 Ok, mes amis, même les meilleures choses ont dû arriver à la fin et cela fait que la célébration de Poincaré et nous sommes approchés du début de la fin de cette conférence pour aller bien avec la célébration de légendaire Henri Poincaré. Nous avons ici un professeur légendaire, Percy Diaconis, de Stanford, connu, bien sûr, pour ses bonnes habilités de mathématicien et aussi pour ses bonnes habilités de magie dans la vie du passé, que peut-être ce n'est pas trop passé. Donc, Percy va nous parler de Poincaré et de la théorie de probabilité. Merci. Merci. C'est un bon lunch, Cedric, et votre travail est de s'assurer que tout le monde s'occupe d'un de ces pièces de papier. C'est une main-d'oeuvre. Est-ce que c'est quelque sorte de réveil ou quelque chose ? C'est un bon directeur, il sait qu'il a des problèmes. Donc, bonheur, bonheur. Ce que je veux vous dire aujourd'hui, c'est deux choses. La première, c'est que je veux essayer de vous donner de la vie, un peu de ce que Poincaré a fait quand il a fait de la probabilité. Et la deuxième chose que je veux faire, et je vais essayer de le faire, c'est que c'est encore très bien réveillé par Poincaré. Il y a de la théorie, il y a des problèmes à l'open, il y a tous les idées intéressantes. Et je vais essayer de le faire. Ne t'inquiète pas trop, dans le sens suivant, Poincaré n'a pas de probabilité de sa main-d'oeuvre. Il a été intéressé dans le sujet quand il a commencé à apprendre sur le sarbonne. Et il a finalement donné une course en probabilité qui a été note, qui a été écrit en 1896. Et puis, il a publié la deuxième édition du livre en 1912, l'année que il a mort. Et donc, ce que nous avons à travailler avec n'est pas une collection de papiers, ou d'un gros corps de travail, mais ce texte qu'il a écrit. Et je vais parler de trois topics qui sont dans le livre, qui sont des topics que j'ai travaillé aussi. Et le premier est roulette. Et je pense que c'est utile de commencer avec une discussion de roulette réelle. On va bientôt faire de la math. Mais il y a aussi une roulette réelle. La roulette réelle, il y a un gros roulet. Et il y a un balle qui sort du roulet à l'extérieur. Et puis il y a un roulet à l'intérieur, qui est le spun qui sort de l'autre côté. Et le balle éventuellement sort du roulet à l'extérieur, qui sort du roulet à l'intérieur, et qui sort d'un numéro. Et il y a un table que vous pouvez mettre. Et bien, de cette description, et comme vous le savez, roulette est un jeu physique. Il y a un objecteur réel. Et quand j'étais un professeur d'assistance à Stanford, trois de nous ont réalisé que roulette est un jeu physique. Nous pouvons cliquer. Et donc, ce que nous avons fait, c'était de faire un petit gadget, qui était à la taille de ce gadget. C'est un gadget qui est connecté à mon pocket. Et ça va se rendre능 par rapport à ça. Donc, nous serions dans le cas gentil. Et le supérieur le balle. Et quand il y a un point de fixation dans le ruban, vous couvrez un bouton. Et quand il y a un bouton, vous couvrez de nouveau. Et maintenant le gadget bang Through-the-Move se défend. Et quand zéro de l'intervier entre la courte, vous taperez. et le tapis, et puis le tapis d'aujourd'hui, et le gageit sait comment vite le rouleau est coupé, et puis le gageit fait une calculation et il vous dit où la balle va finir. C'est seulement l'accuré d'un demi-rouleau. Mais si vous êtes en train de faire 35 à 1, sur un 18 à 1, c'est comme avoir un vacuum cleaner dans le casino. Et on a construit un gageit, et on a utilisé le rouleau, et on a fait de l'argent. C'est juste de la pensée, on n'a pas touché tout ça. Alors, à partir de la façon de mettre un peu de sentences en réalité, la première chose que nous avons fait, c'est de faire les équations. C'est une sorte d'intéressant math pour faire les équations. Et puis je rentre sur un rouleau rouleau, et on a essayé de le faire. Et tous nos équations étaient complètement d'accord. Et on a réalisé que, comme vous le voyez, on a fait des frictions, et c'est de manière raisonnable. Et il y a trois types de frictions qui ont été maté ici. Il y a la friction de rouleau, la friction d'air, et le type qui a vraiment été maté était la friction d'air, pour que, quand on a fait les conditions d'air dans le casino, toutes les équations ont changé. Ce n'était pas utile d'avoir fait la math. En fait, la forme des équations, on pourrait juste laisser leur forme avec des paramètres, et aller dans le casino et les tourner, et estimer les paramètres basé sur 20 ou 30 spines de rouleau, et avoir des assises assez précises de la rouleau. Et donc, il y a un monde de vrais roulettes, et c'est un jeu physique, c'est une pensée pour vous. Donc, pourquoi est-ce que la roulette est réelle? C'est une question, c'est-à-dire, qu'est-ce que c'est réellement réellement réelle? Et Poincaré a été avec cette question dans une chapter de son livre, et il l'a dit de cette façon, comme nous le faisons tous, il a fait une roulette de rouleaux, et il a dit supposé que quand la roulette est réellement réellement réelle, elle se tourne, et elle vient de l'aider avec une certaine probabilité de distribution, donc, f de, je vais l'appeler Theta, c'est la probabilité que la roulette s'étend à Theta, et donc, on a pris la roulette et l'a ouvert, c'est OK, c'est 0, c'est 2 pi, et f est une fonction, et bien sûr, si le dealer est très accurate, peut-être que f est très pique, mais en tout cas, c'est f, et Poincaré a parlé de la probabilité de la roulette, et donc, vous pouvez penser sur la roulette, les roulettes et les bleus alternent, et donc, il a pris une serrage, où c'est une petite radius h, et il a raison que la chance de la roulette est l'arrière sous la roulette sur les squares de la roulette. Et donc, c'est une propensity pour la roulette pour la roulette. Maintenant, notice, c'est simplifié, un grand, grand, il n'y a pas de roulettes anymore, il n'y a pas de bouchons, c'est la chance que la roulette s'étend à Theta, mais encore, il a pris les assumptions qu'il a fait. Et puis, il prouve, et nous allons aussi, que la probabilité de la roulette si h est petite, tend à 1,5 comme h est à 0, ce qui naturellement fait le sens, mais c'est le théorème de Poincaré de son livre de probabilité. version de son argument, il a dit, regardez, on peut bien voir que la probabilité subh de rouge minus la probabilité subh de noir et c'est la différence entre l'arrière entre les squares de rouge minus la différence entre les squares de noir et donc c'est moins que l'inégal de 0 à 2 pi de f de theta minus f de theta plus h de theta. Donc la différence entre les deux zones peut être boundée par prendre l'absolute valeur à l'intérieur et puis Poincaré dit que c'est moins que l'inégal de l'alcool omega subf de h, où l'alcool omega subf de h est le modulus de continuity, c'est le maximum sur les différences de moins que l'inégal de h de f de x minus f de y, c'est ce que l'inégal de la fonction est. Et il a connu que pour les fonctions continuantes, le modulus de la continuity s'adresse à 0 et donc cela s'adresse à 0 et la probabilité de rouge et la probabilité de noir sont équiles, dans les limites que l'alcool omega subf de h s'adresse à l'alcool omega subf de h. Ce sont des raisonnements utiles et c'est naturel de tenter de demander comment est-ce que l'alcool omega subf de h est si petit. Comment est-ce que l'alcool omega subf de h est si petit. Si vous avez des problèmes calculs, vous pouvez essayer de prouver la probabilité de rouge minus 1,5, je vais le faire de cette façon, c'est moins que l'inégal de h de 8 fois l'inégal de 0 à 2 pi de f prime de theta, de theta. Vous savez, en orderant à prouver qu'une sorte de théorique de cette sorte, quelqu'un qui pourrait être très accurate en roulant et cette densité peut être très très pique sur un certain point. Donc, de toute façon ou d'autre, vous devez dire quelque chose sur comment la distribution est pique. Et pour les personnes qui ont des problèmes calculs, ce n'est qu'une sorte de théorique de h de 8. Dans l'autre direction, il y avait une littérature suivant tout que Poincaré a écrit, il y avait une littérature suivante et Poincaré a assumé la continuité et les gens m'ont dit que vous pouvez le faire avec moins. En fait, aujourd'hui, c'est facile de voir que cette théorique, cette théorique est faite pour les fonctions majeures. Vous n'avez pas besoin d'assumptions, parce que la translation est continueuse en L1. Si vous aimez cette sorte d'argument, ça fait 10 à 0. J'aime cette sorte d'argument, qui me donne une sorte de rate. Cette méthode a été abstractée, et je vais vous parler d'une ou deux sentences plus sur ça. Je vais voir si j'ai... OK. J'ai une. Oui, j'ai beaucoup de choses. Une image de la randomité que nous parlons de tout le temps c'est ce, Pylophase, et pourquoi fait-il coin tosser, qui est une image basée de la phénomène de la randomité, pourquoi est-ce que c'est la randomité? Et Joe Keller a écrit un très bon papier en lequel il a dit, «Listen, quand on flippe un coin, si je savais combien de fois c'était allumé, et je savais combien de fois c'était allumé, Newton me dit, je savais combien de fois ça se tournait. Je savais combien de fois c'était allumé, et on se verra ce que ça va faire. Le coin tosser est physique, c'est pas random. Donc, vous pouvez faire la physically. Et le hands-out, j'ai une picture que je devais Ins communities, mais c'est OK. Je sais ce qu'il y a dans le hands-out. Thank you, kind person! So first, this is the picture from Poincaré's book, if you want to see it, you can see his version of the argument. This is the phase space of coin tossing, so let me just tell you what it's like. This is velocity, how fast the coin is going up, and this is omega, how many times a second the coin is spinning. Et donc, chaque fois que vous flippez un coin, dans cette version, dans laquelle c'est sort de flippant autour d'un accès à la centre de la gravité, un coup de poignée correspond à un coup de poignée sur cette picture. Comment est-ce que le coin est en train de se passer, et à quel point est-ce que c'est en train de se passer? Et donc, par exemple, un coup de poignée ici signifie que c'est en train de passer avec beaucoup de vitesse, mais très peu de spin, donc il va se passer comme une pizza. Vous pouvez imaginer qu'il y a une région, et c'est en train de se passer par une hyperbola quand vous faites ça, où, à des points sous cette région, le coin ne se tourne pas tout à l'heure. Par exemple, ici, mais aussi ici, le coin peut aller avec un grand nombre de spin, mais pas beaucoup de vitesse, donc il ne se tourne pas tout à l'heure. Et puis, il y a une région, une prochaine région, où il se tourne pas tout à l'heure, et puis, deux fois, etc. Et la picture sur la droite est une picture de cette spécificité. Si vous regardez la picture, ce que vous verrez, c'est que, quand vous avancez de l'arrière, les régions s'arrêtent plus près, et donc, intuitively, les petites changements en conditions initiales font la différence entre les hautes et les tails, et tout le monde sait que c'est là qu'il y a de l'arrière. J'étais intéressé en regardant les coins réels, où sont-ils sur cette belle picture ? Et donc, j'ai fait beaucoup d'expériments, et dans les années, dans un de ces, on a un strobe-lite tunable, et, vous savez, on a peint la poignée bleue et la poignée blanche, et on a flippé, et on a essayé d'adresser le strobe, et de la voir où nous sommes sur cette picture. Et dans les unités de cette picture, 1, 2, 3, 4, 5, dans la vitesse, la vitesse, les coins, la façon dont vous flippez-les normalement, vont à peu près 5 milles d'heure dans mes unités. Dans les unités de cette picture, la vitesse est un cinquième. C'est 5, c'est 1 à 5, c'est assez près de 0. Mais, la poignée blanche, quelque chose comme 40 révolutions par seconde, et une poignée blanche typique fait environ 1,5 secondes. Et donc, dans les unités de cette picture, la variable de vitesse est 40 unités. Donc, en fait, la picture ne dit rien sur la poignée bleue. Mais la poignée blanche, vous dit comment, la poignée blanche est serriée. Et bien sûr, si vous voulez savoir sur la poignée blanche, ce que vous avez, c'est une distribution de pictures ici. Et si les régions se mettent près, à peu près de la poignée blanche se mettent sur la poignée blanche et à peu près de la poignée blanche. Et donc, une peut faire une version de Poincaré's argument, ces types d'arguments pour ce type de set-up. Et si vous voulez apprendre plus de cela, en fait, les coins réels font quelque chose de plus compliqué. Quand vous flippez un coin réel, en fait, ils s'expriment. Et la référence du papier avec Susan Holmes et Richard Montgomery, nous avons fait le cas de 3 dimensions. Et donc, dans les 3 dimensions, c'est en fait, dans les mécaniques, c'est 12 dimensions. C'est, vous savez, c'est 12 dimensions. Et donc, vous avez une séiliation de 12 dimensions de l'espace. Et vous voulez savoir quelle portion de cela est en place sur la poignée et sur la poignée. Et il y a des versions de cette théorie pour cette sorte de situation. Donc, c'est Poincaré's roulette argument. C'est souvent appelé, aujourd'hui, la méthode des fonctions arbitraires parce que Poincaré était intéressé dans le fait que, pour une densité de probabilité arbitraire, si l'âge est assez petite, la chance de red est 1,5. Et la même chose dans ces autres conditions. J'ai dit qu'il y avait des choses à faire et l'une des choses à faire qui, comme je le sais, n'a pas été faite, c'est de faire la bonne abstraction de ces théorèmes. C'est-à-dire, j'ai juste dit à vous un couple de spéciales cases. Vous savez, cette 1, 2 dimensionnel, 3 dimensionnel. Qu'est-ce qui est le bon état ? Qu'est-ce qui est la bonne abstraction de cette théorie ? Donc, vous avez une scale, je ne sais pas, en fait. OK ? Comment, personne ne s'est réveillé et fait ça. Si vous êtes intéressé, Poincaré, à l'end of his probability book, has all kinds of things he's talking about which relate to that. OK, so that's my first topic of what's in Poincaré, the method of arbitrary functions. One more sentence. This method was brilliantly developed by Eberhard Hopf who began a classification of low-water differential equations. And if you have uncertainty about the initial conditions or the coefficients of them, how does that propagate into uncertainty and Hopf who I think called it method of arbitrary functions. And if you want to see references to the literature, there's a section on it in the paper that I wrote with Susan and Richard Montgomery. And for what it's worth, I put all my papers on my home page. If you type Percy into Google, you'll find me. And if you spell it correctly. And so it's easy to find. OK, so that's my first topic Poincaré on the method of arbitrary functions. This was a very, very innovative idea. What he was trying to establish was that there were certain situations in which the exact details of the probability distribution didn't matter. He had three examples in the book and this is one of them. The second example I'm going to talk about is a little less widely discussed, but hope that'll change after this talk. And so again, remember, my job is to try to bring what's in Poincaré alive to you. And so I'm going to tell you the way we think about it now and then I'll tell you what he did same way I just did. So I call this subject, Bayesian numerical analysis. And just to start, suppose I had a function on 0, 1, say on 0, 1. I have a function f of x. And I'm going to tell you what it is. It's e to the cosh of x squared plus cosine x plus x to the fifth plus cinche x. OK, that's my function. And so there it is, my function. And now suppose I also had a water pistol. Pistol, though. How do you say water pistol in French, say? Thank you. And I have my water pistol and I say, Cedric, what do you think the integral of that function is between 0 and 1? And he says, and I go, Cedric. So there's a question, what does it mean to know a function? Just seeing a formula for a function doesn't mean you know very much about it. You don't know about the maximum, you don't know about its integral, you don't know lots and lots of things about it. Well, one thing you could do and we're going to do is to own up to that fact and say, hey, you know, just seeing a formula, I know something, I don't really know all about the function. So let me try this. So since I don't know, I'll put quotes around no, the function, I'll own up to that fact and I'll put a probability distribution, a prior distribution on functions. Now, I'm not a fancy French probabilist so I only know one probability distribution on functions, Brownian motion. So say, I think, or I assume for a moment that f of x, I don't know what it is exactly, I have this formula, sum a plus b times b of x where b of x is Brownian motion. It's crazy, but bear with me for a second. Now suppose I observe the function at a few points, we see yi equals f of xi, one less than equal to i, less than equal to n. So I observe the function at some points. So now by calculus, I have an a posteriori distribution. I assumed it was like Brownian motion. I see where it is at a few points. I have a new distribution which is Brownian motion constrained to go through those points. Then you could ask the question, well, if that's correct, what's the optimal quadrature rule? What's the best guess at the integral with this information? So the best guess under squared error is lost. And I'll call it the integral of f of x. The x is f hat, I'll call it i hat. And what it is is the expected value, the average value of the integral of this b of x, the x given what you're given, the conditional distribution. It's just Bayes' theorem in this setting. And what this is, is the trapezoid rule, the trapezoid rule. That is, I have my function. I know it at some points. I connect the points up by straight lines and I use the area under that straight line approximation as that's what this calculation gives you. Well, seeing a classically used rule, the trapezoid rule is very often used, comes out of this crazy set of assumptions, makes me wanna try to use this board properly. So you might say, well, look, that function that you wrote down there, horrible as it may seem, it's nothing like Brownian motion, it's much smoother. Well, you might therefore say Brownian motion isn't the right prior. You could try some smoother prior. You might try say f of x is distributed as the integral from 0 to x of b of t dt or some other smoothing procedure. But let's take once integrated Brownian motion, that's a measure on the space of all functions. And if you use this prior and go through that calculation, you'll find that the Bayes estimate, the optimal estimate, is the cube-expline interpolate of your function. If you integrate k times, you get 2k plus one ordered splines. So seeing standard numerical analysis procedures coming out of this crazy assumption maybe makes you wanna think about them a little bit more. You can ask, going backwards, is there some measure on the space of all continuous functions say, which gives you even order splines? We don't know, open problem. Let me ask. The fact that I was using the quadrature rule that I got out of an assumption as some information about the quality of my prior assumption leads to the math problem is Brownian motion, the only measure on functions which gives you the trapezoid rule. Well, the answer is no. So let me write it down as a question. It's Brownian motion, the only measure on functions giving the trapezoid rule. And the answer is no. Any independent increments process, a Poisson process, gives you the same rule. Any increments, well, any independent increments gives the same. But independent increments processes have jumps. And if you wanna say I have a measure not on functions, but on continuous functions, then that gets rid of that problem. The second issue is, well, twice B of T, twice Brownian motion predicts the same way as Brownian motion. And there's nothing special about two. Two could be sigma and sigma could be random and independent of Brownian motion. But there's a beautiful theorem of David Williams which says that's it. That is if you have a measure on continuous functions which predicts in the same way as Brownian motion, that is gives the trapezoid rule as it's quadrature rule, then it is a rescale version of Brownian motion. And so that's an example of how little math problems come about. Now, nobody needs me or us to tell them how to integrate one dimensional functions. But if you're integrating in higher dimensions, this idea of working with our priori distributions and seeing what their Bayes' rules gives is a very, very healthy way of doing various tasks and numerical analysis. And so this little subject is called Bayesian numerical analysis by me. Now, who did it first? Well, this is a talk on Poincaré. So I'll tell you what Poincaré did. Poincaré did it first. And more or less exactly in this way. So here's what Poincaré did. He has a chapter in his book which is called Interpolation. And what's he doing? Well, he says, suppose you have a function f of x on say 0, 1 on r, on r, say. And I don't know it, don't know it. But we observe it at some points. Observe yi is equal to f of xi. One less than equal to i, less than equal to n. And he wants to know what's the best guess at the function at another point. So the best guess at f of some x star, some other point. And he says, I'm going to do this problem by the method of causes. The method of causes was 1900's language for Bayes theorem. And he puts a prior distribution on functions by assuming that f of x had a power series expansion. i equals 0 to infinity of some coefficients. I'll call them ai times x to the i. Where, according to Poincaré, ai was assumed to be Gaussian, independent from i to i. Normal with mean 0, say, and variance sigma i squared. Variances would fall to 0. And then he says, well, if that's true, then this unknown function is itself a Gaussian variable. And we're being told, in linear functions of a Gaussian process, and we're being asked to calculate it, it's mean at another point. Nowadays, that's a standard problem. When Poincaré was working, there were no reproducing kernel Hilbert spaces, he just does it anyway. And he figures out what the Bayes rule is. And I'll tell you, what his estimate is. Poincaré, he made assumptions on sigma i squared, falling off properly so that this exists for all. Poincaré looks at, let me call it phi of x, which is the sum of sigma i times x to the i. Equal 0 to infinity. This is an ordinary function. These are just the variances. And says that f hat at x, the best guess of f at a given point is given by the mean of the posterior, doing the same calculations I was doing, but he figures out what that is. And it's the sum from i equals 1 to n of epsilon i phi of x star times xi. This is phi. These are the xi's that you observed at. And epsilon i are parameters that are chosen to enforce this condition. So epsilon i, you have to solve a linear system. And he works out what this is, in the case in which this phi is a polynomial, in which case he's getting polynomial interpolation. And so that's quite an original thing to have done in 1890s. And it's an original thing now. And if you haven't seen it before, it may seem strange, but it's all the rage in certain parts of numerical average case analysis of this type is being used in all kinds of high-powered computations. And I put one reference on the references, which is a survey article by Andy Stewart. And if you want to see serious numerical analysts using this kind of Bayesian approach to solve very complicated problems, and this article is a wonderful place to start looking. But many, many serious numerical analysts are doing computations in this way. So that's my second example of Poincaré doing interesting things. And if you read that chapter, there are other parts of that chapter. And as far as I know, nobody's ever read that chapter. No, he didn't. He doesn't do any examples. He does work out exactly what this is in the case of a certain choice of the sigma i's vanishing, but that's as far as he goes. And I wrote a survey article long time ago called Bayesian numerical analysis, which traces other people who have done this kind of thing. And you can do it for any task in numerical analysis. You can try this as a way to go. Okay, so that's a second example. And so the last example I want to talk about is Poincaré's work on bâtage des cartes and shuffling cards. And so shuffling. And that was an example that Poincaré used in several articles for the public where he was trying to explain the onset of randomness and how things that start out not random could wind up being random. And I want to take you through a little bit of what he did and didn't do. So let me start where he started. In anything he wrote until the last edition, the second edition of his book, he said, well it's too complicated to do even with three cards. Let's do it with two cards. Okay, well bear with me for a second. So let me work with a deck which has n equals 2. And so there's two arrangements. The cards can be in order 1-2 or they can be in reverse order 2-1. And what we're going to do is to repeatedly mix them and the mixing is we're either going to do nothing with probability p1, this is the identity permutation, or we're going to switch the two cards with probability p2 and of course p1 plus p2 is equal to 1. And Poincaré says, let's consider the following bet. We're going to shuffle K times and if the cards are in their original order, 1-2, you get a dollar or Frank and if they're in the reverse order you have to pay me a dollar or Cedric. And so let's think about that. So shuffle K times. And if after K shuffles it's in 1-2 you get say 1 unit, if it's 2-1 you give 1 unit. So let's do a few calculations, little baby calculations. Suppose you shuffle once. If K is 1, what can happen? Well, if you don't shuffle and the cards are in the original order you get the dollar. If you happen to switch them they're out of order and you lose the unit. And so the average value is p1 minus p2. If you happen to leave them alone you get a unit. If you switch them once you lose a unit. Let's take K equals 2 now. You shuffle twice. What's the average payoff? Well, if you've done two shuffles you might do nothing twice. The average is p1 squared. Or you might switch them and then switch them again. In that case you also get a dollar p2 squared. But you might not do anything and switch them or switch them and not do anything. In both of those cases you lose a unit. So minus 2 p1 p2. That's the average payoff to you. And of course this is p1 minus p2 squared. And for general K the average is easy to show is p1 minus p2 to the kth power. And Poincare says you can see that that goes to zero so that the game is fair if you shuffle a lot. If K is large. And so that's Poincare trying to get people to understand that if you shuffle a lot you mix it up. You mix up the cards. Beautiful math has to go. But it will. Poincare must have felt a little guilty for having written four or five times. The case of general K is too hard. So in the second edition of his book which was completed in the last year of his life he adds a section and it's quite a long section maybe 15 or 20 pages of that length in which he analyzes the problem of shuffling cards for general deck sizes. And I want to at least tell you a little bit about where he got to and how he did it the kind of math that he used in order to solve this problem. So he treats the problem so for general deck sizes he says well there the different ways of arranging a deck of cards and let's label them Sigma 1, Sigma 2, the different orders Sigma and factorial all the different possible arrangements and our shuffling scheme will be starting at the cards all in order starting at the identity and we'll have a certain probability distribution over the allowable permutations so we work with repeated shuffles I'll call it PI is the probability of Sigma I some way of shuffling and and he says in order to work with this problem I need to use some algebra and this is 1912 and but still so he doesn't quite have the language but or he didn't care about the language so he works in what we now would call the group algebra which is he calls it generalized complex numbers but which is the set of all linear combinations of permutations I calls 1 to n factorial so my probability assignment I can associate to an element of you know it might only be it might be that PI is one for transposition and it's a half for a transposition and a half for an n cycle and zero for all the other cases so PI is some probability distribution on on these guys and he understands shows that if you square this element so if probability of Sigma after two shuffles after two shuffles well of course it's equal to the probability of Sigma I times the probability of Sigma times Sigma I inverse over all I the chance of winding up at Sigma after two shuffles you have to have done something your first shuffle and then chosen the permutation that gets you to Sigma after your second shuffle is the coefficient of Sigma so if you take this element in the group algebra and square it and look at the coefficient of Sigma that's the chance of Sigma after two shuffles and so on and he says what we want to prove and he says theorem under assumptions that I'll make specific in a second P to the kth power if you shuffle k times it tends to one over in factorial times the sum over all Sigma so and that says that if you shuffle the cards a lot they get all mixed up I mean all arrangements become equally likely that's what he wants to prove now that theorem the way I've just said it isn't true because your shuffling could be you just switch the top two cards and that's it well you're not going to mix them up the method that Poincaré uses to prove this theorem is methods that Frobenius and Cartan had derived about algebras and so Poincaré uses the following the idea of the characteristic vector or eigenvalue idea that so P is an element of the group algebra X is another element of the group algebra and Cartan had shown that we nowadays we understand that there are eigenvectors in algebras and this is an eigenvalue and this is the eigenvector so he's going to use these ideas and he proves that that this convergence holds if and only if the support of P i is not in a coset of a subgroup so you need the way that you're shuffling to be able to get to all possible arrangements and you need to avoid parity problems and he's aware of that and in these cases he shows that any eigenvalue is less than or equal to 1 and that the vector this element of the group algebra is an eigenvector with eigenvalue 1 he knows that and he proves that all of the assumption that all of the other eigenvalues is strictly less than 1 and he therefore concludes that this converges to that he's aware and has to deal with the fact that this operator, this element P might not be diagonalizable so you need something like the Jordan form of an operator and he worries about that and deals with it and does manage to prove this theorem and it's quite an elaborate argument with a lot of details in it obviously somebody was bugging him about it maybe he was bugging himself about it but for Poincaré there's an awful lot of detail there some things he doesn't do doesn't one is mention Markov that's not surprising but Markov 1906 at any rate had developed the theory of Markov chains and if you want an interesting answer to a good question ask me during questions why did Markov develop Markov chains it's an interesting story Markov developed Markov chains for some interesting reason and he had two examples one of which was rhyming patterns in Eugene O'Nagan gathering the data actually and the second was shuffling cards and so Markov has an elaborate description of shuffling cards and a much more general setup Poincaré's argument really uses the fact that he's in the group algebra and it wouldn't work for more general Markov chains the second thing he doesn't do is anything quantitative anything quantitative tell you two quantitative results so you can see what I mean there's the usual way that people shuffle cards you have a deck of cards you cut them about in half and you go like that you riffle shuffle them together and Dave Bayer and I showed that three halves log to the base two of n plus c get you e to the minus c close so if k and so this is says about seven shuffles required to mix up 52 cards if you riffle shuffle on the other hand if you do the other method of shuffling you know this method of shuffling you know what I mean by that you have a deck of cards it's n squared log n versus n squared log n for overhand shuffling and so when n is 52 n squared is 2500 log 52 is about five four and a half so it's 10,000 so it takes 10,000 of these versus seven of the others right so there's an interesting world of quantitative theory Poincaré was interested in the fact that for any method of shuffling as long as it wasn't stupid you would converge at infinity and that's what he was set out to do and what he did and Tempus Fugit so I won't I won't say more about shuffling I promise you I could the last thing to say is something for all of this but in particular for me the math problems that I'm working on right now are trying to understand what I call shmushing cards that's this method of shuffling you know what I mean people put cards on the table and go like that and how long does it take to shmush I'm not going to give a talk about that but the way I'm doing it is using fluid dynamics treating the cards as a fluid and if you look at the last 30 pages of Poincaré's appendix, the last chapter it's all about mixing in fluids and there's just an awful lot of math as far as I know nobody's ever looked at that I'm looking at it pretty hard he doesn't get very far but you know he's Poincaré and he gets some place what I tried to tell you in this talk is that first of all there's a lot of life in Poincaré and the second thing is that there's still gold in them their hills taking them up in any area and trying to read them ok, any questions questions comments, yes here microphone please we have to get a microphone it's coming hello again Percy do you are your preliminary results show that smushing is a little between riffle shuffling basic shuffling the preliminary results show that it's an amazingly fast way of mixing just to say the first thing I did was get some data so I got some graduate and had her smush cards for a minute and then record the permutation and do that a hundred times so I had a hundred permutations if you didn't smush enough why wouldn't they be random you might think there were too many cards which were originally together that are still together for example we found we made ten test statistics of that sort smushing for a minute passed all tests smushing for half a minute passed all tests 15 seconds started to detect it's a pretty efficient it's a pretty efficient way of doing it it's a standard way of shuffling in Monte Carlo it's not just me who's interested too it's the standard way they shuffle in poker tournaments so it's not ok so it seems to be more effective than I think and we have some preliminary results but there are miles to go so I first have a comment and then a question my comment is that all to the credit of he had an added disadvantage that in France we have this beautiful card game it's a shuffle so he actually did that without actually working with shuffling so much and I have a question why did Markov invent Markov chains what a good question did you find this question I will answer that because it's a nice story so around the turn of the century but actually it lasted quite a long time in the Soviet Union a kind of mathematical it took over the mathematical community it was it had a religious overtones it was one of the main themes of it was free will and the Russian analysts in the cities to which this was going on were sort of instructed to not work on smooth functions because they didn't have enough free will but that's somehow where the soviet school of more general functions arose you may think this is crazy nobody can tell us what to work on if they break your windows beat you up fire you from your job and it all happened all of a sudden it's a problem so there was some crazy people who were espousing this free will there's an interesting book about it there was a czar of probability and he was called necrosoft like Microsoft and he declared that the only things that were fit to work on for probabilists were sums of independent random variables because only they had enough free will in order to so that the basic laws of probability would hold Markov couldn't stand this story and he invented Markov chains as a political statement Markov chains are dependent and he proved the law of large numbers in the central limit theorem for Markov chains and the last line of his paper is thus free will is not necessary to do probability so they were invented as a political example not because it's a wonderful story and there's a beautiful article about it by Eugene Sinida and question there wait for the microphone please I may be wrong but I thought that Poincaré's book was a textbook actually and it was part of lectures I was wondering why he was asked to give a lecture about probability and if there is anything known about the reaction to his course and who the students were and if they did something with it right so I meant to say this at the beginning and those of you who have my handout had I done it I should have I would have written it down but there's a very very nice question by M A Z L I A K it was given just recently Laurent Masiak thank you good and he traces very carefully the answer to that question I'll give it briefly Poincaré didn't have a fancy job but a chair of probability and physics came open and he that was the job he wanted it was at the Sarbonne and his first couple of years he didn't teach any probability whatsoever during that time he was teaching thermodynamics and he got involved in controversies with the English school Poincaré I'll say wasn't a believer in which many people didn't believe in before the turn of the century after all Boltzmann killed himself and he in the middle of that controversy decided maybe I better figure out about probability and he started to give these lectures they were first you know written down in longhand and then they were used as used as a textbook and so I think the main the main were students but it did become the standard textbook in probability for about 15 20 years following a book by his advisor Bertrand and I don't quite know it's not so easy to read Poincaré although this is pretty friendly as those things go I was told by somebody in this room that Poincaré was a criminally bad lecturer so maybe they read the books very carefully but I think I find interesting I'd like an answer from you after this is over the probability that Poincaré put in his book is the kind of thing I was talking about here their concrete real world problems what we would nowadays call applied probability given that he was the major force in mathematics major probabilist of his time how did French probability wonderful healthy French probability not knocking it at all how did it go in such an opposite direction of what Poincaré set out to do I really don't understand that it's just a completely different subject and it's an interesting question to me how that happened here question well it's not a question currently towards the end of the process of acquitting Dreyfus so Poincaré was involved in the Dreyfus scandal and did write a position paper making terrible fun deserved fun of Bertillon the fingerprint guy not the fingerprint guy anyway and if you look at the document it's a very sketchy document it's three or four pages and I wasn't so impressed by Poincaré's work on that case now he might have done a lot of work in private or something like that and he did very forcefully defend Dreyfus but anyway it was real applied probability and he certainly did work at that so I don't know but in the document it's on the web if you just type in Poincaré and Dreyfus you find it and it was pretty sketchy it was a lot of name calling and rhetoric there were a few calculations but it didn't seem like he decided I think to try to argue from on high that's the way it looked like to me from what I understood there is a longer published document in the family Poincaré about this problem in which he computed probabilities did study a lot the written handwriting and so on so that would be something that one would have to look at, he does have some numbers so it wasn't just a quick action job but it didn't I wasn't I was expecting a more thorough investigation given what a big deal it was I mean it was a big deal other question ok I have one Percy could you I didn't follow when you were explaining about difference between the two kinds of shuffling over hand and the other what are the two different ones ok so the riffle shuffle I meant to bring cards I didn't the riffle shuffle is you have a deck of cards you cut it about in half according to the binomial distribution you start dropping the cards with your thumbs according to a rule for example if you have a cards here and b cards here the chance that you drop your next card from here is a over a plus b so you drop cards proportional to packet size and they riffle together ok that's one shuffle and then you can ask how many repetitions of that shuffle does it take to get you with an epsilon close to random and some metric on probabilities and I could say it as carefully as you want but in order to get e to the minus c close you have to do this many shuffles the second way of shuffling I won't do it but I can't but if you went like this you kind of took random packets they do that in casinos they do strip shuffling but we with cards here's a face down deck of cards here's a clump clump clump clump clump and so a mathematical model for that is in between each pair of cards you put a coin with probability theta say a half and then you drop off all until the first head and all until the second head so you drop little packets and in that model it's a theorem of Robin P. Mannell's that it takes n squared log n repetitions another question in the first part of the talk you talked about guessing the integral for random functions an approximation yes you threaten me and you didn't talk about rates of convergence about how typically what is the rest of approximation or things like this I didn't but people do in fact House dwarf and and Banach wrote about that kind of thing and you can find an appendix by House dwarf and Banach's book about putting measures on L1 and rates of convergence there has been some work but by and large this was something that Poincaré did that I noticed in the 1980s was an interesting subject a few students developed it then it went away again and now it's come back with a vengeance in numerical analysis so I think there's that awful lot to do there so there's a lot of nice things to do and they haven't been done so it's a good question but something to do for all of us ok let's thank Percy again and then I'll make a few announcements so thank you again Percy so first Percy as all the lecturer gets Poincaré medal this is your medal Percy for those of the lecturers who didn't get it just come to see me I will give you on one side there is Poincaré on the other some representation of some of the Poincaré great actions great deeds there we have Poincaré medals we have Galois medals that we can provide for those who want them of course if you didn't lecture you need to pay a bit for the medal that's how life goes now the year we celebrated Galois this was the 200th anniversary of his birth now this year it's the 100th anniversary of Poincaré we are trying each year to organize some commemoration or celebration that has some wide interest like this one it's great that we can see here there's a lot of people we would like to have more young people in this kind of organization please advertise for young people to participate in this kind of meetings that are not so specialized I think we think at the Institute Poincaré and many of us I think too that it is very important to open your mind to be exposed not only to the talks in your particular specialty now all the talks in this conference will be available we will have been recorded and will be available at some point not far from now on our web page next year probably there are two natural things to celebrate one thing is 2013 is the Mathematics for Planet Earth a year all around the world Mathematics Institute will be hosting and discussing about things relating mathematics to the earth or human populations or astronomy we have two of our trimesters will be devoted to these kind of things one about ecosystems and there will be one about astronomy and 2013 will also be the 200th anniversary of the death of Lagrange so of course there will be something about Lagrange as you know Lagrange is claimed both by French and by Italian people and if you go to Turin you can see the house where Lagrange was born you can see the statue of Lagrange and so on so we will definitely celebrate Lagrange and the last thing I will advertise for is that tomorrow is the end of this week so we started the last Saturday with the Broad audience celebration at Sorbonne for young people, for general audience so on, next was the conference for mathematicians that was concluded by the beautiful talk of Percy and tomorrow there will be the Poincaré seminar with Olivier Darigol, Alain Chanciner today this was the two-card problem tomorrow it will be the three body and so on there will be Laurent Masliac there will be François Beguin it will be concluded by a movie by Philippe Vombs this is also the opportunity to advertise for the Poincaré seminar which has been doing a great job for years now to produce thematic days that are also intended for Broad audience for young people, this is great way to get hold on some subject I encourage you to participate in the one of tomorrow if you are available, that's all thank you very much for coming and we are always happy to host you thank you