WEBVTT 00:00:00.804 --> 00:00:05.216 What is it that French people do better than all the others? 00:00:06.454 --> 00:00:08.380 If you would take polls, 00:00:08.404 --> 00:00:10.211 the top three answers might be: 00:00:10.235 --> 00:00:14.274 love, wine and whining. NOTE Paragraph 00:00:14.298 --> 00:00:15.599 (Laughter) NOTE Paragraph 00:00:15.623 --> 00:00:16.782 Maybe. 00:00:17.530 --> 00:00:19.868 But let me suggest a fourth one: 00:00:19.892 --> 00:00:21.082 mathematics. 00:00:21.760 --> 00:00:24.613 Did you know that Paris has more mathematicians 00:00:24.637 --> 00:00:26.438 than any other city in the world? 00:00:26.801 --> 00:00:29.295 And more streets with mathematicians' names, too. 00:00:30.215 --> 00:00:33.664 And if you look at the statistics of the Fields Medal, 00:00:33.688 --> 00:00:36.181 often called the Nobel Prize for mathematics, 00:00:36.205 --> 00:00:40.137 and always awarded to mathematicians below the age of 40, 00:00:40.161 --> 00:00:44.048 you will find that France has more Fields medalists per inhabitant 00:00:44.072 --> 00:00:45.240 than any other country. NOTE Paragraph 00:00:46.286 --> 00:00:49.240 What is it that we find so sexy in math? 00:00:50.153 --> 00:00:53.357 After all, it seems to be dull and abstract, 00:00:53.381 --> 00:00:56.864 just numbers and computations and rules to apply. 00:00:58.518 --> 00:01:00.630 Mathematics may be abstract, 00:01:00.654 --> 00:01:01.805 but it's not dull 00:01:01.829 --> 00:01:03.558 and it's not about computing. 00:01:04.178 --> 00:01:05.925 It is about reasoning 00:01:05.949 --> 00:01:08.209 and proving our core activity. 00:01:08.513 --> 00:01:10.035 It is about imagination, 00:01:10.059 --> 00:01:12.078 the talent which we most praise. 00:01:12.102 --> 00:01:14.203 It is about finding the truth. 00:01:15.613 --> 00:01:18.350 There's nothing like the feeling which invades you 00:01:18.374 --> 00:01:20.545 when after months of hard thinking, 00:01:20.569 --> 00:01:23.867 you finally understand the right reasoning to solve your problem. 00:01:25.042 --> 00:01:28.618 The great mathematician André Weil likened this -- 00:01:28.642 --> 00:01:29.793 no kidding -- 00:01:29.817 --> 00:01:31.406 to sexual pleasure. 00:01:32.197 --> 00:01:37.538 But noted that this feeling can last for hours, or even days. NOTE Paragraph 00:01:38.804 --> 00:01:40.657 The reward may be big. 00:01:41.325 --> 00:01:45.189 Hidden mathematical truths permeate our whole physical world. 00:01:45.680 --> 00:01:48.350 They are inaccessible to our senses 00:01:48.374 --> 00:01:51.098 but can be seen through mathematical lenses. 00:01:52.092 --> 00:01:53.684 Close your eyes for moment 00:01:53.708 --> 00:01:57.183 and think of what is occurring right now around you. 00:01:58.337 --> 00:02:01.830 Invisible particles from the air around are bumping on you 00:02:01.854 --> 00:02:04.587 by the billions and billions at each second, 00:02:04.611 --> 00:02:06.674 all in complete chaos. 00:02:07.049 --> 00:02:08.200 And still, 00:02:08.224 --> 00:02:12.912 their statistics can be accurately predicted by mathematical physics. 00:02:13.715 --> 00:02:16.507 And open your eyes now 00:02:16.531 --> 00:02:19.841 to the statistics of the velocities of these particles. NOTE Paragraph 00:02:20.510 --> 00:02:23.750 The famous bell-shaped Gauss Curve, 00:02:23.774 --> 00:02:25.955 or the Law of Errors -- 00:02:25.979 --> 00:02:28.701 of deviations with respect to the mean behavior. 00:02:29.550 --> 00:02:33.852 This curve tells about the statistics of velocities of particles 00:02:33.876 --> 00:02:36.415 in the same way as a demographic curve 00:02:36.439 --> 00:02:40.280 would tell about the statistics of ages of individuals. 00:02:40.884 --> 00:02:43.534 It's one of the most important curves ever. 00:02:44.137 --> 00:02:47.323 It keeps on occurring again and again, 00:02:47.347 --> 00:02:49.750 from many theories and many experiments, 00:02:49.774 --> 00:02:53.055 as a great example of the universality 00:02:53.079 --> 00:02:56.631 which is so dear to us mathematicians. NOTE Paragraph 00:02:57.694 --> 00:02:58.921 Of this curve, 00:02:58.945 --> 00:03:01.994 the famous scientist Francis Galton said, 00:03:02.018 --> 00:03:06.542 "It would have been deified by the Greeks if they had known it. 00:03:07.064 --> 00:03:10.415 It is the supreme law of unreason." 00:03:11.818 --> 00:03:18.420 And there's no better way to materialize that supreme goddess than Galton's Board. 00:03:19.774 --> 00:03:22.971 Inside this board are narrow tunnels 00:03:22.995 --> 00:03:27.623 through which tiny balls will fall down randomly, 00:03:28.295 --> 00:03:33.682 going right or left, or left, etc. 00:03:34.139 --> 00:03:37.390 All in complete randomness and chaos. 00:03:38.085 --> 00:03:44.165 Let's see what happens when we look at all these random trajectories together. NOTE Paragraph 00:03:44.189 --> 00:03:49.624 (Board shaking) NOTE Paragraph 00:03:49.648 --> 00:03:52.492 This is a bit of a sport, 00:03:52.516 --> 00:03:57.386 because we need to resolve some traffic jams in there. 00:03:59.715 --> 00:04:00.865 Aha. 00:04:01.313 --> 00:04:04.900 We think that randomness is going to play me a trick on stage. NOTE Paragraph 00:04:07.609 --> 00:04:09.072 There it is. 00:04:10.382 --> 00:04:12.965 Our supreme goddess of unreason. 00:04:12.989 --> 00:04:14.508 the Gauss Curve, 00:04:14.532 --> 00:04:20.984 trapped here inside this transparent box as Dream in "The Sandman" comics. 00:04:22.623 --> 00:04:25.321 For you I have shown it, 00:04:25.345 --> 00:04:30.630 but to my students I explain why it could not be any other curve. 00:04:31.128 --> 00:04:33.998 And this is touching the mystery of that goddess, 00:04:34.022 --> 00:04:38.723 replacing a beautiful coincidence by a beautiful explanation. NOTE Paragraph 00:04:39.027 --> 00:04:41.360 All of science is like this. 00:04:42.213 --> 00:04:47.561 And beautiful mathematical explanations are not only for our pleasure. 00:04:47.585 --> 00:04:50.245 They also change our vision of the world. 00:04:51.040 --> 00:04:52.277 For instance, 00:04:52.301 --> 00:04:53.451 Einstein, 00:04:53.476 --> 00:04:54.626 Perrin, 00:04:54.651 --> 00:04:55.801 Smoluchowski, 00:04:55.826 --> 00:04:59.385 they used the mathematical analysis of random trajectories 00:04:59.409 --> 00:05:01.446 and the Gauss Curve 00:05:01.470 --> 00:05:06.398 to explain and prove that our world is made of atoms. NOTE Paragraph 00:05:07.524 --> 00:05:09.326 It was not the first time 00:05:09.350 --> 00:05:12.740 that mathematics was revolutionizing our view of the world. 00:05:13.555 --> 00:05:15.767 More than 2,000 years ago, 00:05:15.791 --> 00:05:18.401 at the time of the ancient Greeks, 00:05:19.502 --> 00:05:20.981 it already occurred. 00:05:21.827 --> 00:05:23.113 In those days, 00:05:23.137 --> 00:05:26.420 only a small fraction of the world had been explored, 00:05:26.444 --> 00:05:29.486 and the Earth might have seemed infinite. 00:05:30.034 --> 00:05:31.801 But clever Eratosthenes, 00:05:31.825 --> 00:05:33.242 using mathematics, 00:05:33.266 --> 00:05:38.377 was able to measure the Earth with an amazing accuracy of two percent. NOTE Paragraph 00:05:39.969 --> 00:05:41.385 Here's another example. 00:05:42.238 --> 00:05:46.043 In 1673, Jean Richer noticed 00:05:46.067 --> 00:05:52.979 that a pendulum swings slightly slower in Cayenne than in Paris. 00:05:54.350 --> 00:05:58.750 From this observation alone, and clever mathematics, 00:05:58.774 --> 00:06:01.080 Newton rightly deduced 00:06:01.104 --> 00:06:06.645 that the Earth is a wee bit flattened at the poles, 00:06:06.669 --> 00:06:08.270 like 0.3 percent -- 00:06:08.843 --> 00:06:13.256 so tiny that you wouldn't even notice it on the real view of the Earth. NOTE Paragraph 00:06:14.276 --> 00:06:18.204 These stories show that mathematics 00:06:18.228 --> 00:06:22.990 is able to make us go out of our intuition 00:06:23.512 --> 00:06:26.997 measure the Earth which seems infinite, 00:06:27.021 --> 00:06:29.315 see atoms which are invisible 00:06:29.339 --> 00:06:32.720 or detect an imperceptible variation of shape. 00:06:32.744 --> 00:06:36.591 And if there is just one thing that you should take home from this talk, 00:06:36.615 --> 00:06:37.809 it is this: 00:06:37.833 --> 00:06:42.211 mathematics allows us to go beyond the intuition 00:06:42.235 --> 00:06:46.484 and explore territories which do not fit within our grasp. NOTE Paragraph 00:06:47.609 --> 00:06:50.608 Here's a modern example you will all relate to: 00:06:51.362 --> 00:06:53.029 searching the Internet. 00:06:54.037 --> 00:06:55.379 The World Wide Web, 00:06:55.403 --> 00:06:57.207 more than one billion web pages -- 00:06:57.231 --> 00:06:58.905 do you want to go through them all? 00:06:59.660 --> 00:07:01.462 Computing power helps, 00:07:01.486 --> 00:07:04.672 but it would be useless without the mathematical modeling 00:07:04.696 --> 00:07:07.259 to find the information hidden in the data. NOTE Paragraph 00:07:08.491 --> 00:07:10.870 Let's work out a baby problem. 00:07:11.872 --> 00:07:15.679 Imagine that you're a detective working on a crime case, 00:07:15.703 --> 00:07:19.491 and there are many people who have their version of the facts. 00:07:20.032 --> 00:07:21.777 Who do you want to interview first? 00:07:22.681 --> 00:07:24.596 Sensible answer: 00:07:24.620 --> 00:07:26.057 prime witnesses. 00:07:26.878 --> 00:07:28.112 You see, 00:07:28.136 --> 00:07:32.356 suppose that there is person number seven, 00:07:32.380 --> 00:07:33.531 tells you a story, 00:07:33.555 --> 00:07:35.569 but when you ask where he got if from, 00:07:35.593 --> 00:07:38.629 he points to person number three as a source. 00:07:38.653 --> 00:07:40.721 And maybe person number three, in turn, 00:07:40.745 --> 00:07:44.441 points at person number one as the primary source. 00:07:44.465 --> 00:07:46.126 Now number one is a prime witness, 00:07:46.150 --> 00:07:49.388 so I definitely want to interview him -- priority. 00:07:50.148 --> 00:07:51.299 And from the graph 00:07:51.323 --> 00:07:54.551 we also see that person number four is a prime witness. 00:07:54.575 --> 00:07:57.018 And maybe I even want to interview him first, 00:07:57.042 --> 00:07:59.401 because there are more people who refer to him. NOTE Paragraph 00:08:00.354 --> 00:08:03.018 OK, that was easy, 00:08:03.042 --> 00:08:08.288 but now what about if you have a big bunch of people who will testify? 00:08:08.864 --> 00:08:10.216 And this graph, 00:08:10.240 --> 00:08:15.859 I may think of it as all people who testify in a complicated crime case, 00:08:15.883 --> 00:08:19.905 but it may just as well be web pages pointing to each other, 00:08:19.929 --> 00:08:22.000 referring to each other for contents. 00:08:22.878 --> 00:08:25.214 Which ones are the most authoritative? 00:08:25.587 --> 00:08:26.921 Not so clear. NOTE Paragraph 00:08:28.091 --> 00:08:29.991 Enter PageRank, 00:08:30.015 --> 00:08:32.551 one of the early cornerstones of Google. 00:08:33.337 --> 00:08:37.579 This algorithm uses the laws of mathematical randomness 00:08:37.603 --> 00:08:41.460 to determine automatically the most relevant web pages, 00:08:41.484 --> 00:08:46.546 in the same way as we used randomness in the Galton Board experiment. 00:08:47.341 --> 00:08:49.682 So let's send into this graph 00:08:49.706 --> 00:08:52.556 a bunch of tiny, digital marbles 00:08:52.580 --> 00:08:56.329 and let them go randomly through the graph. 00:08:56.353 --> 00:08:58.020 Each time they arrive at some site, 00:08:58.044 --> 00:09:02.210 they will go out through some link chosen at random to the next one. 00:09:02.234 --> 00:09:03.987 And again, and again, and again. 00:09:04.358 --> 00:09:05.986 And with small, growing piles, 00:09:06.010 --> 00:09:09.763 we'll keep the record of how many times each site has been visited 00:09:09.787 --> 00:09:11.732 by these digital marbles. NOTE Paragraph 00:09:12.243 --> 00:09:13.394 Here we go. 00:09:13.418 --> 00:09:15.266 Randomness, randomness. 00:09:15.811 --> 00:09:17.259 And from time to time, 00:09:17.283 --> 00:09:21.235 also let's make jumps completely randomly to increase the fun. NOTE Paragraph 00:09:22.471 --> 00:09:23.687 And look at this: 00:09:24.358 --> 00:09:27.143 from the chaos will emerge the solution. 00:09:27.483 --> 00:09:29.968 The highest piles correspond to those sites 00:09:29.992 --> 00:09:33.503 which somehow are better connected than the others, 00:09:33.527 --> 00:09:35.800 more pointed at than the others. 00:09:35.824 --> 00:09:37.546 And here we see clearly 00:09:37.570 --> 00:09:40.602 which are the web pages we want to first try. 00:09:41.507 --> 00:09:42.658 Once again, 00:09:42.682 --> 00:09:45.142 the solution emerges from the randomness. 00:09:45.775 --> 00:09:48.026 Of course, since that time, 00:09:48.050 --> 00:09:51.757 Google has come up with much more sophisticated algorithms, 00:09:51.781 --> 00:09:54.061 but already this was beautiful. NOTE Paragraph 00:09:54.981 --> 00:09:56.457 And still, 00:09:56.481 --> 00:09:58.092 just one problem in a million. 00:09:58.734 --> 00:10:01.004 With the advent of digital area, 00:10:01.028 --> 00:10:06.044 more and more problems lend themselves to mathematical analysis, 00:10:06.068 --> 00:10:10.433 making the job of mathematician a more and more useful one, 00:10:11.166 --> 00:10:13.888 to the extent that a few years ago, 00:10:13.912 --> 00:10:17.691 it was ranked number one among hundreds of jobs 00:10:17.715 --> 00:10:21.683 in a study about the best and worst jobs 00:10:21.707 --> 00:10:24.682 published by the Wall Street Journal in 2009. NOTE Paragraph 00:10:25.445 --> 00:10:27.297 Mathematician -- 00:10:27.321 --> 00:10:28.754 best job in the world. 00:10:29.646 --> 00:10:32.714 That's because of the applications: 00:10:32.738 --> 00:10:34.877 communication theory, 00:10:34.901 --> 00:10:36.721 information theory, 00:10:36.745 --> 00:10:38.005 game theory, 00:10:38.029 --> 00:10:39.475 compressed sensing, 00:10:39.499 --> 00:10:41.061 machine learning, 00:10:41.085 --> 00:10:42.652 graph analysis, 00:10:42.676 --> 00:10:44.418 harmonic analysis. 00:10:44.442 --> 00:10:47.082 And why not stochastic processes, 00:10:47.106 --> 00:10:48.736 linear programming, 00:10:48.760 --> 00:10:50.788 or fluid simulation? 00:10:51.292 --> 00:10:55.187 Each of these fields have monster industrial applications. 00:10:55.211 --> 00:10:56.362 And through them, 00:10:56.386 --> 00:10:58.385 there is big money in mathematics. 00:10:59.400 --> 00:11:01.440 And let me concede 00:11:01.464 --> 00:11:03.941 that when it comes to making money from the math, 00:11:03.965 --> 00:11:07.789 the Americans are by a long shot the world champions, 00:11:07.813 --> 00:11:12.432 with clever, emblematic billionaires and amazing, giant companies, 00:11:12.456 --> 00:11:15.736 all resting, ultimately, on good algorithm. NOTE Paragraph 00:11:17.091 --> 00:11:21.063 Now with all this beauty, usefulness and wealth, 00:11:21.087 --> 00:11:23.371 mathematics does look more sexy. 00:11:24.399 --> 00:11:26.016 But don't you think 00:11:26.040 --> 00:11:30.160 that the life a mathematical researcher is an easy one. 00:11:30.959 --> 00:11:33.700 It is filled with perplexity, 00:11:34.347 --> 00:11:35.497 frustration, 00:11:36.172 --> 00:11:38.617 a desperate fight for understanding. NOTE Paragraph 00:11:39.955 --> 00:11:42.095 Let me evoke for you 00:11:42.119 --> 00:11:46.499 one of the most striking days in my mathematician's life. 00:11:46.523 --> 00:11:47.674 Or should I say, 00:11:47.698 --> 00:11:49.435 one of the most striking nights. 00:11:50.713 --> 00:11:51.864 At that time, 00:11:51.888 --> 00:11:55.039 I was staying at the Institute for Advanced Studies in Princeton -- 00:11:55.063 --> 00:11:57.202 for many years, the home of Albert Einstein 00:11:57.226 --> 00:12:01.654 and arguably the most holy place for mathematical research in the world. 00:12:02.878 --> 00:12:06.722 And that night I was working and working on an elusive proof, 00:12:06.746 --> 00:12:08.124 which was incomplete. 00:12:09.304 --> 00:12:11.512 It was all about understanding 00:12:11.536 --> 00:12:15.359 the paradoxical stability property of plasmas, 00:12:15.383 --> 00:12:17.341 which are a crowd of electrons. 00:12:18.423 --> 00:12:21.159 In the perfect world of plasma, 00:12:21.183 --> 00:12:22.961 there are no collisions 00:12:22.985 --> 00:12:26.643 and no friction to provide the stability like we are used to. 00:12:27.392 --> 00:12:28.543 But still, 00:12:28.567 --> 00:12:31.600 if you slightly perturb a plasma equilibrium, 00:12:31.624 --> 00:12:34.312 you will find that the resulting electric field 00:12:34.336 --> 00:12:36.675 spontaneously vanishes, 00:12:36.699 --> 00:12:38.674 or damps out, 00:12:38.698 --> 00:12:41.992 as if by some mysterious friction force. NOTE Paragraph 00:12:42.728 --> 00:12:44.563 This paradoxical effect, 00:12:44.587 --> 00:12:46.064 called the Landau damping, 00:12:46.088 --> 00:12:49.077 is one of the most important in plasma physics, 00:12:49.101 --> 00:12:52.103 and it was discovered through mathematical ideas. 00:12:52.970 --> 00:12:54.121 But still, 00:12:54.145 --> 00:12:58.375 a full mathematical understanding of this phenomenon was missing. 00:12:58.399 --> 00:13:03.185 And together with my former student and main collaborator Clément Mouhot, 00:13:03.209 --> 00:13:04.701 in Paris at the time, 00:13:04.725 --> 00:13:08.811 we had been working for months and months on such a proof. 00:13:09.832 --> 00:13:11.167 Actually, 00:13:11.191 --> 00:13:15.937 I had already announced by mistake that we could solve it. 00:13:15.961 --> 00:13:17.686 But the truth is, 00:13:17.710 --> 00:13:19.857 the proof was just not working. 00:13:20.196 --> 00:13:24.545 In spite of more than 100 pages of complicated, mathematical arguments, 00:13:24.569 --> 00:13:26.259 and a bunch discoveries, 00:13:26.283 --> 00:13:27.550 and huge calculation, 00:13:27.574 --> 00:13:28.743 it was not working. 00:13:29.290 --> 00:13:30.971 And that night in Princeton, 00:13:30.995 --> 00:13:35.296 a certain gap in the chain of arguments was driving me crazy. 00:13:35.658 --> 00:13:40.251 I was putting in there all my energy and experience and tricks, 00:13:40.275 --> 00:13:42.017 and still nothing was working. 00:13:42.553 --> 00:13:46.435 1 a.m., 2 a.m., 3 a.m., 00:13:46.459 --> 00:13:47.767 not working. 00:13:48.545 --> 00:13:52.866 Around 4 a.m., I go to bed in low spirits. 00:13:53.915 --> 00:13:56.375 Then a few hours later, 00:13:56.399 --> 00:13:57.550 waking up and go, 00:13:57.574 --> 00:14:00.931 "Ah, it's time to get the kids to school --" 00:14:00.955 --> 00:14:02.106 What is this? 00:14:02.130 --> 00:14:04.272 There was this voice in my head, I swear. 00:14:04.894 --> 00:14:06.807 "Take the second term to the other side, 00:14:06.831 --> 00:14:08.750 Fourier transform and invert in L2." NOTE Paragraph 00:14:09.257 --> 00:14:10.408 (Laughter) NOTE Paragraph 00:14:10.432 --> 00:14:12.134 Damn it, 00:14:12.158 --> 00:14:14.271 that was the start of the solution! NOTE Paragraph 00:14:15.519 --> 00:14:16.670 You see, 00:14:16.694 --> 00:14:18.977 I thought I had taken some rest, 00:14:19.001 --> 00:14:22.389 but really my brain had continued to work on it. 00:14:23.008 --> 00:14:24.605 In those moments, 00:14:24.629 --> 00:14:27.230 you don't think of your career or your colleagues, 00:14:27.254 --> 00:14:30.944 it's just a complete battle between the problem and you. NOTE Paragraph 00:14:32.098 --> 00:14:33.426 That being said, 00:14:33.450 --> 00:14:37.399 it does not harm when you do get a promotion in reward for your hard work. 00:14:37.808 --> 00:14:42.968 And after we completed our huge analysis of the Landau damping, 00:14:42.992 --> 00:14:44.607 I was lucky enough 00:14:44.631 --> 00:14:47.661 to get the most coveted Fields Medal 00:14:47.685 --> 00:14:50.552 from the hands of the President of India, 00:14:50.576 --> 00:14:54.496 in Hyderabad on 19 August, 2010 -- 00:14:55.453 --> 00:14:58.704 an honor that mathematicians never dare to dream, 00:14:58.728 --> 00:15:01.127 a day that I will remember until I live. NOTE Paragraph 00:15:02.366 --> 00:15:03.813 What do you think, 00:15:03.837 --> 00:15:05.978 on such an occasion? 00:15:06.002 --> 00:15:07.152 Pride, yes? 00:15:07.791 --> 00:15:11.431 And gratitude to the many collaborators who made this possible. 00:15:12.304 --> 00:15:14.516 And because it was a collective adventure, 00:15:14.540 --> 00:15:18.682 you need to share it, not just with your collaborators. 00:15:19.548 --> 00:15:25.240 I believe that everybody can appreciate the thrill of mathematical research, 00:15:25.264 --> 00:15:29.582 and share the passionate stories of humans and ideas behind it. 00:15:30.494 --> 00:15:35.268 And I've been working with my staff at Institut Henri Poincaré, 00:15:35.292 --> 00:15:40.473 together with partners and artists of mathematical communication worldwide, 00:15:40.497 --> 00:15:45.084 so that we can found our own, very special museum of mathematics there. NOTE Paragraph 00:15:46.537 --> 00:15:48.314 So in a few years, 00:15:48.885 --> 00:15:50.462 when you come to Paris, 00:15:50.486 --> 00:15:56.144 after tasting the great, crispy baguette and macaroon, 00:15:56.168 --> 00:15:59.831 please come and visit us at Institut Henri Poincaré, 00:15:59.856 --> 00:16:02.371 and share the mathematical dream with us. NOTE Paragraph 00:16:02.395 --> 00:16:03.546 Thank you. NOTE Paragraph 00:16:03.570 --> 00:16:10.570 (Applause)