New Frontiers in Mathematics: Professor Cédric Villani, “Optimal Transport Theory”

Published by Darron Toy on

our last speaker of the symposium is Professor cédric Villani elect member of French parliament see Jake is an exceptional mathematician in every way he worked on partial differential equations Riemannian geometry and mathematics physics in 2010 he was awarded the Fields Medal for his outstanding work on the kinetic theory of both my equations and nonlinear Landau damping he said he's my personal hero he's a equally well-known for ultimate transport on which he rode the two very beautiful books is a very unusual for mathematician to be elected to a parliament for many years to Drake was the director of the Institute Henri Poincare a made a lasting positive impacts he is a great spokesman for mathematics to which to whom we are grateful he also wrote her a popular book the birth of a theorem today we are very pleased to listen to him on the topic to be announced by himself [Applause] thank you thank you she may everybody hear me Asia including at the back yes thanks a lot and it's a really a pleasure to be here part of this event also to be part of an event which is associated with the name of de moivre and I will also explain why he's one of my the example that I used a lot in my public speeches and first let me explain why the reason of the image you know I had scheduled to give a handwritten talk and when I arrived I discover that there is only this and they told me to put beautiful image so I put this image which I liked her for several reasons also first it shows that it reminds us that mathematics can read you everywhere because it was during a mathematical conference in Algeria that there was this beautiful trip in the Sahara Desert and actually that's one of the great things of mathematics that it makes you travel so much and also it will be this talk will be about the journey you know the road and however about the fact that you should not have too precise plans of where you are heading to when you are doing research maybe it would to take your time or to go on a sideway or whatever let me also insist that one of the reasons why I decided to go for handwriting's first I love the old-fashioned handwritten talks and secondly the beautiful computer talks like we've seen today require a lot of time in preparing you can do a talk that is 10 times better with the computer but it will require you hundred times 100 times more preparation time and you know when you're a politician in Parliament you have so little time so during this week I had to go to China you know we was part of this presidential visit in in China in total of three days I slept a total of seven hours you know there are all kind of things to do everywhere it's impossible to focus on the talk that you have to prepare when you are in such a precipitation you know so it had to be a talk that is not very much prepared it will be a bit of a messy talk but I hope it will not be a big problem and hopefully it will also be institution for a question interaction also I understand that there are some very young fellows around there yes from the high school yes now people you are the hope of the nation of we count on you you are the future etc you are the best and this talk will be impact in Russia aimed at you so that I will try to be not too technical but I would insist I will tell you about the subject but also about how I came into the subject how I change direction etc so title could be sub optimal Road to optimal transport the subject is optimal transport okay I will have to write big so that you cannot see right optimal transport is a program by which you want to have some object that you want to transport from one place to another and you want to do it in the way that is most economical in the original formulation which goes back to the 18th century it was something like this assume that you have some file of some earth to dig in and you have to reconstruct it in some kind of way you know and how will you transport all the material these from here these from there etc in such a way as to spend the least amount of energy for the very practical problem it was an engineer's problem but it was an engineer's problem which was set by a guy with a very strong mathematical spirit this was Gaspar mandra the same guy who was a played an important role and science at the French Revolution who was one of the founders of the culprit technique etc and it's one of the example of great mathematicians who were able to think in a very theoretical ways but also with very concrete problems in mind and that has been very enriching so that is the subject I will talk you about I wrote two books on the subject which you may was very kind to mention and I'm very proud of having written these books enormous amount of effort of synthesis but I will tell you about a sub-optimal road to go into this subject because the directions which he took for me were never of direction which I intended at first and let me explain you about this so some of the key words that we'll be going through this talk are the power of meetings international travels travels local and international interactions the power of chance you know in research is advice for the young people one of the most important things is how you recognize when you are being lucky and how you explode your pieces of luck I had enormous pieces of luck in my in my career and the fact that I was able to spot them and use them was a completely instrumental unexpected developments will be important and also a theme which I like is always teach even if it's for research it's good to teach including teaching stuff that you don't master yourself so let me go through this and let me give you a more precise definition but only slightly more precise so imagine that you have a distribution that's it's the same picture except that the you know on this side instead of doing it right inside the ground it will be something like up the ground something like we are transporting an amount of material like this doing something like this let's call this a probability measure mu zero and this is probability measure mu 1 on some space whatever and assume that you have a unit of mass here you it you have to give a certain if a certain amount of work to transport it to another position and that will be the cost of transportation maybe here this is position x0 this is position x1 and this will be the cost to go from x0 to x1 people in the back row is it ok if I write at this side yes thank you so and everything yeah I always if is number one thing if you don't if people don't understand what you are saying this is the talk the talk is lost so and each amount of matter that you transport you will pay this cost this is the intuitive way to formulate it there are civil rigorously real mathematical ways to formulate it one way is like probability to say that you want to minimize the expected value of the cost to go from the of the this is the cost see this is random variable X naught this is random variable x1 and you want to do this of all possible couple realizations of X 0 X 1 such that the load of X 0 is equal to mu 0 so that is a distribution of X 0 and the law of X 1 is equal to mu 1 that is one possible way to mention it to state it in trouble istic terms let me give you another way which is equivalent this will be let's write it this way minimum of double integral of C of X Y square PI of DX dy well this is a joint probability measure imagine that this is a the space of X 0 this is the space of X 1 a consistency I should write this like X 0 X 1 DX 0 DX 1 will be better an imaginary probability measure here it's like you know some continuous distribution there when I integrate in this way I see mu 0 sitting here when I integrate in that way I see mu 1 sitting here and I think of this drug distribution measure as the plan it's my if I am engineer this is the transportation plan telling me every time I have some mass x0 here how much I should move from this place x0 to this place x1 so that the transport be as efficient as possible okay there are other there are other possible ways to reformulate this problem but so far you have the the idea so it tells you that it's about the optimal way to transport one probability measure to another so it's part of probability measure if you want but it's also part of optimization because you want to make something as small as possible and it turns out it can also be recast as a problem in geometry if you put some geometric features here in the cost x0 x1 so for instance if you work in a twilight to the square here sorry there's no square but very often the cause that you will put will be something like x0 minus x1 square square distance if it's a norm if you are working in the Euclidean space or if you are working in a geometric setting it may be distance from x0 to x1 or distance from x0 to x1 with a power tool or with a power P and these problems has turned out to be extremely efficient also to work out in geometric context actually in geometry you know maybe the most basic concepts of geometry that goes beyond Euclidean context is one of geodesic what is geodesic you have one point another point and you ask which is the shortest path to go from this to that point so it's an optimization problem it's part of calculus of variations that the basis of geometry so also here is an optimization problem in which I am minimizing and actually you can see this thing as a kind of geodesic problem in which probability theory is incorporated so mixing the geodesic idea with the idea of probability measure how did I go into this subject at first let me tell you because as you mentioned I was trained in partial differential equations and the kinetic theory which is theory of gases and and plasmas and it turns out that in the seventies Japanese mathematician named Tanaka had showed the following ok never mind it doesn't distract you too much this no ok so Tanika had said the following it was in the 70s if you define that will be to of measure one measure two to be the square root of the optimal transport cost to go from Museum to mu1 you know that is solving the problem of transporting mu zero to mu1 most efficiently possible when the cost is equal to x0 minus x1 square square of distance that is you pay a very high price when your particles are far apart and this is in our dish maybe R is equal to three and this you have to think of in physics as the space of velocities then when you look at a certain kind of Boltzmann equation which is the equation describing gases you see that as one of your solutions of the Boltzmann equation evolves with time let's say mu zero T as time goes and the other will be mu one T as time goes to different solutions evolving with time through collisions making the probability distribution change from time to time then you will see that the distance between mu zero T and mu one T is a decreasing function of time so that your two solutions become closer and closer to each other and why was the NACA interested in this to give a new solution a new proof of one of the most basic facts in k-tec theory that is when you let gas act through collisions between the particles then the probability distribution of velocities converges to a Gaussian velocity to Gaussian distribution so he was able to give a new proof that for any solution of the Boltzmann equation probability distribution of velocity as time go to infinity it will converge to this familiar bell-shaped curve like the Gaussian you know we call this curve cows but who was the first to discover it dumar yes and that the connection with de moivre here and do have discovered it like it was very unexpected theorem like the maybe the first very advanced theorem of probability theory when you look at say a game of heads and and tails and you throw up your coin in the air and you count the head and the tail you know and I've read it will be 50% 50% but you can look at the you can look at the error the error will be smaller and smaller as a number of trials increases when we let see what is the distribution of that error if you make a zoom so that it gets fixed as the number of trials increases you will see a beautiful cows distribution a beautiful curve like this with an equation that is something like constant exponential minus v square divided by something and the mob was the first to get this very particular function you know when we learn the beginning of probability theory we were told about the law of large numbers it's not obvious to prove but that this is intuitive you know if I threw my coin in the air many many times it will be 50% 50% a child can believe this but why does this come from no child will understand where it comes from it's really advanced and unexpected and that's one of the great things the Maya did apart from that the world is a great example because he was a French you know you know from the Protestant and he had to escape from France after the when the non treaty was revoke 18 by louis xiv and they religious persecution and it's a great example of the talent that you can lose when you have an approach which is not tolerant and when you start segregating people and so on so it's a lesson we should never forget when you try start to put barriers and prevent the freedom you are losing talent to the outside world okay that is for demova let's go back to Tanika and Boltzmann equation so far so good I took it would be little bit confused but anyway so I was I was a student and very interested in this and I happen to discuss with my with the supervisor of studies in economics appear here when I was his name was Ian Bruni Asia and it turned out that goony was interested in this same kind of optimal transport things but for completely different reasons he was an expert of fluid mechanics and particular incompressible fluid mechanics working on the Euler equation and trying to construct some real solutions of Euler equation and he had found out that by using in a clever way this optimal transport problem of most he was able to construct unexpected solutions of the Euler equation and so we started discussing with this and he told me wow that is great and it's very funny because there is lot of work currently on this optimal transport problem and he showed me a paper I told me you know this is a young German guy his name is Otto he is working on this kind of problem also from partial differential equation but from a completely different perspective related to heat diffusion equations then we started discussing with this body at some point he said why don't we organize a conference together in which we will discuss about various aspects related to the dis optimal transport thing I said yes of course of course I had to organize a lot of things but that is very that is very teaches you a lot also and I did not present my own work instead I rather explain the work of Tanaka even though it was decades old and I explained how it was interesting in perspective etc and Otto presented his stuff about you can use optimal transport to get a new point of view on the heat equation and I listened to his talk carefully it was it seemed a bit abstract it seemed a bit real it was different from all the other the other talks and we had a good discussion and now a great piece of luck which one of the best pieces of luck which P occurred to me later was that a few weeks after that I was happening to be reading a course on probability theory by initial LeDoux from tourism one of the advisors in my jury specialist of Boltzmann equation had told me take a look at the law to work it's not your PhD theme exactly but maybe it's related you will find some interesting example in there so I started to read the little book and I thought wow this is fantastic a problem and so on and at some point I see one of the children that was explained by Lulu it was through him by telling her about so-called concentration of measure and it was a certain relation in certain spaces it could be probability measure relation with the optimal transport problem with the entropy problem which is a way to measure the disorder in the Boltzmann framework and with the so called measure concentration problem which is when you have a probability space and a certain set C for a B T 1/2 and when you enlarge this set how fast do you increase the probability measure and there was a little by telegraph which was you know explaining how it was related giving some relation between this and I looked at this result and I thought there has to be a connection with the talk by auto which I heard a few weeks ago I am sure there is the connection it's often in mathematics that good ideas come from connected to connecting to things and they indeed it took me about ok it took me quarter of hour to find the connection but it was wrong it's often the case that our first fries are wrong then a bit more work and I found the connection right even it was formal and then I rediscovered with Otto and we published the paper together which was not about a new theorem but about a new proof of a theorem which was already known press some press some extras stated in this way it was not a big deal I am exaggerating a little bit there was so quite some work but it was you know work of a week's worth of work something like this and my friend this is my most quoted paper of all my career not that it is the most important you know one should not it's not that your paper is the most quoted that it is automatically the most important neither way wrong neither neither way also but still I think his paper one had some importance and it came from piece of rock and from the fact that I was attending these two events none of which was directly connected with my PhD but having some some kind of relation another thing which was important in the little toe in the little course in that this is the first place in which I heard about the lazy gum off is a parametric inequality which I will explain about and which I was fascinated by immediately so let me before I go into this lady guava let me mention the sequel and again it would be the kind of joyful Celtic randomness that that can happen to you in the or in the career we found with Otto this proof of new theorem I will not tell you exactly which theorem it was but anyway at a short distance from that I was invited in Atlanta by Eric Allen who was member of my jewelry and expecting me to present some results about Boltzmann equation and when I arrived at the seminar I talked of Boltzmann equation but I also talked about why the work that we did with Otto it was a good piece of chance that in the audience there was Wilfred gumbo who was a specialist of optimal transport and who thought wow this thing is completely new and he was an expert on the subject but he was very much excited by this and he thought there was there was something that we have to make further and we got along well together we were freed so he arranged that I could come again and spend one semester in Atlanta for being a visiting professor I said of course yes I want to come and what should I do and he told us it would be great if you could teach us a course on optimal transport now to make sure at that time I only had one small contribution to optimal transport theory but the great majority of the theory I could not claim to be an expert in anyway so the natural situation would have been something like you know I'm not an expert in optimal transport let me to make a course on Bushmen equation and I will perfect myself in optimal transport with you and then later we can see what we can do but instead I said yes I will teach optimal transport my friends teaching is by far the best way to learn a subject so I spent a semester working out this course on Atlanta about optimal transport preparing my course at city Ryan in the end this course became my first book on the subject which became the first reference textbook kind of in the in the shield and after teaching this course I could say ok now I'm an expert I've read these dozens and dozens of papers I know the various proofs and making developments etc and the story doesn't finish there after that there were a few developments but you see on this example the power that can have on your career the various encounters the randomness the exchanges and the ability to be ready to to teach now let me tell you a bit more about lady come up and this will be the transition to the sequel of the talk you many of you maybe most of you maybe all of you know about the isoparametric inequality is a parametric inequality the classical one is the parametric inequality it's something that feeds you what say you are in ahran ahran and it's you kiddin space on and you are considering a certain set with a certain measure so maybe the set is a and it has a certain lebesgue measure and you will consider the ball the euclidean ball which has the same measure let's call this alpha' and let's assume that this is the ball bish with the measure that is equal to alpha volume is alpha then the volumes are equal then the measure of the surface of this the surface of this has to be at least the surface of this so let's write it like let's write it like surface of a is bounded below by the surface of bitch it's a very powerful principle and you know it's used to explain about the shape of bubbles because in the bubble the quantity of air inside is fixed so the volume is given but the energy is minimized when the surface is as small as possible so that's why you have round bubbles etc it's one of the most fundamental inequalities in geometry so prototype of geometric analysis on and it's the motivation of many many developments now this is a theorem about Euclidean geometry what can we say about non Euclidean geometry then there are a number of possibilities of ways to generalize this but one of the most satisfactory is the following and is known as the lady gamma is a parametric inequalities okay so what does it say take and dimensional geometry Romanian manifold as we say so you have this geometry with the the surface or generalized surface maybe in many dimensions let's call it let's call it Emma and it has dimension N and on there there is a way to measure distance which is the geodesic distance there is a way to measure the volume there is a way to measure the surface etc and you are taking a certain set let's call it a you know maybe it's a messy set it's a part of your manifold and let's assume let's write let's assume that the volume of a divided by the volume of your manifold M is equal to alpha so it may be if alpha is half of the set M in the sense of volume alpha will be equal to 1/2 and okay you want to compare it with some other sets in some other geometry if your manifold is something very irregular whatever it doesn't mean anything illusion of a particular ball okay you may define a ball as a set of points which lie at a certain distance from a point but it may be very messy too instead of the past which non neutral and geometry has chosen to go is by comparing it with the sphere so this will be a sphere and what you will try to compare the situation to situation occurring on the sphere even the idea that the sphere is maybe the most simple non Euclidean geometry that you can think of now to define us here you need to know about the dimension and you need to know about the diameter of the sphere equivalently you need to know about the curvature of the sphere you know if it's very small curvature will be high if it's large the curvature will be low and here the diameter that you like the radius that you want to take will be square root of n minus 1 divided by K and this K here you take as a lower found on the curvature here curvature of M being bounded below by K here let me not be very precise about this curvature the current not even the correct meaning of curvature here is the so-called Ricci curvature which is a famous for its tied with general relativity and also the fact that it appears everywhere in certain branches of intersection of probability theory and geometric theory actually whenever you have problems with some kind of is a parametric content you can be sure the natural curvature involved would be the Ricci curvature so and what you do is this you consider the ball here B that is set of points which are at distance which is no more than a certain diet radius from a certain point here and you chose your ball in such a way that the volume of the bore divided by the volume of this here is equal to alpha also okay so it's the same as having the same volume that we saw before and then what the theorem says that if this situation is correct then the surface here of this set is not less than the surface here of this ball so the surface of this divided by the total volume okay of course I should have made some compromise about the yes this way it will be okay maybe so the surface let's write it this way of a divided by the volume of M is bounded below by the surface of the ball B divided by the volume of the ball so you see it really sounds like these are parametric inequality that we saw in the Euclidean case but it's a non Euclidean version it's a sharp in many respects it's really let me say that it's really a beautiful inequality first guessed by a lady in the 50s probably with a poo that was non-rigorous reproved by Gromov in the which proof that was rigorous but that was a bit crazy personal question yes hello Sigma oh this is s Thank You Pascal ok contradiction is important in our field and it's the basis for a number a number of developments actually Livi was interested in this precisely for same kind of stuff that al Adha was interested in problem about measure few ish because for instance once you know once you have an information like this concentration of measure once you have an information if you know an information about the volume of the set a it implies the information on the surface then when you increase the set by enlarging it the information of the surface will in fly in time information on how far how fast the volume of the set will increase also so it will be part of the same problem of concentration of measure which lady is considered to be the founder of many applications so in probability theory etc the proof by romoff was beautiful but natural in a way but resting and very very sophisticated tools tools about geometric measure theory that you don't want to hear about let me just say that the first step of the proof is to find a set a which is optimal which achieves the worst possible situation in the inequality and it's like finding a solution to the use of parametric problem and these problems are not to known to be extremely technical relying on huge proofs and ok we can use these big black boxes and be happy with it but we may also say come on if I use a very sophisticated tools it will be difficult to generalize it will be difficult to optimize or change the parameters it will not adapt to modern or situation maybe it's not for instance if you replace the smooth geometry that is here by a non-smooth geometry the proof breaks down for instance eeeww if you want to put some errors it will break down and so on and so you would like also to find the better proof better in the sense that it is more elementary more elementary meaning that you control all the parameter that you can generalize it to a non smooth situation etc and from the start for many years it was really considered let's try to find a solution of this which is based on optimal transport fumes now I never myself try to be involved in this kind of thing first by myself until in 2004 in Berkeley I met American mathematician John Lott I will say it again kind of circumstances which is interesting I was there in Berkeley as a young professor visiting professor under the auspice ality of the mirror Institute with a beautiful visiting program in which there was no research obligation new teaching obligation and no administration obligation all I had to do in the mirror Institute was to have lunch once a week with the other fellows of the mirror Institute and tell them about what I was doing yes for this I was paid three times my French salary and a kind of good conditions in which dv4 understand that it's good to have you near the place to participate in this and not to put too much constraints however Berkeley environment was tough for the young professor that I was because very high profile research department many famous researchers very very busy all the time and I was there wondering what can I do what kind of collaboration can I do in the end I ended up collaborating with another guy who was also a visitor there not one of the Berkeley people he was there by chance her name was Charlotte and one day he arrived in my office and told me oh I read your paper with Otto it's very interesting what you did and we are going to do some fundamental work in geometry with this with a new definition of the Ricci curvature that is beta optimal transport I said what really I didn't know of all his problems and I had to learn about what he was saying this was the start of me going into romanian geometry as you said very important or so is that by that time I had spent a few years already in the economic superior de Leon research environment in which I moved in the around 2000 in which the spirit was much more multidisciplinary in a way than what I had known before in in Paris and the fact that I was able to go into geometry the fact that I went into this was also psychologically at least due to the fact that I was discussing in your very often with the geometers much more than I had been discussing with the parrot geometers advice for young people change environment from time to time it's not just the specialized skill that you get there it is different spirits some places will be very good at some kind of collaboration some places will be good at other things and depending on where you are you feel differently what you are able to do so together with lot we started developing things and the theory of how to deal with curvature in a way that is not based on the fancy differential geometry but based on probability theory basic probability and here is the definition which we worked on together with other people about the curvature let me give you this definition imagine that you have the space this is a metric space so this is a distance and this is a measure okay a reference measure like the Euclidian measure maybe if we are nuclear in space or the volume measure if we are in a non Euclidean geometry and we want to know if the space is curved unless I say that we want to know if the curvature of the space is non-negative okay and we want to do it in a very elementary way we may be able to explain with two physicists without using any sophisticated notion of differential calculation or whatever just a bit of measure Furion probability so what here is a way to do here is a way to define it we will say that imagine that you have a transport to do on this pace with one probability measure mu 0 another probity measure mu 1 recognize the situation and I transfer the measure I move the mass mu 0 to the mass mu 1 you know and in a way which is as economical as possible as efficient as possible with a cost function that is C of x0 x1 is the distance from X 0 to X 1 to the square ok and all around the way I do this interpolation you know going from M 0 to MU 1 so it's as if I am choosing the optimal way and transporting you know continuously from time 0 to time when these things each particle going from the initial position to the final position with a certain longer Jodie's equation and all along the way the probability measure will change so maybe assume this is viewed from the let's say this is used from the top I mean maybe I put here this is my geometry here and I'm seeing here maybe there are places of high probability low probability and this is mu1 and all along at intermediate time between T equals 0 and T equals 1 at maybe half time it will be a different distribution of mass because all the mass has gone halfway and I look if this map is spread or not spread according to the good all the laws of few kinetic theory of gases which tell me if the distribution of velocities is very much scattered or not and for this we know that the good formula is the Boltzmann formula that is the entropy of the distribution mu is the integral is minus the integral of Rho log Rho D nu where Rho is the density of U with respect to Nu the density of the gas so I look at this quantity which see when it is very high it means that the gas is very disorder that this motion is very much spread when it is low it means that the gas is quite order distribution is concentrated and I look at the evolution of this measure of disorder over time and if the curve of S is always a concave curve of time it means that I am in a curved space a space that has non negative curvature and the intuition behind this is that in a space of non negative curvature Jodie's it typically start by separating out and then they converge again so that the gas can afford to be more spread at intermediate times because the geodesics are more spread and that is the basic intuition behind it you can translate it in complicated rigorous mathematics etc but all that is technique most important is the first elimination like we are going to use this way to encode the curvature and now that is interesting because this is very robust notion for instance you can define it even if you have no differentiability and if you can't write down the differential equations of the geodesics and there it's very stable if for instance you make some perturbation the picture will be part of the game and if you have a limit situation of geometries which are curved this way it will also be a curve this way so in this way together with a John knot also at the same time with capital Ostrom we could prove some basic problems which were which were not known before some something like for instance assume that you have a family of geometries converging to a limit geometry let's see like this and you know that at each step the Ricci curvature of this is non-negative does it follow that this also has a non negative curvature even if the convergence is very very wild answer is yes but to prove that the answer is yes we have to first reformulate it using the formalism of probability measures optimal transport and and the entropy and this is an example of how a problem can be solved in a completely unexpected way you think it's a problem of just geometry and then the people will come with a solution that uses the entropy the intuition coming from the kinetic theory of gases etc that's an example of unexpected Road taken by the research ok where am i with the time yes beautiful we will keep a bit of time for questions now this field this connection turned out to have a lot of potential many people starting to develop it dozens and dozens of researchers started to explain to explore the connection between the optimal transport curvature it has become very difficult for me even to follow the developments last year or maybe a couple of years ago I gave a seminar on one of these developments which was precisely the solution of the problem I was talking you about using optimal transport to prove to give a new proof of the latecomer is a parametric inequality and the proof which was exactly as you wanted it to be that is a proof that was robust Elementary giving way to generalizations opening paths to new understanding and so on I wish it was my proof but that was not the case and the proof was given by cavaletti and MALDI know after some important very important connection by kata and it's a long story but full of connections but one too just to tell you one thing that was a key so all these guys these guys what they did is give a new proof of living off so based on optimal transport approve the press robust general applying in on smooth context when you try to come up with such a thing you think about it and think how and soon you discover that it doesn't work it can't work you see the classical proof of the lady gum of inequality is based on first finding a set which is best and then looking like you are making a imagine you are starting from this and then you will make a Fourier tion a kind of partition of your space based on this interface for every point which is here you look at the point on the boundary which is closest let us say that this is like this and this point here you see what is the point where it is closest think of this like the octagonal projection on the boundary and you will like cover your entire space by this kind of lines in which you have these all through these cruces approximation what is the classical and then you ask how am i able to connect this with optimal transport and it doesn't seem to work out because this is like a partition of the space in two lines along with something occurs an optimal transport whew it is known that such a partition occurs when the cost function is the distance it was already known to more but it was also known by expert and then we well-known that all the good applications in Romanian geometry came from cast function to be distance X Y Square and it's not the same thing of course and so your reason did that like factorize this it well but this not either until cavaletti for another purpose understood that in many in some situations solving both problems are equivalent and that you can kind of reduce to the theorem reducing only to cases in which the two are equivalent just this psychological changing there is a situation in which both programs are equivalent and it's efficient to work out an expert who knows this would be able to go through the proof to reconstruct it etc but just this information was very important and very imaginative and the other thing which was due to class tag was a connection with another of two pictures you see in the motion picture it's like let's do transport and transport will always occur through lines which don't intersect each other it was known from the start that if you do transport car say in errand with course which is like X minus 1 then the lines can never cross otherwise you know if this is transported to this and that is transported to that it would be better to do this and this so in all people studying cost function like this you were doing this picture and pour in convex theory had something that they called needle decomposition which is a way to study some convex inequalities doing the partition of things into lines we did not intersect my friend in the same picture but to realize that it was the same picture it took somebody who knew about this and that and that was the case of was chaotic and he understood that you could use this problem to do an intrinsic way of this needle decomposition that the people in geometric convex see we were very much used to it another example of the fact that having one foot in one field and another another field can give you keys eliminations about what to do now after these two strokes of very great imagination technique could come and come and come and come and then the problem was solved and then it occurred that unexpected developments of a field in which I was involved for various reasons turned out to solve that problem now these are just some examples of the many many application that optimal transport theory had over the past decades and which are you know surveyed in my books some of these fields are computer vision or financial math or meteorology or at some point for instance people found out that you could find some optimal transport interpretation behind the famous theorem and proof of the Poincare conjecture Peter tapping in the UK is the one of the main expert in these for instance many many things and it's so turned out that it's even more even even may have been this for quite a long time I'm still amazed by how much it goes into so it will be the connection with my current activities and with one of the themes of this thing's is the connection with the very famous artificial intelligence business so let's talk a bit about artificial intelligence I've been mission by the government for a report about the French strategy of artificial intelligence which means that every day for the past month three months let's say I've been eating artificial intelligence everyday eating meaning and thus discussing about people understanding what the economy is about how applications here and there I just had a wonderful discussions here with people from Imperial and people from deep mines here to understand about that and let me say that I like most of the other people I did not expect at all that it would become such a price you know the artificial intelligence when I was a kid when I was a teenager I was a huge fan of Douglas Hofstadter I'm sure some of you have his books like good Arabic and so on and I was fascinated by these pieces in which he was talking about artificial intelligence program in Lisp and whatever probably part of my passion for some parts of mathematics came from there but later when I went into research I went into PDEs I went into mathematical physics and like many other people I thought that artificial intelligence was quite soft not many big results etc actually when one of my collaborators on the optimal transport again early he moved to artificial intelligence I told him are you sure what you're doing you know it doesn't seem to be very efficient artificial intelligence these days of course was completely wrong and I was completely wrong and I had not expected and seen the turn that it had taken and as we know it became more and more famous and 2015 I had the surprise – so let's also say that when it whenever I had a problem about some artificial intelligence to explain about rode audience or whatever I would take out my phone and call done and the scheme can you explain me but this or this and that this is also one way by people who move from one field to another are very interesting and 2015 I was invited by the court people the computational learning theory people for the big meeting that they were organizing in Paris I received the invitation I thought gosh why are they inviting me I am NOT an expert at all so I called John why do they do you think they are inviting me and he told me well you know we don't know but these are very good people or if they are inviting you they have some plan whatever and said maybe it's because of the similarities with some of the keywords etc and of course I accepted the invitation even though I knew nothing about learning huge but my talk sounded familiar to them their talk sounded familiar to me I recognized many ingredients when you go into the technique convex optimization things probability theory some of the of the big tools and so on they really looked familiar I had good discussions many possibilities of intersection and collaboration maybe that they had to taken and there and a couple of years after there they are not that I did anything about it but the connections are there and it is my great surprise and great pride to see that my books are now quoted in many papers about learning theory in which the connection was made with optimal transport let me say that this is very recent research actually a bunch of papers appeared in proceedings or in print in 2017 some of the names some of the people there are ok at first it was precursor about by Hammacher and Luka who are the generative advance of your network I will tell you a bit later about the idea behind this and some of the people who worked on more explicitly the connection and optimal transport are drafts key chintala button mademou Lakeville go Hong Ling young lee soo cui yahoo yeah with the ocean to the famous geometer all the people who did the explicit link are either Chinese or American or Canadian which are three great nations about artificial intelligence and I very much liked the idea behind so let me to just give you an idea of what it is like let me sketch the thing so yes absolutely I'm a you know politicians become very talkative now in you know when you have a neural network the big thing is you want to reconstruct some unknown function you have a function which Maps some input to some output you know a lot of pairs like input output and you want to guess what the function is and you want to guess a function which doesn't depend many parameters just a small number of parameter that you want to explain okay and in the generative adversarial Network a lesser or learning which again you can describe to me as one of the very best ideas which occurred in the field in the past ten years or so it's like you have two new role networks one which tries to reconstruct let's say you have here this is something which depends on a small number of parameters let's call this Z it's a small space here you have a huge space many many parameters may be in some observations there and you want to construct these observations say as a function of this which is transporting some given measure on this space F into some probability measure there which you want to be as close as this to this as possible that is you want to reconstruct the characteristics of the observation from a small number of parameters and now when it is adversarial you have a second neural network which is acting against you trying to to chop you and which is trying to recognize whether this is different from this and trying to recognize from a sample whether it comes from the true sample or from the copy that you are trying to make and they are trying to chop each other this guy is trying to make seen that there are these three undistinguishable as possible and the other guy is trying to find the different season focused on there it's a bit like you know when you have the teacher and the student the student with blue try doing his best to reproduce the technique that the teacher has given and the teacher will figure out some exercises that will shut the student what is something that will resemble the thing but there is a trap somewhere and little by little the exercise and go to the optimal and now the idea which was which states routine work by Luca but then was developed explicitly by these people into the now and so on is that you can measure here the distance between these variations to do it with the transport make sure approach with one of these W functional that I told you and that in fact er this problem when you take certain cost functions to measure Hill discrepancy given in taking the geometry into account these problems here the adversary and the original they are like a dual problem exactly as in the optimal transport problem we know there is a primal and the dual problem one is about minimizing the transport cost the other is to about finding how to find prices which are optimal in some sense in the transportation way let me show you this will be my only use of the computer let me show you how these papers look like okay this is one of these early papers about the energy base generative etc one of the beautiful thing about this technique is the images that it produces quite amazing this kind these are kind of images that look weird but they look realistic when you look at them from a distance you think they are okay when you look at them clearly you see there is something which just doesn't work we will see some example with animals in other paper and here is one of these you see visage tan is one of the key word for this optimal transport kantorovich rubenstein duality is another problem is another way for the optimal transport thing and this is this paper that one of these papers Facebook also is involved in which they use the optimal transport problem to do the to do their adversarial network training let me try to get you yes this is a geometric view of optimal transportation and generative model so printing together the two feet this is the picture I was telling you versus train generative adversarial networks and why is this interesting Seifer they say they have a method and they show that their method is more efficient in some sense and one was done before secondly it gives you a connection connection gives you a way to eliminate both sides and also gives a new ways for people to enter into this let me see with these gosh I wanted to show you some of these crazy images okay yes these are some images that Yin Luke and other people for instance show in there Oh resolution is not so good okay well this I told you this talk would be a bit messy anyway please those of you who have not seen it Google app or quanta up generated adversarial networks images and you will see these creepy images in which you have shaped up animals that from a distance they look like familiar and when you look closely they look like they are not wrong something is wrong these are images that have been divided especially to trap the neural network which is trying to recognize the animals so I don't know exactly what is behind in these people in more details but I tell you my friends next year I will teach this ok ok that's all for thank you [Applause] thank you very much for the fascinating you know one of my problem is that I want to work on everything and at the end but since at the end I am also working in depth in the end I can do much much less than I would like to do but this is subject I was intending to start a PhD thesis in collaboration with the Facebook Paris AI research we had talked about this with some of the guys in Fair algorithmics of solving the optimal transport is a is a bit of a mess it's a bit of a mess and there is quite a contrast bit first it's only recently that some efficient methods have appeared and I am not enough expert to answer your question with certainty but so far there are not really sure of that I mean they are iterative methods as in many ways when you have a jority problem in which you go back and work back back and forth between the primal and dual problem some of them uses some gradient type technique and so on but it doesn't sound like reinforcement learning to me so far but you never know one of the young people there yeah yeah I can see they are working on that yes yes hold on hold on let me come get closer oh my gosh gosh it's my way – now it's my duty to think of all kinds of features and so on yeah yeah this I was telling you about development of the Facebook research lab it has never been my I've never been involved in contracts with Facebook and so on of course as we know there are many issues about the way we the way these big American companies now we can handle them at political level fiscal level control level ethical level whatever many many ways and in with the current extremely powerful status that they have acquired very very tricky problem in which one has to find a good solution you know we've been discussing a lot in France about fiscal issues and fact that it should be a taxation about the see for the fair gosh how Jose Caban how do you say this in English yeah and so on and not not the usual rules for the tax also all these discussions within the new European regulation the dinner data protection regulation about portability of data about the right of not being profiled without being informed etc etc these are car be car have become huge issues it's we should on the other hand of course not consider them as enemies because it would be stupid and we should consider that there are it also gives us information about ourselves I mean for instance discussion with the researchers who went from universities to Facebook by the way my friend Yan that I was talking about generally he went from university to – to the Facebook research and many other people and when you ask all these people why do you move to the Facebook clubs instead of remaining in the academia many of them tell us things like oh now that I have moved to this lab I am much more free for my research this is very weird we use to the fact that if you go to the private sector you have more money but less freedom but actually it's not always true and this shows just the fact that we don't have enough freedom within our research our research units and freedom can be also very pragmatic things on how to start experiment how to buy material whatever and they also tell us we have it's much more easy for me to do some experiments and start big computing to try my theories also that's something we should be able to provide so talking and seeing with this with the realities and with there will be a very important way to make us improve in comparison and one important part of the mission one of the subject that I will be discussing with Pascal Osia from saneras and that the people is how to set up the good facility so that our researchers in this very highly competitive area which has strong links with the industry also are given some very attractive conditions for working in our laboratories and in our public universities and when I see attractive it's not just a question of salary it's a question of having just the tools to do your research as you want yes it indeed is unreasonably effective you know when people talk about unreasonable effectiveness often it is about the fact that it gives so accurate precision in particular in some fields of mathematical physics if you go to areas such as control fusion in which nobody knows how to make it work and in which the experiment doesn't match with the theory people will not find it so effective but when you go in areas such as the predictions by quantum mechanics it's so amazingly effective that it makes you a wonder I like to think also that it's amazingly effective and reasonably effective when you see that a certain tool in mathematics writing mathematics can go here there there there are many other things and sometimes you see thing you see oh it looks like the right concept you don't know why but then it turns out to be useful in many different contexts so then it's a question of personal choice some people will say that because we humans are exposed to many situations that become good at finding the concepts which are sensual other people will say that is because the world is written in mathematical language and so on I like this second point of view but you know I don't want to impose my view on the others and let us also say that there's another quote which I like which is about the interaction between mathematics and life sciences and that other code by famous mathematician says something by giancarlo jota says something like the lack of contact between mathematics and life sciences is either challenge or a tragedy or a shame something like that and he said it's hard to choose which one this was said some time ago now there are much more links between mathematics and life sciences but still one big question is that while the interplay between math and physics has been so extremely strong that physics has contributed mathematics with many deep concepts which have in turn been used in other problems of physics in contrast there is very few there are very few concepts which have been inherited in mathematics from biology neural networks is one of them even though the neural networks reasonable vaguely our a neuron examination but one may say it's one of them but there are very few of these concepts so if you're it's important that when one says that recall that this quote about the reasonable effectiveness of mathematics was coming from a mathematical physicist I guess vigna and somebody involved in the biology application of mathematics may have said something completely different last question maybe last the last one okay so here and there here first and then we finish with the young one that is the dream of artificial intelligence people the artificial mathematician that will put us people out of business I think first it's a question that people ask often I think some of the reasons are first that AI people some of them come from science but many of them are also geeks self self-taught in some respect and some of them fascinated by higher math but saying it's not my business to do higher math and playing with the idea that maybe they will be able to recreate this kind of artificial mathematician another reason is that you cannot think of an artificial in the sense that physicists need experiments while mathematics it's only done in the you know from some given axioms and you can try it's one of the field that is in which you can argue that maybe reinforced learning can play an important role by exploring all possibilities and trying to teach and reteach and so on however my personal view is that to take a long long time there have been some experiments so far we know that for instance some programs are able to recover their moves of Hamid classical Hamiltonian mechanics apart from that and some some part of classical Euclidean geometry apart from that I am not aware of any significant strange straying thing it's a debate we have from time to time with my with some of my colleagues particularly with different mala and Stefan mara likes to say there will be a new change in science and artificial intelligence and data big data theory et cetera will allow us to discover new theorems new new scientific field etc but for the moment it has not occurred and you can even tease those guys by saying you know that statistical learning was developed in a sense are the way developed in a sense was removing meaning rather than putting meaning in it was like let's forget about how things work and which is the cause let's just put a bunch of examples and see the thing decide by itself by analogy what is the what is the output and it could be a view that from purely scientific perspective it is a regression I think this only time will tell if there are some significant contributions in a new new science fields based on AI there are have been some new discoveries in science based on algorithms mixing probability optimization whatever for instance recently one example which I like recently discovery by some biologists that there are actually four species of giraffes not just one was based on big Monte Carlo Markov chain simulations be easy an approach running through genetics of the of the giraffe so it was really a mathematically oriented sophisticated mathematically oriented discovery in genetics and it is very difficult for classically trained biologists say to explain how this discovery was made but it was the problem was specified it was let's see how many clusters the genomes of the giraffes can separate into it was not like let's find a new theory for biology evolution or something like this okay last last question there okay okay okay that question first one thing to say is that you guys in a French course abroad are privileged usually it works much better the training in these foreign schools in main French scusi outside France than the average inside France secondly I have another mission currently beside the mission artificial intelligence and part of the mission which is about the renovation of mathematics teaching in France now my friend it is very painful to recognize that in a country which boasts some of the great mathematicians in history and great pieces of mathematics our education has become so poor that we are at the bottom of the international rankings within ocde and in particular with the training of the young kids for instance when you look at meter high school performance about mastering of fractions you know France has the lowest score of all ocde countries UCD countries it's it's a real shame by the way we also have disaster results in english-speaking and these are the results in in reading about English for sometimes we are competing with the Italians and the Spanish to be at the bottom but I think we won the competition now and yes yes nothing to be proud of so to come back to the math question we will soon be giving back our conclusions and recommendations to the minister and we will make some very tough recommendations it's a whole system which is sick and some of the problems are about the mathematics teaching some programs are modern rule the more how to see the most pain food is to see that solutions are known and not being used it's a horrid mistreat administration system the whole governance which is very very sick so then I think we will go through some energetic I hope we'll go through some energetic reforms which is the tendency that the Minister has expressed he would do it rather start and the most important thing being to realize that France even though its education system istic has a lot of goodwill including a lot of passionate teachers a lot of pride and currently is in the mood of reforms and improvement so I am very optimistic that we will do it thank you [Applause]

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