The Extraordinary Theorems of John Nash - with Cédric Villani

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Very interesting. I am just a little in, and already being blown away with the type of problems Nash solved. I thought Nash's biggest accomplishments were in game theory, but this isometric embedding he did seems uber relevant to a lot of fields.

👍︎︎ 3 👤︎︎ u/abomb999 📅︎︎ Nov 05 2016 🗫︎ replies
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[APPLAUSE] John Nash was a most peculiar man-- an extraordinary man. And it will be the purpose of this talk to pay due attribute to this man. And recall, as suggested by this title, that what he did was amazing or, to put it in the words of the great geometer Mischa Gromov, speaking of an accomplishment of Nash, "It could not be true, and it was true." Let's explain what Gromov meant by that. Here's a short biography of John Forbes Nash. Born in Virginia, he starts as an engineer at heart, studies chemistry, but eventually goes to mathematics. In 1948, he becomes a student in Princeton. 1949, 1950, he defends his PhD, including in particular the famous Nash equilibria. It's a two-page paper. From 1951 to 1958, he does a first-order world-class mathematical career in MIT, Courant Institute, in New York, and Princeton. And then he proved some theorems which are extraordinarily famous among analysts and geometers-- embedding and continuity. 1959, history-- a sad history of acute paranoid schizophrenia. And, around 1990, spontaneous, progressive healing. 1994, Nobel Prize in Economics, while lived in Princeton. This slide was written a few years ago, as I was for the first time lecturing on Nash. And later on I will update this, at the end of this talk. Here is a paradox. Maybe 95%, I know it's subjective measurement of Nash's fame is due to game theory and his work on the so-called Nash equilibria-- famous in economics, famous in biology. But is that not, to use his own words, the least of his accomplishments? Should not Nash be thought of as one of the greatest analysts in this century? As Gromov said, he was combining extreme analytical power with geometric intuition. And the goal tonight is to focus on the less-known story-- less known than the game-theory stuff-- to ask ourselves, what makes a value of a mathematical result, and to get behind the scene of a famous and dramatic paper of mathematical research. Let's get 60 years in the past, at the moment when young Nash arrives in Courant Institute. He's 28 years old. His a, you know, strongly built guy. He arrives in New York. And in New York, he arrives in this famous-- already famous-- institute for advanced, uh-- pride mathematics, in some sense-- Courant Institute, a place that, after the Second World War has reason to extreme mathematical fame and which is part of the world history of mathematics. As he arrives in Courant Institute, he's young but already preceded by his legend. And Louis Nirenberg, one of the most important people in Courant Institute, has a problem for him. Hey, young Nash is coming. I know what I will ask. What was it that had made Nash famous already? Let's do a little bit of flashback and talk about that. What had made Nash famous was the so-called isometric embedding theorem. To motivate this, let us say that geometry can be done in several ways. Historically, geometry is measuring the Earth or making a map of the world. "Map of the world," we think of something which is two-dimensional. You know, on a sheet of paper, let us draw the shape of the continents and the rivers and whatever. But, as we understood, as ancient Greeks already understood, the Earth is not flat. The Earth is spherical. And, as was first observed at the time of space exploration-- you know, the '60s-- the Earth is indeed round. And somehow that second view is simpler. The flat view is more complicated. Why more complicated? Well, you have all kinds of problems and paradoxes when you want to represent the world on a flat space. Problems of the shapes and, you know, the areas. If you look at the usual map of the world, in a rectangle, like, Antarctica looks like it's huge! And the countries in center of Africa, they look small, when in fact they are huge and Antarctica is not so big. It's distortion, due to the flat representation. There are other problems which are interesting. This is a picture which I took some years ago, in a hotel in Palestine. And, as people who traveled in some places in the world in which dominating region is Islam, when you arrive in a hotel you always have this. A qibla. That's an arrow indicating the direction of Mecca. Because, if you're a good Muslim, that's the direction in which you should pray. Now, my friends, what does it mean, "the direction of Mecca"? Well, viewed from here, it's OK. South, east. But the answer is not so OK, not so easy, if you are in any part of the world. If you are in North Pole, what does it mean, "direction of Mecca"? It actually was a problem. It was one of the motivations for development of geometry in the Arab world-- at a time in which European mathematicians were complete amateurs, compared to the Arabic and Persian mathematicians. And here is a problem, for instance, that really was a problem when mosques started to be built in the United States. Viewed from, say, New York, which is direction of Mecca? And the answer is easy, if you think of the Earth as a sphere. Put it on your globe. Say you take some elastic string, and you put one end in Mecca, the other end in New York, and you see what is the trajectory, there, of the string. But if you look on the planisphere, it's not obvious at all. And the answer, if you are in New York and you want to go in direction of Mecca, you have to face, somehow, northeast. This is a thing that people, some people, could never understand. Actually, there were some incidents with this in which some embassy was involved and, you know, saying, you are crazy. Put it direction of south, because it's south-- No, no, no! But it's geodesic-- it's shortest path-- and so on. I don't understand your thing. It's south-- it should be south-- whatever. Now you know, with the development of aviation, it's obvious. You look at the trajectories to go, say, from Paris to New York. Has to be this way, you know? It's a curved trajectory. And we know it's the best way. It looks as a curve, it looks as a path which is not economic on the plane, in two dimensions. But if you think of it as a sphere, it makes sense. It's obvious that's the way it should be. So spherical geometry, in a sense, is more simple than plane geometry, for that reason. On the other hand, plane geometry is more economical, because you just use two dimensions. While spherical geometry, you need the three dimensions to put your sphere in something, in our space, which is three-dimensional. So it seems that it demands more information, the sphere. So this is the kind of dilemma that you have. Either work on a simple geometry that requires "many" dimensions-- in this case, many three-- or work with a complicated geometry that requires few dimensions-- and, in this case, "few" is two. This is a problem that can get much more complicated than this, but this is the essence of this problem, which geometers called "embedding." Embedding is like the spherical representation, because we think of the sphere as part of the three-dimensional space, as embedded in the three-dimensional space. And, on the contrary, the plane representation, geometers would call it "intrinsic," because it only uses the two dimensions which are intrinsic to the surface of the sphere. So, intrinsic, or embedded. What is geometry? Should it be intrinsic, or should it be embedded? It takes us back to the problem of what we want to do in geometry. At first, geometry developed, you know, with the ancient Greeks as a Euclidean geometry, drawing figures that you can draw in the sand or on paper or on the flat stone or whatever. But then people understood, when you travel long distance, and so on, you need to understand spherical geometry. And then some people say, after all, you could do geometry on any surface, not specially a sphere. Could be some weird thing. And then people said, let's work on any possible geometry and devise a theory that can encompass all of them. And let's try, also, to do things which withstand deformation, things which are as intrinsic as possible. You know, if you take a sheet of paper, and you make a drawing on your sheet of paper-- this is my sheet of paper. I do some drawing on it. If I bend the sheet of paper, from an embedding point of view it will be very different, but the figure, here, will remain the same. So, if I look at it intrinsically, it should be the same object. And geometry made huge progress when it started to think in intrinsic terms, even for curved geometries. This is related to the development of the non-Euclidean geometries in the 19th century and, in particular, the so-called hyperbolic geometry, which was devised by people like Gauss and Lobachevsky. Hyperbolic geometry, like the Euclidean geometry, was a geometry in which all points were equivalent. No privileged point. A sphere, also, is such a geometry, you know? On a sphere, from any point the sphere looks just the same. There are very few geometry which have this property of looking the same. And the hyperbolic geometry is one like this. But somehow it's the inverse of the sphere. While the sphere is always positively curved, the hyperbolic plane is always negatively curved. And it has some weird properties. Here is an example of a picture, in the hyperbolic plane, of these lines which are all geodesic lines, meaning "shortest path." And we are making it rotate, like we would do to rotate a figure by rotation in the plane. See how unusual it looks. Here is a famous rendition of the hyperbolic geometry by MC Escher, the famous artist. And, you see, hyperbolic geometry is a geometry in which units of length will change from place to place in a plane representation. I told you, intrinsically every point is the same. But when we represent it on the plane we are forced to make it having dimensions that change from place to place, so that, in this, each fish has the same length as any other fish. This is hyperbolic geometry. And you may ask, OK, but now, can I represent this hyperbolic geometry in such a way that the distances are not distorted, like I would do on the sphere? Can I embed this hyperbolic geometry in our three-dimensional space? And this is a problem which already tormented Gauss, the early 19th century. And he thought, well, there's a problem in there. There's a problem. And it was a huge progress when Riemann, the student of Gauss, said, let's not worry whether it's embeddable or not. Let's go on and work with the intrinsic properties. And we'll make progress from this. And he defined the so-called curvature which, to this day, has remained the most important tool used by geometers to study non-Euclidean geometries-- the Riemann curvature. Without Riemann curvature, general relativity would never have existed, and an enormous amount of everything that we see in image processing could not exist, either. And Riemann set up the axiom that we still use, to this day, to describe a non-Euclidean geometry-- the concept of Riemannian manifold, which is a geometry in which each point is in a neighborhood resembling a distorted Euclidean space. This was a marvelous achievement and one in which you say, let's forget whether it can be embedded. So we can treat much more general situations. OK. Looks much more general. Is it, really? Is it really more general than embedded geometry? Hmm! Can one embed hyperbolic space, for instance? Can we have a representation of the geometry that I showed you which would be set in our three-dimensional space without distortion? Answer is, no. We can have approximations. We can have some partial geometries with some so-called singularities, like crests, like on these representations. This one, on the right top, is called the "pseudosphere." Below it, it's called a Coons surface. And, on the left bottom corner is called a "hyperbolic crochet." [LAUGHTER] I have one here. Let's hand it over to you. And you may pass it from people to people, OK? [LAUGHTER] That's a hyperbolic crochet. You can distort it, and so on, and reflect on the fact that every point of this has the same geometry properties as the others. And that they should try to continue this for knitting and knitting. There are recipes from this. You'll find this online, how to do hyperbolic crochet. If you try to make a very large one, you will fail. Such a knitting, such a crochet, has to be rather limited. And it was proven by some of the famous mathematicians of 20th century. Hilbert, for instance, proved it's impossible to have a very large hyperbolic crochet. So, OK, this seems like, yes, the embedded geometry is restricted. And there are some geometries like the hyperbolic space that we cannot embed. No, no, no. And the mathematician will tell you, OK, we cannot embed it in our three-dimensional space. But who said we need three dimensions? Let's dream! Let's embed it in a space of four dimensions, maybe, or five, or six-- whatever. It was proven by Blanuša that you can embed hyperbolic geometry, infinitely large, in a six-dimensional space. So, in the end, it is embedded geometry, all the same. But we just needed to enlarge the point of view. And now you may say, is it the same for any geometry? So here's the rule. I give you an abstract geometry-- an abstract surface, for instance. Can you find a Euclidean space, a straight space, but possibly with 10 or 20 or 100 dimensions, in which I can embed my geometry such that it would be a part of that geometry, as the sphere is part of the three-dimensional geometry. And this was a problem that stood open, ever since the time of Riemann. An old, respectable problem. Now, let's go back to Nash. You have to understand that Nash was not very humble, as a young man. [LAUGHTER] Nash was rather, you know, annoying. And when he arrived in MIT, one of his colleague, Ambrose-- a quite good mathematician-- became so annoyed with Nash's arrogance-- like, you know, I am a genius. I'm the best in here. OK, Wiener is good, but I think I'm even better. And so on. And once he was so angry, he told him, well, if you're so good, why don't you solve the isometric embedding problem? Nash's reaction was, what? What is this embedding problem? What is this about? And Nash was, like, thrilled. Oh, is this a difficult program? Maybe I can become famous by solving it. OK. So he checks, you know, asking people whether this is really a problem that can make him famous. [LAUGHTER] Starts working on it. Spends more than two years on it. Wow, he said, I will solve it, I will solve it, et cetera. Here is my idea, here is this, and so on. [EXHALE] Ambrose laughs at him. We have a letter of Ambrose in which he says to one of his colleagues, well, there's Nash. "We've got him, and we saved ourselves the possibility of having gotten a real mathematician. He's a bright guy but conceited as hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y." [LAUGHTER] This is mathematician humor, you know? [LAUGHTER] OK. But now the thing is, Nash did solve the problem. Not only one proof, but two amazing proofs-- two amazing theorems that he got from it. Nonsmooth embedding, smooth embedding. People were not even aware there was an interest in nonsmooth embedding, here. He proved both. Let me explain a little bit what it was about. First amazement was the method of proof. Here was an abstract geometry question, you see-- general question. When you have a general question, in math, it's natural to think it will be solved by general reasoning. Abstract question will be solved by abstract proof. Not at all. He solved it by concrete analysis, getting his dirty hands into big calculations. And we'll get back to that. It was to anticipate a little bit the same amazement as 50 years later, when Russian genius Grigori Perelman solved the most famous Poincare conjecture, about all possible shapes of the three-dimensional universe-- a very general question-- by some very hands-on and technical calculations and reasonings. Of Nash's proof, Gromov said, it's "one of the first works which made Riemannian geometry simple-- an incredible change of attitude on how to think of manifolds. You could manipulate them with your bare hands." This leads us to the question, what is an analyst? We asked the question, what is geometry? Now, what is an analyst? What is analysis? You know, mathematicians is not a single species. You have several [INAUDIBLE]. You have several subspecies and subsubspecies and so on. Analysis can be compared to fine cuisine. In Japanese it's the same word, by the way. It needs fine tuning, precise control. Analysts pride themselves on the strength and sharpness of attack with simple and powerful tools. We like to study, in great detail, functions-- signals-- often unknowns, because they are solutions of problems. And they ask how fast they change. is it a small variation, a fast variation, a big variation? Like stock exchange, will it fluctuate very much, and so on? Et cetera. Here are some examples of functions. Some of them are smooth-- slowly varying. Others are wild, and so on. Analysts spend their life on this. Well, the main-- the first tool of analysis are the derivatives-- the differentials. This is the peak of variation. This is a slope. And inventors of this famously were Newton and Leibniz, who were both geniuses and engaged in a horrible battle, for the shame of mathematical community, in a sense. But it was brilliant what they did. You know, look at this graph, there. If I drew the tangent to the graph, which is the line which touches the graph, and I look at the slope of this line, it will tell me instantly what is the variation-- if it's growing fast, or decreasing, and so on. And when I have this slope, I can, for any value of the variable, plot the value of the slope and then look at the slope of the slope. Which is a variation of the variation, the second derivative, and so on. And I can continue. And the study of these successive derivatives gives me strong information about the way these functions change. This is what people call the "regularity." Derivative is easy to understand. You know, 1% interest rate-- we understand this easily. Second derivative, not so easy. Starting from third derivative, it's really difficult to get it. Well, actually there is a famous example of third derivative in political speech. This was Nixon, 1972, when he announced publicly that the rate of increase of inflation had started to decrease. [LAUGHTER] Which, as you may guess, was not such great news. You know? And which, as you see, was a good way to say things are improving in a way that nobody can understand. [LAUGHTER] But for analysts, it's no problem. We're used to this. We use arbitrary numbers of derivatives-- or dimensions, for that matter. Even fractional number of derivatives is no problem. And the more derivatives there are, the more the function is smooth. This is our daily bread, and it can change the conclusion of a problem. For instance, if you're looking at a problem about fluid mechanics, whether you're looking for smooth solutions or nonsmooth solutions, it will change not only the mathematical attack but the physical conclusion of the problem, and so on. And, as a true analyst, Nash revealed that, in that geometry problem, the regularity was very important. And, depending whether you're looking for smooth embedding or nonsmooth embedding, the answer could be completely different. Geometers had no idea about this. The proofs are incredible. For instance, to construct his nonsmooth embedding, Nash started by grossly reducing distances, you know, in a way that was certainly not an embedding and then increasing them back, progressively, by some progressive process, by spiraling. Looked like crazy idea. And from the smooth embedding, he attacked an incredibly difficult system of equations-- solutions, in a sense, loses derivatives. That may not say anything about this, except that it was identified as a nightmare. And he understood that you can counteract this nightmare by a numerical method which had been devised by Newton to solve equations in a very, very fast way. In one problem, it's a regularity issue. In the other, it's like a numerical problem-- finding solution. But he understood he could play one against the other. Nobody had a clue about this. The methods were founding. And the tools that he introduced gave rise to new, powerful theories which later would be taught in mathematical classes. Even without referring to the geometry problems that they had been introduced to. It was not only solving problems but finding new techniques to solve these problems. And the conclusions were powerful and amazing. Here is one of the things that Gromov referred to as "impossible." You may take a sphere and crunch it, without altering its geometry-- its intrinsic geometry-- neither getting bumps, you know? Not like taking a hammer and-- boc, boc, boc, boc. There will be no bumps-- still, it will be crunched. This is contradicting our experience. We know in some sense that the sphere is rigid. But that transform defies experience, because it is hardly smooth. Here is how it looks like. This is the way that Nash-- or, rather, this is how, a few years ago, a team of mathematicians in Lyon represented what Nash had proved to exist-- a way to embed the flat torus. What is a flat torus? It's a geometry that is well known to people of a certain age who used to play Pac-Man. [LAUGHTER] You know Pac-Man-- it's like a square, and there is some labyrinth, whatever, and there is your Pac-Man, your body. And it goes, goes-- and when it goes out in some direction it enters back through the other side, you know. And when it goes this direction and gets over the top, it comes back from the bottom. This is known to mathematicians as the "flat torus" geometry, a geometry in which you identify left side and right side and up with the down. And that geometry, if you try to do it on a sheet of paper-- hmm-hmm, OK-- let's identify this with this. Easy. I do like this. OK. And I glue this. And indeed, if my little guy will go there, it will enter the other side, et cetera. But now you try to glue this to that. Mm-hmm! [LAUGHTER] Does not work. And you can prove it doesn't work. Well, Nash [INAUDIBLE], my friend, it doesn't work if you'll go for a smooth embedding. But if you do it in a clever way, and that is the clever way, it works. So that thing, here, which nowadays is called a "smooth fractal," looks like this. You see how it has tiny structures and micro structures and micro micro structures and so on. But still it's not irregular as a fractal. There is a derivative at every place in there. And it is flat. If you are a tiny, tiny, microscopic being, living on the surface of this, you would not distinguish-- you will not feel that it is curved. For you, it will be perfectly flat. It took years before geometers digested these new things of Nash. And then came, two years later, the great embedding theorem. If you take an abstract geometry, you may embed it in a very smooth manner, much smoother than this, but provided you put enough dimensions. In this, you only needed three dimensions. If you get it smooth, you cannot make it in three dimensions, but you can do it in a large-enough number of dimensions. This was great achievement. And there, it solved a problem which had been asked something like 80 years before-- whether the point of view of Gauss, embedded, and the point of view of Riemann, intrinsic, were actually equivalent. Answer? They are. That was good. That was good, and, as a result of this, Nash had demonstrated that he mastered regularity better than anyone. And that's the reason why Louis Nirenberg, when he saw Nash arriving in Courant Institute, thought, that's the guy to solve my problem. It was for a problem of regularity-- regularity of partial differential equations. What are partial differential equations? Partial differential equations are equations about derivatives-- tangents. But you know, in real life a function doesn't depend on one variable. Depends on many variable. Take temperature, for instance. If you're interested in meteorology, temperature depends on the time-- you know, this morning it was quite cold, today it's better, and so on. It depends on the latitude and the longitude and the altitude. Depends on four parameters. So you may compute the derivative with respect to any of these parameters. Is it getting warmer by the minute, or colder by the minute? Is it getting warmer when I get to the south-- et cetera. Partial derivatives capture these, these tendencies with respect to the parameters. And this is a big discovery of people, starting from the 18th century, that almost any phenomenon you can think of eventually is modeled by these partial differential equations-- related to tendencies of the functions. For instance, temperature in this room is a problem of partial differential equations. Electric potential in our brain? It's a problem of partial differential equations. Whatever. You know? Control, understanding the motion of fluids? It's partial differential equations, et cetera. Here are some of the most famous partial differential equations. I don't want you to understand them, but first you may appreciate how beautiful they are. [LAUGHTER] This here is Boltzmann equation. Gosh-- I spent 10 years of my life on this! [LAUGHTER] This describes the evolution of a gas. Was first devised by James Clark Maxwell and Ludwig Boltzmann and changed the face of theoretical physics. This here is the Vlasov equations. Tells us about the evolution of a galaxy, for instance, over billions of years. This one, on the contrary, is so, so small. Schrodinger equation, the basis of quantum mechanics. This one, actually, when you go to Paris you may visit a public place in which it is engraved. It's in the Paris subway. The sculpture, in station Chatelet-Les Halles-- I like it, because every day probably hundreds of thousands of people pass near this without noticing whatsoever that there is the Schrodinger equation on there. [LAUGHTER] You know? Like a metaphor of the fact that we are surrounded by these marvelous equations around us without us noticing this. Here is some other couple ones. These are the equations of fluid mechanics. Euler equation, Navier-Stokes equations. They changed everything in our technology, or many things. They are solved every day to predict the weather. They are used at enormous length by the Hollywood industry to make all kinds of special effects in the movie theaters. Go to see Titanic or whatever-- it's full of resolutions of these partial differential equations of fluid mechanics. OK. And here's another one, which was solved by other mathematical genius, Alan Turing, to understand the problems of pattern formation on the skins of animals. And here is another one, which was used by Joseph Fourier, in the 19th century, to understand the evolution of temperature-- say, in a block of iron. Here it is. Heat equation. It's a very famous partial differential equations and, when you take courses in partial differential equations, one of the first that you study. It's about temperature in a block of metal, conducting. On the left is the time derivative-- the tendency of the temperature with respect to time. Is it getting warmer, or colder? And on the right, there is the space derivative-- and, actually, two space derivatives. The tendency of the tendency, in respect to the spatial variable. And there is a coefficiency. It's the conductivity. Or, to be more rigorous, thermal diffusivity. They are in relation of each other. Because, you know, some materials-- in some materials, the heat is easy to transmit, in some others difficult to transmit. OK. When you have a conductivity that changes from place to place-- say, in a mixture of two metals-- the distribution of heat can become quite complicated. The equation is more complicated than the solution, though. You cannot compute it exactly, then, but you can study it. Let me show you some examples. Here's an example in which you have a heat distribution in a metal bar-- hot, cold, hot, cold. You see this is x. This is, like, the distance to the origin. And initially you heat some places and see how it evolves with time. You let it cool down. Look carefully. This is the evolution of temperature, as time goes. At the beginning, I had several bumps. Now I only have one. It's getting a bit boring, because the evolution at first was very first and now it's very slow. That's one of the first things that we learn when we study heat equation. Starts first and then becomes slow. Other things that we learn is that, even if globally it's cooling down, some places are cooling down-- others are not, initially. Look at this hot spot and this cold spot. After just a moment, the hot spot is becoming colder, but the cold spot is becoming warmer-- until there is some kind of equilibration, and then they start to get decreasing. OK. Another thing that we learn in these courses is that heat equation regularizes things. It makes you smoother. Look here. Is it smooth? Well, not much. You see these strong variations, wild variations, of the slope-- changes completely. But look after a fraction of a second. It has a smooth. You know, as a mountain gets eroded, heat equation does some kind of erosion on the data. This is the regularizing effect of the heat equation. OK. This simulation, here, is for a constant-- the homogeneous metal bar. Let's now look at another one, in which it would be a mixture of various metals. And let's also start with initial data of temperature that is crazy-- you know, hot, cold, hot, cold-- completely crazy. Look what happens after a few seconds. Ah, not so smooth, but better. And, you know, here it was quite discontinuous. But here, continuous. And better and better. We see it's not as regularized as it was with the homogeneous metal bar, but still not so bad. So you have some regularization effect. At least, that was the conjecture. And that is exactly what Nirenberg wanted Nash to prove with mathematics. Take any alloy-- you know, any mixture of metals-- in any geometry, in any dimensions, and any distribution of heat, initially. And let it act for a few seconds. Will it become smooth? It may seem like a very specific problem, but this was an important problem because it was a key problem to understand a whole class of related problems, you know? And so Nirenberg explained this to Nash, and Nash-- ha! How to do this? OK. This is what I was just explaining to you. And this is the mathematical statement. So, if temperature of time and x is a heat distribution in a medium with discontinuity conductivity-- you know, any mixture-- and a discontinuous initial distribution of temperature, after one second, will the temperature be continuous? This is the problem. And Nash is very interested-- also check that he can became famous for this. [LAUGHTER] And starts working on this. Starts working on this, gets started, and works on the problem. It's fascinating, because we have accounts and testimonies on how he worked on it. Going and meeting the people, you know, betting back to Nirenberg-- tell me more about it! I want to know this and this. And is it true that this and that and that? Nash was not a specialist at all of that-- of mathematical physics-- diffusion equation. Had no idea. And then he went to see other people-- people in Princeton, people in New York. Hey, I heard that you are specialist of this. Can you explain me this, that, and that? And at first his questions were quite stupid, you know? Like, he was an outsider, not knowing about this. And Nirenberg was starting to wonder, hey, is this guy as smart as they said? And, little by little, questions were becoming more and more to the point. And he was putting everybody through contribution as a conductor, you know? Hey, my friend, I need you to prove me this and this. I think that you are the expert, and you can give me this. I can use it to prove something more-- and so on. As a conductor who would give assignments-- you know, here, you're the violin player-- will play this and this. You are the trumpet. You will play this and this. Each one does their part. Nobody understands the great plan, except when the orchestra starts to play. And Nash had the overall plan for this. And everybody was amazed when, after six months, the problem was solved. Putting all people to contributions. And, again, the solution was amazing. Let's examine this famous paper that he wrote from there-- one of the most famous papers in the 20th century-- partial differential equations. "Continuity of Solutions of Parabolic and Elliptic Equations," by John Nash. And now you understand what it was about. 1958, 24 pages-- rather short. By current standards, very short. It's interesting to read what he thought about this. "The open problems in the area of nonlinear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. It seems clear, however, that fresh methods must be employed. We hope this paper contributes significantly in this way and also that the new methods used in our previous paper will be of value." That was absolutely true. "Little is known about the existence, uniqueness, and smoothness of solutions of the general equations of flow of a viscous, compressible, and heat-conducting fluid." It's still true. Almost 50 years after what he wrote, we are still in the dark about some basic features about this. The style was informal and informative-- very interesting. You know, he doesn't suggest this and that and that. He also says about key results. Also tells you what he had to work hard on. And he also says things like "This is dimensionally the only possible form for a bound"-- reasoning a bit like a physicist. He speaks of "dynamic inequalities." What does it mean, a "dynamic inequality"? Well, you know, it's to convey some impression. It's precious. Or "powerful inequality"-- et cetera. "The methods here were used by physical intuition, but the ritual of mathematical expression tends to hide this natural basis." He doesn't want just people to see right through. He wants them to understand how he got into this. Very generously, actually. The notions were unexpected. He was thinking of the solution of the heat equation in terms of statistical mechanics, with temperature being like density of matter. As if there was such a thing as atoms of heat. Doesn't exist, the atom of heat. But let's think like it's this way. And he used the entropy solution of disorder used by Boltzmann, and later by Shannon, to measure the disorder or microscopic uncertainty of that temperature. From physical point of view, this quantity minus integral of T log T makes no sense. It would make sense, you know, if T was, like, a density of matter. But there's no such thing, as I said, as matter-- as a particular associated to temperature. But it works. In a radically unusual context. And he used this notion in a context which is very different from the one it was introduced for. And he demonstrated, with his proof, the power of differential inequalities, that these inequalities between slopes of various quantities, involving simple quantities containing information, in some sense. It was, a bit, the same amazement as the one that Grigori Perelman, again, would generate when he showed that, to solve the Poincare problem of geometry, it was useful to introduce some entropy. . This was a masterpiece in several parts. Let me not go into details but very rapidly say that it was like first act was about understanding displacement of what would be something like an atomic sort of heat. Like a Brownian motion, like a particle of temperature. And, through them, the temperature was neither too low nor too strong. That contributions of sources of temperature would overlap, in some sense. That, if two point sources are closed, then the resulting heat distributions are closed too, and the decontinuity. And each of these steps has precise mathematical statement. You know, in the ideal of mathematics the Holy Grail is to have a nice, beautiful proof. And we get this concept as a legacy of the Ancient Greeks, and geometry in particular. This implies this from that, et cetera. They like playing with this, but the-- like, the bricks are not triangles and lines. These are, like, qualitative properties of the heat distribution. Let me skip this, even though these were some of the inequalities. And let's focus on one thing, here. You see, what is written here is Nash's inequality, in the middle of this slide. Everybody in analysis knows this is Nash's inequality. Truth is-- and it's clear, if you read the paper by Nash-- Nash did not prove this inequality. He asked one of his colleagues, named Stein, to prove the inequality. Stein was an expert in this kind of things. You want this inequality? Yeah. Let me prove it for you. Here is how you do it. Thank you. And Nash showed how to use it in that problem of heat distribution. He was genius in this kind of integrating the various parts. And of on. Let's skip this-- also this. He got from another guy, actually, this one, from his colleague Carlson, who introduced him to the concept of entropy after he had studied that from his colleague [INAUDIBLE], who had been specialist of Boltzmann equation. That was the style of Nash, taking this idea here, that idea there. Mmm-- I think they are linked. I will use them. And so on. This was brilliant. [INAUDIBLE] Hollywood and movie, after that would be the big celebration. I don't know-- Nash would have married a beautiful heroine or whatever. It was not this way. The celebration was lost. In 1957, Nash heard that a young, unknown Italian mathematician, Ennio de Giorgi, proved the same result by a different method. To all mathematicians, this would become as the, you know, schoolbook example, textbook case, of simultaneous discovery-- by different people, at the same time, with different methods. Di Giorgi was completely unknown at that time. He was eccentric, he was monachal, he was genius. He would become a living legend. His proofs set the standard for generations of experts. Nash, in spite of being so bright, was in pathological need of recognition and attributed to this coincident his failure to get the 1958 Fields Medal, which went to French mathematician Rene Thom. The epilogue was sinister. Nash's paper was accepted in Acta Mathematica, arguably the best journal in the world, in those days-- maybe nowadays, also. And after it was accepted he withdrew it. Which is unheard of, you know, if your paper is accepted in such a journal, to withdraw it. And he sent it to the American Journal of Mathematics-- maybe the most famous paper in the history of American Journal of Mathematics-- in the hope of getting the Bocher prize, 1959, a prize which had to go to a paper published in there. OK. In vain. It was Nirenberg who got the prize that year! For some other work. Hmm! Nash actually is already undergoing paranoid delirium in there-- the start of a long tragedy and more than two decades of going mental hospital, at times, having some relapse and some clear thoughts, at other times being completely miserable-- haunting the corridors of Princeton, talking nonsense, whatever. Meanwhile his papers will make their revolution. To Louis Nirenberg, in 1980, somebody asked, do you know mathematicians that you can consider geniuses? And Nirenberg answered "I can think only of one, and that's John Nash." In 2011, to a young, brilliant Princeton mathematician, John Pardon, somebody in an interview asked, who is your favorite Princetonian, living or dead, and he answered "Probably John Nash." In 1994 as he had been out of his mental condition, Nash was awarded a Nobel prize in Economics for his PhD, for that two-page paper in which he defined the Nash equilibria. Here is the situation. OK, you know it's not really the Nobel prize. It's the Sveriges Riksbank Prize in Economic Sciences in memory of Alfred Nobel, awarded jointly to-- et cetera, et cetera-- including John Nash "for their pioneering analysis of equilibria in the theory of noncooperative games." Looked like Nash's genius had been recognized at last. 2001, it was the movie Beautiful Mind. So this is Russell Crowe, playing the role of Nash. I'm not sure he understands anything of what's written on the blackboard. [LAUGHTER] I can recognize these are the equations of isometric embedding. The film beautifully manages to massacre everything in the life of Nash. [LAUGHTER] The chronology's wrong, the nature of the mental condition is wrong, the nature of the science contributions is wrong. It's amazing how the film manages to get it wrong on all points. Brilliant. Nash hated every single bit of the movie. But interestingly, his wife-- his wife Alicia-- found it was lovely to be played onscreen by Hollywood beauty Jennifer Connelly. [LAUGHTER] Now, for mathematicians, it still remained as a problem, you know, that Nash's genius had been recognized in economics but not for his truly beautiful mathematical work related to the geometry and analysis. And little by little came this public recognition. 2009 was a big splash, when Nash's ideas led Camillo De Lellis and Laszlo Szekelyhidi, bright young mathematicians from Italy and Hungary, respectively, to construct some impossible solutions of Euler's equation-- crazy solutions. Imagine a fluid that would be at rest initially, then start to agitate like crazy, then be at rest again, without any force acting on it. Something that made us rethink the definition of what is a solution to a fluid equation. In 2012, I depicted my emotional encounter with Nash in "Birth of a Theorem." This was, like, autobiographical book, or at least talking about how it is to prove a theorem. All the ups and downs, difficulties, mistakes, traveling, meeting with people-- whatever. And one chapter is devoted to my encounter with Nash in Princeton, at the T. And I was so, you know-- he represented so much to me. I was so, how to say, impressed that I did not even dare to speak with him on the first encounter. And 2015, Nash at last was awarded the Abel Prize. Abel Prize is, together with Fields Medal, for sure the most prestigious prize. Abel Prize is certainly more difficult to get than Fields Medal. Abel Prize is younger-- started, like, 15 years ago. Abel Prize typically goes to the living, old legends in their 70s who made contributions that everybody knows and is amazed of. And I sat on the committee on the Abel Prize, in those days. It was very emotional for me, also, as being on that committee the person who was scientifically closest to Nash-- you know, the one in charge of defending the work of Nash. I will not say more about the discussions, because, of course, it's secret. But this was very interesting discussions we had about this. 19 May 2015, in Oslo, the Abel Prize was awarded jointly to John Nash and Louis Nirenberg-- you know, these guys, they have a long history together-- for striking and similar contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis. Very emotional moment. You have to imagine the ceremony, also-- King of Norway was there, Nash gave speech, and so on. There was public speech of Nash, in which Nash recollected some of his older work, some other work that he discussed with Einstein, related to general relativity. I had the honor to be his chairman. This was our fourth encounter, actually. And for all the community, everybody thought, at last-- Nash got this reward. And it was quite due. So it was the mood-- after this prize, he will at least-- he can relax-- recognize, money, whatever. Not so. 23 May 2015, back from the ceremony, en route to Princeton, he died, together with his wife, in a taxi crash. We met again with Nirenberg, later on, trying to make sense of it, whatever. Nirenberg [INAUDIBLE] Nash ever did anything like anybody. Extra-terrestrian. Always gives these proofs that no one can understand at first. He himself was unable to explain them-- always saw things that people think is impossible. Gets Nobel Prize for his PhD work, after having becoming insane from an illness that you're supposed never to recover. And, in the end, dies like nobody else. Anyway, this is tragic. But the beauty of it, of course, is the methods. And many mathematicians feel that Nash is part of their family, in the sense that he brought so many ideas and techniques. If I think of my own work, I can see clearly relations with Nash's work. The taste statistical-mechanics problems, the taste for entropy, the key role of regularity-- which I uncovered with my collaborators in similar problems. We, for instance, in the problem that was one of my-- our-- most noticed papers, about behavior of plasmas-- we uncovered how critical the role of regularity was, in a way that had not been understood before by physicists or mathematicians and this. And, also like Nash, I admire his talent take pleasure in this sharp, massive attack of given simple problem, in some sense-- with simple and well-calibrating tools, always trying to uncovering new connections. This idea, that idea, let's put it together, and so on. And so, in a way, the Nash legacy still lives on. And I will conclude with this. Putting on, flashing, just this bibliography, recommending to you, by the way, the Beautiful Mind book by Nasar-- well, especially chapters 20, 30, 31, in which you will find the stories behind the scenes for the mathematical achievements-- which, for us, is even much more important than the mental in the story. Nash wrote four big papers, you know. That's nothing, compared to a big list of papers. But, of these four papers, three, probably, in retrospect, would have deserved the Fields Medal. On my web page, on my blog, on date of 26 December 2015, you will also find, in French, a short article-- or not so short-- in memoriam John Nash, entitled "Breve rencontre"-- "Brief Encounter." And, with this, I will conclude this discourse. Thank you. [APPLAUSE]
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Channel: The Royal Institution
Views: 583,157
Rating: 4.8630838 out of 5
Keywords: Ri, Royal Institution, john nash, math, mathematics, maths, cedric villani, game theory, geometry
Id: iHKa8F-RsEM
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Length: 59min 51sec (3591 seconds)
Published: Wed Nov 02 2016
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