Professor of Mathematics, Université de Lyon, Director, Institut Henri Poincaré, Paris, Winner of the Fields Medal 2010, France.
Breaking the Walls between Economics, Physics and Geometry. How Optimal Allocation of Resources and Entropy Meet in the Non-Euclidean World
Hello everybody. It is a great pleasure and great honour for me to speak in this session after these great talks we have just heard. After Ingrid, I will take you back for a short while in the wonderful world of mathematics. Now, there is some kind of curse on the mathematician, like in the legend of the Lady of Shallot. The mathematician is condemned by this curse to look at this world only through its reflection. In this case, this is abstract reflection of mathematical formulas and concepts. And the world is complicated, as we know, and full of walls; and so is the mathematical world: also full of mysteries, of riddles and of great walls waiting to be taken down.
If you ask a mathematician what is the greatest wall of all, maybe, the most famous, there is a good chance he answers: “the Riemann Hypothesis”. Here, I wrote: deepest scientific mystery of our times – of course, this is pretentious exaggeration, but if you talk to a mathematician, that is kind of the idea. The Riemann Hypothesis is related to the distribution of prime numbers, how they arise among numbers, and because these numbers, in turn, play the critical role in encryption,The Guardian a few years ago fantasised that maybe a solution of this mathematical Holy Grail could bring disaster for the internet. Of course, that is exaggeration, spectacular, but still there is no need for that exaggeration. The Riemann Hypothesis is a monster with many implications. In fact, there is a million dollar bounty on the head of the Riemann Hypothesis; if you solve it, you get one million bucks. But as Andrew Wiles once put it: most mathematicians would gladly give away a million dollars to have the honour to solve this problem.
Now, Riemann did not only leave us with unsolved problems, but also he solved great problems. One of those, which he left us, is the non-Euclidean geometry, called Riemannian geometry in his honour. In this world of non-Euclidean geometry, at each location, units of length and angles may change – like in the picture on the bottom here in this artist’s view of the hyperbolic geometry, in which each fish has the samelength, just because the unit of length changes from place to place in this disk. And in this world, the shortest path between two points is not any longer a straight line, but it may be a curved line – like, in fact, on the surface of the Earth. Planes that want to minimise the time they spent in the air don’t follow a straight line; they follow these shortest curves, which are called geodesics.
Riemann also left us with a great way to distinguish between some of this geometry. This is called the curvature. In this world, some curvature – if it is curved in the positive way, as we say, as is the case on the sphere – this is on the bottom left here, then triangles are fatter, and geodesics get closer. Look at this outrageously fat triangle drawn on the sphere with three right angles, adding up to 270°. On the contrary, if you live in a negatively curved world, then triangles are skinny, as on the right, in this hyperbolic geometry.
The negative curvature world is often very elegant and has been the source of inspiration for artists. Look at these beautiful shapes, negatively curved hyperbolic surfaces, arising in several problems. The one on the right there, by sculptor Hiroshi Sugimoto, is currently under display at a major exhibition of contemporary art in Paris – where else could that be?
But, Riemannian geometry is not only good for art. It is also a tremendously efficient tool. When Einstein developed the theory of general relativity, he used the formulism of Riemann. You see him on the blackboard? The equation that he is writing here: RiK “R” here stands for Riemann. It is the Riemann in curvature – Ricci-Riemannian curvature. This also has had applications in your daily life. Whenever you use GPS, there are some calculations in there that are related to general relativity. So, there is a bit of Einstein and a bit of Riemann in your GPS – as you can check next time you open it.
Now, Riemann did not do only this. In fact, when you look in an encyclopaedia, the list of concepts to which Riemann has attached his name is just tremendous, in contrast with the brevity of his life. All this makes Riemann a kind of romantic hero. He would be the Chopin of mathematics in a way. He changed completely the whole thing in a short amount of time. In fact, I know of a world famous rock singer who told me that she goes from time to time to meditate on the grave of Riemann – maybe getting some inspiration. Why not?
I too, like to go over graves. Here's me in the central cemetery, in Vienna, on the grave of one of my heroes: Ludwig Boltzmann, Austrian physicist – one of the greatest. Boltzmann developed the kinetic theory of gases, in which you are interested in the statistical properties of the gas, like the air around us, or any kind of situation in which you don’t want to look at all the precise positions of the molecules of the gas, but the statistics of it – like the curve that is there on the right on the top. Just below that curve is the Boltzmann equation, which describes with extreme precision the evolution of the distribution of particles.
Think about this, because this is amazing achievement. The particles all around us, these are all molecules – billions and billions of them, all bouncing on each other in a kind of chaos, like in a fury, crazy really. In spite of all this mess, the Boltzmann equation can predict with great accuracy what will become of these statistical properties, and you can use it. Boltzmann further devised the formula for the entropy: S = k log W, which is now written on his grave. S = k log W: “W” here is the number of configurations in which you can arrange your gas. For instance, if your gas is confined in the half box, this gives you much less possibilities than if it is spread all around the box. How many less possibilities? It is a huge, huge number – much, much bigger than these attoseconds we heard about recently – or than the debt crisis, of course.
Now, Boltzmann also gave practical formulas for this. The formula that you see at the bottom: -∫ f log f, is a practical formula for computing the entropy. This is a lot for a single man, and at the time of Boltzmann, not all his ideas were well understood. But later, his followers hailed him as a hero, like Einstein, Perrin or Smoluchowski: people who used his work to prove the existence of atoms – that now we can visualise thanks to technology.
On this example also there are important applications that came later. There are some people probably in the audience whose car engine was optimised by some code based on Lattice Boltzmann equation – to look at the air movements in the engine. Now, in the case of Boltzmann, like in the case of Riemann, we see that applications can come about 100 years after a scientific discovery. But, it is not always the case, and sometimes applications come in a very straightforward way. This is the case, most spectacularly, in the work of one of my other heroes: Leonid Kantorovich. If there ever was a devoted scientist for the public good, this was Kantorovich. His work is emblematic of all mathematics with these apparent contradictions and diversity. He worked on very abstract subjects like functional analysis of partly ordered spaces, but also very concrete topics like railroad transport, frightening applications like the atomic bomb, but also very benevolent ones like helping people to escape from the siege of St. Petersburg by the German army through the road of life – or revolutionising the taxi fare, which is an example, which did change the life of people in Russia.
Kantorovich wrote some masterpiece treaties in mathematical economics, which won him the Nobel Prize in 1975. The Nobel Prize! Some of you might know that the father of Alfred Nobel was the inventor of plywood, and in a way, it is plywood that started the way of Kantorovich to the Nobel Prize. One fine day, these guys from the plywood industry knocked at the door of Kantorovich and asked him: “Ok, Professor Kantorovich we have this problem and so on; we are making plywood. We are extracting wood from this and that place, and we want to put our wood to the places where it could be processed, our machine can process that amount of wood and so on. How should we do it? How should we organise a transfer in a way that is most economical, the best possible?”
Kantorovich thought a lot and understood that this problem was of a much more general nature. He turned it into an abstract problem and solved with this a whole bunch of problems, including the problem of optimal transport asked by Monge more than 150 years before him. Now it is called the optimal allocation theory; it tells you about the best way to distribute production to match it with the consumption areas in such a way as to spend the least possible energy in transport. At the same time, Kantorovich developed the theory of prices, which was a very dangerous thing to do at the time – in communist Soviet Union. You could well risk death for inventing a rational theory of prices. So, this was really a great man.
Here, in fact, his theory, now called “linear programming”, occurs in a number of problems, like the one I put here, which are solved daily by many industries. This is taken from a tutorial that you get on the web of all these linear programming problems, which all fall in the range of the Kantorovich technique. Linear programming is, whenever it happens, you have to optimise some quantity, which is proportional to each amount of the unknown. It is done very well nowadays by a number of computer tools. So, you see here the power of the abstraction, which allows you to solve a number of problems at the same time once you identify the right formalism.
So, these are three great men. Each of them destroyed a great wall: Kantorovich and his theory of optimal transport, optimal allocation; Riemann and his theory of non- Euclidean geometry, fat or skinny triangles; and Boltzmann with his entropy and the formula. Now, sometimes progress is obtained by destroying a wall against adversity, against the unknown. Sometimes it is obtained by destroying an internal wall, something that separates some fields. This is a case of what I am presenting here today. So, a wall which we destroyed together with some collaborators, between these three fields, recognising a hidden unity behind them and the link that was completely unexpected for us also.
This was made in a team; like many scientists I am just part of a team, just another brick in the wall – the great wall of science – in particular made with my German collaborator Felix Otto, at the time in Santa Barbara, and my American collaborator John Lott. So, one fine day, I was in this ugly building in the math department in Berkeley, on the beautiful campus, and John Lott knocked at my door: “Hello, Professor Villani, I read your paper with Felix, and I would like us to do some great work together about Riemannian geometry and your work on optimal transport.” I was very surprised: what is he going to do? We did a long collaboration and solved this problem and found this link that John felt there was, as a consequence of our work with Felix Otto.
Let me, in just one slide, one thought experiment, as Einstein would have said, summarise this link. Let’s call it the “lazy gas experiment”, and you will recognise the three basic ingredients of the three theories I presented to you. It is a new way to conceive positive curvature. But now it is not about the shape of triangles, but it is equivalent; but it is set in terms of statistical theory – the way you would explain the ideas of Riemann to Boltzmann, if he was still alive, or to his ghost. Now, this is the experiment. You start from a gas; you see here on the top left, there is at t=0, a certain density of gas. There are some regions of high density, low density – and you decide that the gas has to be in another configuration, at time = 1 – say one minute later, you impose this on the gas. And the gas will obey you, and all the particles of the gas and all the molecules will be redistributed to occupy positions at time 1 in such a way that the new density is achieved. And the gas obeys you, but he is lazy. He doesn’t want to spend too much energy; so, he will do this process of relocating the mass in the most economical way, as in the world of Kantorovich. While the gas is doing this, at each moment of time, you measure its entropy, its disorder, in the sense of Boltzmann, using the formulas set by Boltzmann about the entropy s = -∫ ρ log ρ − whatever this means. You compute this entropy, and if the curve is like this, always concaved, always above the line, then you know you live in a positively curved way, in the sense of Riemann. So, in this way, the three concepts are linked in an unexpected way.
Will this have practical applications? Who knows? Maybe 100 years from now, maybe a few years, maybe in some computer program, who knows. In a way, this is the end of my talk, but this was the beginning of a story for me, based on that, in particular, I wrote a thousand-page book, Optimal Transport: Old and New, which was quite a hit in the community and inspired many further researchers. You see we get inspiration also from frequenting these dead people. This also, you may say, you see as soon as you make connections between different fields, all the knowledge that you accumulated here you can recycle there. All of a sudden you are richer in terms of knowledge. So, for the moment, I will just be happy to say that I am happy as a scientist who just discovered a new way to look at the world, a new enrichment in terms of knowledge. Thank you very much.