 So, the study of this nonlinear dispersive equations took off basically in these late 70s and 80s, where many properties of special solutions, such as traveling waves, were discovered sometimes rigorously and sometimes not. And I guess in the late 80s and early 90s with my collaborators, Ponce and Vega, we described a way of using the modern tools of harmonic analysis to study the nonlinear problems as perturbations of the linear problems that could be analyzed by Fourier's method. We extended many of these properties to nonlinear equations. There's been a lot of progress on these equations since. Many people have been involved in this and there are some names for those of you who are mathematicians, Burgen, Tao, Tataru, Kleiner and Macedon and many others. And what this was able to accomplish using basically harmonic analysis is a very satisfactory understanding of the short time well-posedness of these equations and for a long time the well-posedness for very small solutions. And it was at this time that the notion of criticality arose. Criticality is an important notion in nonlinear analysis and it's not a fixed notion. Criticality has to do with the groups of invariances that act on the solutions of the equation and it depends on what property of these groups you look at that you can define a different notion of criticality. And the notion of criticality that arose here had to do with scaling, with looking at things microscopically or telescopically. So this is another promotional slide. The thing that became of interest in the last 15 or 20 years is the understanding of global properties where you cannot use perturbations of the nonlinear equations by the linear equations so that you cannot really use harmonic analysis at least in a direct manner. So what do you do? Well there was a beginning of this in work that I did with Frank Merle about I don't know roughly 12 years ago in which we introduced a method which allowed us to understand in a better way the nonlinear effects that cause the existence of solitary waves. But our ultimate goal in this method was to study the problem of asymptotic soliton resolution. So I'll go back to this now and this will be the next part of the talk where I'm going to speak about a simplification that is effective in this nonlinear problems because of the existence of these traveling waves, these truly nonlinear objects. Okay yes this is correct but this was used for dispersive equations for the first time here. Okay is there any other comment? Okay so what we wanted to study is something that is more of a philosophy than a theorem which originates in the mathematical physics community which is the belief that the long time asymptotic behavior of arbitrary solutions of arbitrary dispersive equations is given by coherent structures and free radiation. And this is a this should describe this long time asymptotic behavior and this belief is what's come to be known as the soliton resolution conjecture. So the soliton resolution conjecture is not really a conjecture but the belief and in every specific instance it becomes an actual conjecture once you formulate it precisely. So what this says is that asymptotically in time the evolution of a generic solution decouples as a sum of modulated solitons which are traveling wave solutions, a solution that is called the free radiation which is a dispersive term which does go down in time in size solving the linear equation. And so why is this a simplification? What happens with these very complex nonlinear evolutions is the solution starts evolving and for a while you for very short time it starts behaving like a linear solution but then after a while it comes into an intermediate regime where you really cannot see what it's doing it's like in a black box that you cannot peer into but if you wait long enough then you can see it come out of this box as a sum of solitary waves plus a solution to the linear equation plus a small error. And so asymptotically in time you get a formula for the solution. You get an asymptotic formula in terms of these nonlinear objects which take care of the part that is nonlinear and then a linear term that takes care of the rest. So that's what this postulates. Of course it's easy to say this but then since we are mathematicians we try to prove something like this. Okay so this is the second simplification I wanted to discuss. So here comes a very, what to me is a very interesting story. What made people think that this was possible? So the story really started at the end of World War II at Los Alamos. So at Los Alamos of course they built the nuclear bombs that they used to end the war in Japan in the east and there was a large congregation of scientists at Los Alamos at the time and one of the things that they did in order to actually manufacture these bombs was construct a computer. So they constructed one of the first computers which allowed them to do the huge calculations that they needed to do. This computer was called the maniac and so after the war Fermi who was still at Los Alamos at the time decided that there should be a good use for this machine that in fact this machine should be it should be possible to use it for theoretical purposes. Fermi was a professor at the University of Chicago by the way. Anyway so what he decided is that he was going to try to conduct a computer experiment of a mathematical problem that tested some physical law but which you couldn't really calculate by hand but by doing this numerical simulation you could get an idea of what you of what was going on and this thought of Fermi is really the beginning of scientific computation and that's that was the effect of this observation of Fermi's and so he with Ulem and Pasta Pasta was a computer expert Ulem was another one of the people who worked on the bombs at Los Alamos he was a applied mathematician and a topologist and so what they decided to try out was understanding how a vibrating string with a nonlinear term with a nonlinear effect behaves and they decided to put quadratic or cubic nonlinearities into this and the the physical intuition that they had was that the nonlinearity should cause some kind of equi repetition of energy which meant that all the Fourier mode if you start out with something which had just one Fourier mode if you waited long enough all the Fourier modes should be touched and the energy should be equally spread out among all these Fourier modes that was Fermi's conjecture and this was called ergodic behavior and they did the experiment and this did not happen and Fermi called this a minor discovery and what happened was that instead the energy got constant kept being concentrated at the same mode so what they what this means is they solved for cosine x and and showed that the solution most of its energy was still concentrated in the first mode as time evolved for a very long time and this was a paradox that they couldn't resolve and they published this in Los Alamos report and by the time this was published Fermi had already died like most of his team died early from cancer and for many years this was not explained now an interesting anecdote is that there was a fourth author of this experiment who is the person that did the programming whose name was Mary Tsingu and her name disappeared from the story so you can draw your own conclusions about this I think there are two possible explanations one is the obvious one the other one is that they didn't think that people who did the actual coding were at the same level as the authors okay so it is unclear which one of the two explanation is is the correct one so this thing was taken up about 10 years later in the mid 60s by Kruskal and then by Kruskal and Zawuski and Kruskal made a very very interesting observation he observed and now by then numerical experiments were much more common and the machines were much better but Kruskal made a theoretical observation he said he the way that they had solved this non-linear vibrating string was by discretization they put in many many points and solved finite differences instead of derivatives and just solved that by computational methods so what what Kruskal was able to do is that if you he calculated the limit as the mesh goes to zero and when you calculate the limit as the mesh goes to zero you actually get the correct debris equation and it's the the existence of solitary waves for the correct debris equation that explains the Fermi-Pasta-Ulan paradox then he conducted in Kruskal conducted another very interesting numerical experiment with Zawuski now here too there was a third party involved who did the coding this third party was a man and the name also has been dropped from history so the jury is out as to what really happened with this anyway in the Zawuski Kruskal numerical experiment two things happened two things that to my mind are very important the first was that they solved the correct debris equation which with the non-linearity quadratic non-linearity for data cosine x just like Fermi-Pasta-Ulan and they observed that for very long time for very long time when you waited very long time the solution decomposed into this sum of translated and rescaled solitons going at different speeds and that's where the soliton resolution conjecture originates in this computation of Zawuski and Kruskal to explain the Fermi-Pasta-Ulan experiment the second thing that they observed is that as time evolved the solitons were traveling at different speeds and some caught up with others and the moment they caught up you would expect that something would explode in some way or the things would crash or something strange but no nothing like this happened the one that was going faster took over the one that was going slower and emerged unchanged so the the solitons interacted as if they were linear their interaction was linear and this is called at the moment of collision and later and this is what we call elastic collisions okay and I lost my thing okay so this is the background for the soliton resolution conjecture these numerical experiments so I think it's actually fascinating how the advent of the computer age led to really interesting mathematical conjectures and led to scientific computing and modeling which is one of the main most important activities nowadays so this soliton resolution conjecture is has been since the 70s one of the holy grails in the subject of dispersive equations now the only cases in which it had been solved until very recently were cases where the nonlinear equations were actually what we call integrable so this means that the nonlinear equations can be reduced to a countable family of linear equations and for those equations in particular for the correct degrees equations of powers two and three like Kruskal and Zavuzki did for the one of the Schrodinger equations and so on people were able to prove the soliton resolution conjecture but even in this integrable case this were very difficult results the one for the Schrodinger equation it was heuristically solved in the 80s and mathematically this was a paper in the archive this year 2018 so anyway but then the quest really became to understand what happens with soliton resolution in non-integrrable cases when you don't have this linear machinery to do this and just to give you an idea in 2006 in his lecture at the International Congress of Mathematicians Avi Sofer said that nobody had any idea how to do this for any integrable equation for any non-integrrable equation so we're very far from proving such a result for any interesting equation so and in a series of works starting around 2013 still ongoing with the Keier and Merle we have managed to actually do this for a very a non-trivial example of non-integrrable model which is the energy critical wave equation so I'll just show you what this is but first let me explain what is believed to be the mechanism for this decomposition so there's a physical mechanism that's believed to take place that makes this decomposition possible and it's the fact that as you evolve energy the energy is conserved so it cannot disappear but it's pushed towards a spatial infinity and that's why this decomposition is happening now this is observed in experiments and for instance in the dynamics of gas bubbles in a compressible fluid you can see this in the lab and in the formation of black holes and the gravitational collapse it also happens but of course you can only see in a computer nobody has seen a black hole okay so and the way we were able to do this is by finding a way to quantify this ejection of energy so I don't have that much time so I will not torture you by showing you a theorem maybe I'll put it up but not say too much about it so here are two theorems I'll give you a soliton resolution the last one is one where we know that this soliton resolution happens instead of for all times for well-chosen sequences of times and we've been working very hard in trying to prove this for all sequences of times but what is interesting in here is the connection with the Zabowski-Kruskal experiment because what we need to do in order to prove this for all times now is to understand the collisions of solitary waves now in non-integral cases the collisions of solitary waves are much more complicated what we expect is that they collide and they reemerge but when they reemerge there's some extra energy that goes into radiation and that's what we need to capture in order to be able to pass from proving this theorem for some sequence of times to all sequences of times and in the two years since this slide was prepared there's been some progress in that we now understand all soliton collisions in radial situations okay so we understand that always energy is moved out in the form of radiation when they collide anyway so I hope that this gives you a flavor of some of these things thank you very much are there any questions Gil if not let's thank Carlos again