take massless quantum electrodynamics in four dimensions, then the axial current, which is formally conserved, is actually, conservation law is removed by certain anomalies, provided that we have some, well, a concreteness, I took the instant on background here. And we have this, I shall explain the meaning of this equation in a moment. And I just want to show you two equations. The meaning of this equation, the meaning of this equation and as you will see the meaning of this anomaly in turbulence is the following. We have to, in quantum electrodynamics, massless quantum electrodynamics, we have to regularize the theory by some cut-off, which we have to assume that some cut-off exists at which the theory breaks down. And this cut-off explicitly breaks axial symmetry. However, as you go to low energy, there are two possibilities. One is that the symmetry still will be violated, or that the symmetry will be intact. ABJ showed that in quantum electrodynamics, as we go to the physical energies, the input of symmetry breaking at high energy still felt and provided that you are in the non-trivial background, which gives you FF dual, that's the field strength. Now, the same is true for turbulence, except it's not, of course, axial anomaly, but it is energy anomaly. We have, at very large scales, we have a fan, which injects energy into the system. It propagates through the scales, reach a physical scale called inertial range, and then dissipates at smaller scales. And that gives the famous Klamogorov equations, which is written here. Now, there is a very, that's, I think, this is important, I think this is important and not used yet to full extent. I will have something to say later about it. In fact, with dissipative anomaly, I shall not explain what is Galilean regime here, but it's such as some special range of velocities. You can derive the reduced master equation to the equation written here. And you can find scaling solution and get the universal answer, which is written here. So W here is probability for having velocity, you have two points separated by the distance y, and V is the velocity difference, velocity increment between two points. And W is the probability of this. Sasha, I'm sorry to interrupt you, but you have five more minutes. What? Five more minutes for you. Ah, yes. I see, how many? Let's say six to seven. Okay, I'm sorry, but thank you. Okay. So, there is a still some unsolved problems here. And there is another solution which can be found for the master equation, which contains not the Bessel function, but hypergeometric function, and it relates to non Galilean invariant, non Galilean invariant range. So that's the exploration of these master equations. It's, I think, very important. And it's only, we only scratch the surface at the moment. And that's one of the reasons I'm decided to talk about this subject is that my is my feeling that there are plenty of hidden flair hidden treasures. I wanted to say hidden pleasures anyway. It's a beautiful subject. And another thing, another concept which seems to be present in turbulence is general covariance. The equations, it appears when we describe the fluid in terms of Lagrange coordinates. And it is related to these master equations, but obviously I don't have time for this. You start you can start the this action, which looks very much like action of string theory is actually gives you the evolution of fluid in Lagrange coordinates and then you can make some conjectures and drop certain terms and the resulting probability to have given Lagrange and X of psi probability is generally covariant. And so perhaps gravitation methods will be the last thing I'm going to this to tell you about. My goal you see, I'm telling you about unfinished theory. And I just found some gems here and there. So I'm telling about the separate findings without really having a full general theory and concept. But that's probably okay. All science, all good sciences like that. Now the last point which I want to stress, which I want to tell you about is that the action, this is the action which you can write down with the turbulence. There is a change of variable related to what is called the hope for substitution, which can be generalized to this problem. And it gives somewhat identity and functional integral and so on. And by change by this change of variables you reduce the action to the action of complexified both a guess, one dimensional both guess. So which is completely integrable system. So it is quite fascinating that turbulence as an example of chaos. In fact, it involves probabilistic description, but probabilities themselves satisfy some are obtained from some integrable system. The integrable is somehow anti-turbulence, means anti-turbulence, still it is related to turbulence. And again, this both a guess is exploration of it just begins. There are plenty of interesting things which I did cover here. Which hopefully will be clarified. Okay, so this was a pitch for turbulence. And I think there are many things left outside the stop. And I think I stop here. Thank you very much.