 The program talk is an actual math lecture. And I'm very pleased to introduce Carlos Kenick. So he's obviously one of the organizers of the program, known to hopefully all of you, but I just want to say a few words about him. So Carlos is the Lewis Block Distinguished Service Professor at the University of Chicago. You know, he's won numerous awards. I'll list the Boucher Prize, the Salem Prize. He was a Guggenheim Fellow and he was a plenary speaker at the 2010 ICM. And was the chair of the ICM in 2014. So he's a man of great accomplishments, both mathematical and really a profound influence in the mathematical world. He certainly has influenced me in his talks and papers, which I've read since the beginning of my career. And I'm really pleased that he was involved in this program at all. And I'm also extremely pleased that he agreed to give this talk. So without further ado, I'd like to introduce Carlos Kenick who will speak on simplification when you're non-linear PDE. Hello, can you hear me? The mic is working, okay. Well, thank you for the nice introduction. Let me just warn you, you might regret this, anyway. So I have to say that it's not an easy task for me to give a general audience lecture and I've attempted it only once, two years ago, as you can see from the dates on the slides. Hopefully it will work. It's very fortunate for me and for the audience that we had two days ago, Frank Ferris' talk, in which he introduced many of the things that I'm gonna start saying. So it's an introduction to these things. So that was serendipitous. Okay, so simplification is, of course, something very desirable. And the general idea that I want to convey is that you may start out with something that looks very complicated, but somehow there's a way, conceptually, to find the simplification. And that will be the first part of the lecture and the second part of the lecture, it will deal with other kinds of phenomena in which what you have to do is wait long enough. And if you wait long enough, then things that look extremely complicated become quite simple. So that's the general moral of the story today. So let me start, oh, I see. Yeah, I was warned I have to aim this. It has to be very precisely, precisely aimed. Okay, so this is the complicated part of the lecture. Okay, so we're gonna be discussing two subjects that are very dear to my heart. The first one is Fourier analysis, harmonic analysis, and partial differential equations. And these subjects have been intertwined since the very beginnings, going back to the work of Fourier, 1768, 1830, so Fourier was an accomplished person in many ways. He was Napoleon's envoy to Egypt, he was governor of Egypt, and he also found time to think about mathematics, physics, and various other things. So let me start this story with the mathematical theory of heat conduction that Fourier originated. And what he did was derive an equation, now known as the heat equation, which describes heat flow in a one-dimensional bar from Newton's cooling law, which says that the flow of the heat through a point is proportional to the temperature gradient at the point. And basically what you want to do is you heat this bar, you keep the temperature constant, and you want to see how the temperature evolves. And that's what the heat equation describes. Okay, so now I have to change pages, so this is a very delicate process. Okay, so the heat equation is this partial differential equation where the partial derivative in T, the first partial equals the second partial in X. And what Fourier did was solve what we call the initial value problem, IVP, for this equation. And what that means is that given the initial temperature of a finite bar, and we keep the ends at constant temperature, we want to calculate the future temperature at any point on the bar. So this was what Fourier set out to do, and in the course of this calculation, he made a claim that has motivated the study of Fourier analysis, harmonic analysis from that point on. And I remind you that our research topic here is harmonic analysis. So this is why this is related to this talk. So what is this claim that Fourier made, which people are studying even today? Fourier made the claim that given any function defined, let's say, on the interval minus pi pi, no matter how capricious its graph is, you can represent it as a sum of trigonometric functions, Cn e to the i nx, or if you want cosine nx plus i sin nx. And not only that, but there's a formula to calculate the coefficient Cn given the function f, and that's this formula as an integral. It represents the coefficients as an integral. Now, when Fourier made this claim, this was very controversial. This was in the first half of the 19th century, in the beginning of the 19th century, he made this claim and people didn't believe it because they said, well, these functions, cosine nx plus i sin nx, they are very nice functions. They're very smooth, they look very nice, but at that point in time, they were beginning to be aware of the existence of some very nasty functions. And so the objection is, how could it be possible to write a very nasty function as a sum of nice functions? This seemed contradictory. And whether this is actually true or not is an issue that is still being investigated. So we don't know the end of the story in this direction. We know many partial answers, and it's one of the main topics of studying harmonic analysis. So the fortunate thing was that even though Fourier's claim is actually false for all functions, it is true in sufficient generality that the method that he used to solve the heat equation and several related equations, like the wave equation that we saw in Frank's talk on Tuesday, this method actually worked, and it quickly had a huge, huge impact on the real world. So there was a huge technological impact of this idea of Fourier, and I can describe some of the applications in the 19th century of this. First, it was then possible to calculate the temperature of the Earth. We know the temperature on the surface of the Earth, because we can measure it, and then we can calculate it in the middle of the Earth without digging a hole to the center of the Earth, and that uses Fourier series. The second idea, the second application I want to mention, and this was fundamental for navigation in the 19th century, was that you could predict the tides using Fourier analysis. So as commerce was mainly done through traffic by sea, this was also of huge importance. The third thing that I'm mentioning here is the construction of what was called the harmonic analyst. So the harmonic analyst is not what most of you think, that is to say the people in this room. In fact, the harmonic analyst was a machine. It was an antecedent to the desk calculators that were used until not so long ago, and it allowed you to compute the Fourier coefficients of certain simple functions. And using that, you could do actual calculations using Fourier's method. The next thing which has a very interesting story was it allowed for the construction of the transatlantic cable. So this was a major achievement. It was Lord Kelvin who did this. So there was a company that wanted to construct a transatlantic cable to have telegraphic communication between the United States and Britain. This is what happened. And so they decided that they were gonna try to put the cable through the Atlantic to be able to send telegraphic signals. And now this was a huge technological challenge because first of all laying down a cable in the bottom of the Atlantic was not simple. But the key question that they wanted to understand is how do you design the cable? How thick does it have to be? And the second thing, and this was the crucial thing, is what does the voltage have to be in order to transmit from one side of the Atlantic to the other. And this company employed two engineers for this project. One was Kelvin and the other one is somebody whose name will disappear from history. And they had a big disagreement as to what the voltage had to be. So the other guy thought that the voltage had to be extremely high. And Kelvin using Fourier analysis decided the best thing was to use very low voltage. So they laid down the first cable and they used the other guy's prediction. And the cable burnt up. So the cable was completely destroyed and so Kelvin persuaded them that he was right. And they laid the second cable and that's how the telegraphic communications between Europe and the United States started. Okay and the third or the last, the last advance was that it was then possible to give a very accurate estimate to the age of the earth. So as you can see, this represented a major pushing forward of everyday life actually. Okay and this was through Fourier analysis. So let me move on and the next thing I want to mention here just briefly is that if instead of trying to study how heat transmits for a finite bar, you look at that infinite rod and this was done by Fourier much later in 1822. This leads to analogous representations of functions on the real line as an integral of e to the i x c or sine x c plus i x c. These functions are called plane waves times some coefficient c of c and the formula for c of c is given here as an integral and this function that takes c to the coefficient of f at c which we now normally write f hat of c is called the Fourier transform of the function f. So the fact that Fourier's formulated conjecture is true for many functions has been instrumental only for these technological advances but also for theoretical advances. After all, I come from the University of Chicago where they sell t-shirts that say, okay, all that is well in practice but what about the theory? Anyway. So this has been instrumental in the theoretical development of linear partial differential equations in the last 100 years. So what is Fourier's actual idea to solve this initial value problem for the heat equation? Why did he postulate this formula for every function? Because he says if I then want to solve the heat equation because of linearity of the heat equation I used to say if you have two solutions and you add them you still have a solution then it's enough to solve it for plane waves for f of x equal to e to the ix c for each wavelength c because if you solve it for that then you can add up all the solutions and you get the general solution with the coefficient c of c in front as you add. So that represents a huge simplification because instead of having to solve this partial differential equation for all functions I can just solve it for very special functions. It turns out that for these very special functions you can solve it right away. So for instance for e to the ix c the solution is given here by e to the ix c e to the minus tx squared over two. So that's the formula and then you add all those up with multiplied by coefficients in c and then you get the general solution. So this is the simplification I was talking about at the beginning, the first simplification where instead of solving for very complicated objects the problem you solve it for special functions and then you add everything up and you get the solution. And I guess I want to really make the point that this besides having practical applications it was instrumental in the development of partial differential equations in the 20th century. Of course I guess one should mention that this idea of Fourier of representing a function in terms of special functions of some kind is also extremely important in its future incantations. For instance in the wavelet decomposition and that any of you who has used JPEG has used wavelets so we all know how important that is. So nevertheless this is strictly linear because we're dealing with equations of which some of two things which solve the equation still solve the equation and we think that this represents the propagation of heat but in fact we are fooling ourselves because everywhere we look there are also nonlinear effects. So we can't restrict if we want to understand the world that's around us we cannot restrict ourselves to only considering linear equations. We have to consider also nonlinear equations. And then this idea of superposition of solutions is no longer valid. So I'm going to discuss another type of simplification that is conjectured to hold for a huge class of nonlinear partial differential equations which represent a huge class of physical phenomena. Okay, so I will concentrate on a class of nonlinear partial differential equations which is called dispersive equations and these equations appearing in connections with nonlinear phenomena of wave propagation, okay. Nonlinear partial differential equations come in many, many different varieties and it is not possible to have a theory that even attempts to cover all of them. So we have to somehow try to put in various buckets different kinds of equations and try to see if we can in some way find a way of dealing with the ones in each bucket at the time, okay. And the bucket that I'm deciding to discuss is the bucket of dispersive equations and the reason for that is two-fold. First, I've been working on that for the last 30 years. So I'm kind of fond of the topic and the second reason is because there's a very interesting possibility of simplification for this class of equations and that's what I'd like to discuss. So I will attempt to describe a little bit the theory of dispersive equations by discussing some of their attributes. So we'll describe them by adjective. There'll be some terms that are a bit technical but don't be too concerned because eventually I'll move on and there won't be so many technical things. So the first thing about this nonlinear and dispersive equations, even the linear dispersive equations is that they are time-reversible. You know, with the heat equation that we discussed earlier, as time goes by, the temperature goes down, right? It becomes, so you cannot go back in time with them. For this class of equations, you can go back in time. So you can move in forward time or in backward time. Nevertheless, the linear dispersive equations can also be treated by Fourier's method and we, in fact, use Fourier's method to treat the associated linear equations. The equations that are of dispersive type and arise in physical phenomena usually have a conserved energy and this gives rise to something that we call a Hamiltonian structure that I will not attempt to describe. But there are some quantities that are conserved in time, both for positive time and for negative time and these are important quantities for the motions that we're gonna study. Okay, so let me first do some, a little bit of propaganda and tell you why these equations are important. They appear in the study of water waves in nonlinear optics. Lasers couldn't have been built without these equations and we know how useful these lasers are because I'm using one right now. They appear in ferromagnetism in nonlinear elasticity, in particle physics, in general relativity and in many models for these difficult physical theories and they also have geometric incantations as flows, geometric flows in various geometries like Kaler and Minkowski geometry. Now, these equations have been studied extensively in the last 40 years and from my point of view, this is one of the most exciting areas in partial differential equations today even though as we will see these equations were really first introduced in the 19th century. So, okay, so here are some examples. If you get cross-eyed by looking at these equations don't worry. The first example is what we call the generalized corrective risk equations which model water waves in a shallow channel and here these are real valued functions that verify these differential equations and I lost my laser, okay? Anyway, so you see in this corrective risk equation there's a linear part, dtu minus d cubed xu which can be treated by Fourier's method and then there's a nonlinear term, u to the k times dxu. And again, we study the initial value problem. The second class of equations are the so-called nonlinear Schrodinger equations that originate in quantum theory and are used to study optics, lasers, and ferromagnetism. And hopefully it hasn't died completely because I need to change the page, not being able to, okay. There's also nonlinear versions of the wave equation which is the same equation that we saw on Tuesday but in higher dimensions and then there are geometric examples where instead of having functions with real values you have functions with values on geometric objects like spheres and they respect the invariances of the targets where they land. And this here, the ones I have here are called wave maps and they are very important in analyzing certain aspects of general relativity and they also preserve this Minkowski geometry. Okay, so I've shown you some equations. Did the battery die out, do you think? Yeah, okay, thank you, yeah, it does work. So why are these equations dispersive? Here I have a kind of long-winded explanation for it and I don't think you should pay too much attention to it, just assure yourself that there's a serious mathematical way of describing this. So these equations have a, first of all, they are dispersive if their linear part is dispersive. And the linear part is dispersive because when you solve it by the Fourier method what happens is two things. Certain quantities get preserved. For instance, the mass gets preserved, it's constant in time. At the same time the support of your function gets spread out. So if you spread out the support but maintain the mass constant that means that the size has to decrease. So the solutions to dispersive equations have to decay in time as time progresses. And this is the picture that you have to have. Things get spread out in support and go down in time but their total mass is constant or the total energy is constant. Okay, and you can ignore my formulas. All you need to remember is what I just said. Well this one works better. Now the interesting thing that, well at least it's interesting to me that happens with these equations. As I said dispersive equations are called dispersive because their linear parts are. Is that for their non-linear versions the dispersion may not occur. And there are certain solutions which are very peculiar that arise which do not go down in size as time goes by. And these are the so-called solitons, solitary waves or traveling waves which propagate just by translation. And these waves have been a mystery since the early 19th century. So these are solutions which do not change in shape and just move along a certain channel and that's how they propagate, okay. So let me tell you the story of solitons. It's an interesting story. So the first recorded discovery of such an object was found by John Scott Russell who was a Scottish engineer. He found them in 1834 going along one of the canals in Scotland on his horse. He was riding along and he saw this wave that was going very fast but did not change in shape. And he followed it on his horse along the canal for many, many miles and nothing happened. It would still remain the same, okay. And he paid attention. Now it turns out that Russell was not the idea of an engineer that we have today. He was a fellow of the Royal Society. He became a chaired professor at the University of Edinburgh when he was 24. He also designed ships. He designed the first Armored Free Gate, Freigate, Freigate, which was called the Warrior and he also designed the largest steam boat that was in the mid 19th century and it remained the largest one until the beginning of the 20th century. So this was a guy who had many, many talents. And so he presented his findings to the meeting of the Royal Society and Aery and Stokes who were fellow members of the Royal Society and were very famous mathematicians at the time said that this was nonsense, that this wasn't possible. There couldn't be such a solution to this water wave problems. And this is because they had developed theories that describe the motion of water waves that were linear. And if you have a linear theory, you will never have a solitary wave like this. And so since they couldn't explain them, they thought that they couldn't exist. Resounds today, okay. Now about 50 years later, Boussinesque and Rayleigh put forward theories to explain Russell's observations. And finally, Boussinesque and a little bit later, Corvac and De Vries formulated this Corvac-De Vries equation that I showed you earlier that described the motion of shallow water waves, such as the one observed by Russell many years earlier. But the essential properties of the solutions to these equations or of these solitary waves were not understood at all. Okay.