0:00:00.804,0:00:05.216
What is it that French people[br]do better than all the others?

0:00:06.454,0:00:08.380
If you would take polls,

0:00:08.404,0:00:10.211
the top three answers might be:

0:00:10.235,0:00:14.274
love, wine and whining.

0:00:14.298,0:00:15.599
(Laughter)

0:00:15.623,0:00:16.782
Maybe.

0:00:17.530,0:00:19.868
But let me suggest a fourth one:

0:00:19.892,0:00:21.082
mathematics.

0:00:21.760,0:00:24.613
Did you know that Paris[br]has more mathematicians

0:00:24.637,0:00:26.438
than any other city in the world?

0:00:26.801,0:00:29.295
And more streets[br]with mathematicians' names, too.

0:00:30.215,0:00:33.664
And if you look at the statistics[br]of the Fields Medal,

0:00:33.688,0:00:36.181
often called the Nobel Prize[br]for mathematics,

0:00:36.205,0:00:40.137
and always awarded to mathematicians[br]below the age of 40,

0:00:40.161,0:00:44.048
you will find that France has more[br]Fields medalists per inhabitant

0:00:44.072,0:00:45.240
than any other country.

0:00:46.286,0:00:49.240
What is it that we find so sexy in math?

0:00:50.153,0:00:53.357
After all, it seems to be[br]dull and abstract,

0:00:53.381,0:00:56.864
just numbers and computations[br]and rules to apply.

0:00:58.518,0:01:00.630
Mathematics may be abstract,

0:01:00.654,0:01:01.805
but it's not dull

0:01:01.829,0:01:03.558
and it's not about computing.

0:01:04.178,0:01:05.925
It is about reasoning

0:01:05.949,0:01:08.209
and proving our core activity.

0:01:08.513,0:01:10.035
It is about imagination,

0:01:10.059,0:01:12.078
the talent which we most praise.

0:01:12.102,0:01:14.203
It is about finding the truth.

0:01:15.613,0:01:18.350
There's nothing like the feeling[br]which invades you

0:01:18.374,0:01:20.545
when after months of hard thinking,

0:01:20.569,0:01:23.867
you finally understand the right[br]reasoning to solve your problem.

0:01:25.042,0:01:28.618
The great mathematician[br]André Weil likened this --

0:01:28.642,0:01:29.793
no kidding --

0:01:29.817,0:01:31.406
to sexual pleasure.

0:01:32.197,0:01:37.538
But noted that this feeling[br]can last for hours, or even days.

0:01:38.804,0:01:40.657
The reward may be big.

0:01:41.325,0:01:45.189
Hidden mathematical truths[br]permeate our whole physical world.

0:01:45.680,0:01:48.350
They are inaccessible to our senses

0:01:48.374,0:01:51.098
but can be seen[br]through mathematical lenses.

0:01:52.092,0:01:53.684
Close your eyes for moment

0:01:53.708,0:01:57.183
and think of what is occurring[br]right now around you.

0:01:58.337,0:02:01.830
Invisible particles from the air[br]around are bumping on you

0:02:01.854,0:02:04.587
by the billions and billions[br]at each second,

0:02:04.611,0:02:06.674
all in complete chaos.

0:02:07.049,0:02:08.200
And still,

0:02:08.224,0:02:12.912
their statistics can be accurately[br]predicted by mathematical physics.

0:02:13.715,0:02:16.507
And open your eyes now

0:02:16.531,0:02:19.841
to the statistics of the velocities[br]of these particles.

0:02:20.510,0:02:23.750
The famous bell-shaped Gauss Curve,

0:02:23.774,0:02:25.955
or the Law of Errors --

0:02:25.979,0:02:28.701
of deviations with respect[br]to the mean behavior.

0:02:29.550,0:02:33.852
This curve tells about the statistics[br]of velocities of particles

0:02:33.876,0:02:36.415
in the same way as a demographic curve

0:02:36.439,0:02:40.280
would tell about the statistics[br]of ages of individuals.

0:02:40.884,0:02:43.534
It's one of the most[br]important curves ever.

0:02:44.137,0:02:47.323
It keeps on occurring again and again,

0:02:47.347,0:02:49.750
from many theories and many experiments,

0:02:49.774,0:02:53.055
as a great example of the universality

0:02:53.079,0:02:56.631
which is so dear to us mathematicians.

0:02:57.694,0:02:58.921
Of this curve,

0:02:58.945,0:03:01.994
the famous scientist Francis Galton said,

0:03:02.018,0:03:06.542
"It would have been deified by the Greeks[br]if they had known it.

0:03:07.064,0:03:10.415
It is the supreme law of unreason."

0:03:11.818,0:03:18.420
And there's no better way to materialize[br]that supreme goddess than Galton's Board.

0:03:19.774,0:03:22.971
Inside this board are narrow tunnels

0:03:22.995,0:03:27.623
through which tiny balls[br]will fall down randomly,

0:03:28.295,0:03:33.682
going right or left, or left, etc.

0:03:34.139,0:03:37.390
All in complete randomness and chaos.

0:03:38.085,0:03:44.165
Let's see what happens when we look[br]at all these random trajectories together.

0:03:44.189,0:03:49.624
(Board shaking)

0:03:49.648,0:03:52.492
This is a bit of a sport,

0:03:52.516,0:03:57.386
because we need to resolve[br]some traffic jams in there.

0:03:59.715,0:04:00.865
Aha.

0:04:01.313,0:04:04.900
We think that randomness[br]is going to play me a trick on stage.

0:04:07.609,0:04:09.072
There it is.

0:04:10.382,0:04:12.965
Our supreme goddess of unreason.

0:04:12.989,0:04:14.508
the Gauss Curve,

0:04:14.532,0:04:20.984
trapped here inside this transparent box[br]as Dream in "The Sandman" comics.

0:04:22.623,0:04:25.321
For you I have shown it,

0:04:25.345,0:04:30.630
but to my students I explain why[br]it could not be any other curve.

0:04:31.128,0:04:33.998
And this is touching[br]the mystery of that goddess,

0:04:34.022,0:04:38.723
replacing a beautiful coincidence[br]by a beautiful explanation.

0:04:39.027,0:04:41.360
All of science is like this.

0:04:42.213,0:04:47.561
And beautiful mathematical explanations[br]are not only for our pleasure.

0:04:47.585,0:04:50.245
They also change our vision of the world.

0:04:51.040,0:04:52.277
For instance,

0:04:52.301,0:04:53.451
Einstein,

0:04:53.476,0:04:54.626
Perrin,

0:04:54.651,0:04:55.801
Smoluchowski,

0:04:55.826,0:04:59.385
they used the mathematical analysis[br]of random trajectories

0:04:59.409,0:05:01.446
and the Gauss Curve

0:05:01.470,0:05:06.398
to explain and prove that our[br]world is made of atoms.

0:05:07.524,0:05:09.326
It was not the first time

0:05:09.350,0:05:12.740
that mathematics was revolutionizing[br]our view of the world.

0:05:13.555,0:05:15.767
More than 2,000 years ago,

0:05:15.791,0:05:18.401
at the time of the ancient Greeks,

0:05:19.502,0:05:20.981
it already occurred.

0:05:21.827,0:05:23.113
In those days,

0:05:23.137,0:05:26.420
only a small fraction of the world[br]had been explored,

0:05:26.444,0:05:29.486
and the Earth might have seemed infinite.

0:05:30.034,0:05:31.801
But clever Eratosthenes,

0:05:31.825,0:05:33.242
using mathematics,

0:05:33.266,0:05:38.377
was able to measure the Earth[br]with an amazing accuracy of two percent.

0:05:39.969,0:05:41.385
Here's another example.

0:05:42.238,0:05:46.043
In 1673, Jean Richer noticed

0:05:46.067,0:05:52.979
that a pendulum swings slightly[br]slower in Cayenne than in Paris.

0:05:54.350,0:05:58.750
From this observation alone,[br]and clever mathematics,

0:05:58.774,0:06:01.080
Newton rightly deduced

0:06:01.104,0:06:06.645
that the Earth is a wee bit[br]flattened at the poles,

0:06:06.669,0:06:08.270
like 0.3 percent --

0:06:08.843,0:06:13.256
so tiny that you wouldn't even[br]notice it on the real view of the Earth.

0:06:14.276,0:06:18.204
These stories show that mathematics

0:06:18.228,0:06:22.990
is able to make us go out of our intuition

0:06:23.512,0:06:26.997
measure the Earth which seems infinite,

0:06:27.021,0:06:29.315
see atoms which are invisible

0:06:29.339,0:06:32.720
or detect an imperceptible[br]variation of shape.

0:06:32.744,0:06:36.591
And if there is just one thing that you[br]should take home from this talk,

0:06:36.615,0:06:37.809
it is this:

0:06:37.833,0:06:42.211
mathematics allows us[br]to go beyond the intuition

0:06:42.235,0:06:46.484
and explore territories[br]which do not fit within our grasp.

0:06:47.609,0:06:50.608
Here's a modern example[br]you will all relate to:

0:06:51.362,0:06:53.029
searching the Internet.

0:06:54.037,0:06:55.379
The World Wide Web,

0:06:55.403,0:06:57.207
more than one billion web pages --

0:06:57.231,0:06:58.905
do you want to go through them all?

0:06:59.660,0:07:01.462
Computing power helps,

0:07:01.486,0:07:04.672
but it would be useless without[br]the mathematical modeling

0:07:04.696,0:07:07.259
to find the information[br]hidden in the data.

0:07:08.491,0:07:10.870
Let's work out a baby problem.

0:07:11.872,0:07:15.679
Imagine that you're a detective[br]working on a crime case,

0:07:15.703,0:07:19.491
and there are many people[br]who have their version of the facts.

0:07:20.032,0:07:21.777
Who do you want to interview first?

0:07:22.681,0:07:24.596
Sensible answer:

0:07:24.620,0:07:26.057
prime witnesses.

0:07:26.878,0:07:28.112
You see,

0:07:28.136,0:07:32.356
suppose that there is person number seven,

0:07:32.380,0:07:33.531
tells you a story,

0:07:33.555,0:07:35.569
but when you ask where he got if from,

0:07:35.593,0:07:38.629
he points to person[br]number three as a source.

0:07:38.653,0:07:40.721
And maybe person number three, in turn,

0:07:40.745,0:07:44.441
points at person number one[br]as the primary source.

0:07:44.465,0:07:46.126
Now number one is a prime witness,

0:07:46.150,0:07:49.388
so I definitely want[br]to interview him -- priority.

0:07:50.148,0:07:51.299
And from the graph

0:07:51.323,0:07:54.551
we also see that person[br]number four is a prime witness.

0:07:54.575,0:07:57.018
And maybe I even want[br]to interview him first,

0:07:57.042,0:07:59.401
because there are more[br]people who refer to him.

0:08:00.354,0:08:03.018
OK, that was easy,

0:08:03.042,0:08:08.288
but now what about if you have[br]a big bunch of people who will testify?

0:08:08.864,0:08:10.216
And this graph,

0:08:10.240,0:08:15.859
I may think of it as all people[br]who testify in a complicated crime case,

0:08:15.883,0:08:19.905
but it may just as well be web pages[br]pointing to each other,

0:08:19.929,0:08:22.000
referring to each other for contents.

0:08:22.878,0:08:25.214
Which ones are the most authoritative?

0:08:25.587,0:08:26.921
Not so clear.

0:08:28.091,0:08:29.991
Enter PageRank,

0:08:30.015,0:08:32.551
one of the early cornerstones of Google.

0:08:33.337,0:08:37.579
This algorithm uses the laws[br]of mathematical randomness

0:08:37.603,0:08:41.460
to determine automatically[br]the most relevant web pages,

0:08:41.484,0:08:46.546
in the same way as we used randomness[br]in the Galton Board experiment.

0:08:47.341,0:08:49.682
So let's send into this graph

0:08:49.706,0:08:52.556
a bunch of tiny, digital marbles

0:08:52.580,0:08:56.329
and let them go randomly[br]through the graph.

0:08:56.353,0:08:58.020
Each time they arrive at some site,

0:08:58.044,0:09:02.210
they will go out through some link[br]chosen at random to the next one.

0:09:02.234,0:09:03.987
And again, and again, and again.

0:09:04.358,0:09:05.986
And with small, growing piles,

0:09:06.010,0:09:09.763
we'll keep the record of how many[br]times each site has been visited

0:09:09.787,0:09:11.732
by these digital marbles.

0:09:12.243,0:09:13.394
Here we go.

0:09:13.418,0:09:15.266
Randomness, randomness.

0:09:15.811,0:09:17.259
And from time to time,

0:09:17.283,0:09:21.235
also let's make jumps completely[br]randomly to increase the fun.

0:09:22.471,0:09:23.687
And look at this:

0:09:24.358,0:09:27.143
from the chaos will emerge the solution.

0:09:27.483,0:09:29.968
The highest piles[br]correspond to those sites

0:09:29.992,0:09:33.503
which somehow are better[br]connected than the others,

0:09:33.527,0:09:35.800
more pointed at than the others.

0:09:35.824,0:09:37.546
And here we see clearly

0:09:37.570,0:09:40.602
which are the web pages[br]we want to first try.

0:09:41.507,0:09:42.658
Once again,

0:09:42.682,0:09:45.142
the solution emerges from the randomness.

0:09:45.775,0:09:48.026
Of course, since that time,

0:09:48.050,0:09:51.757
Google has come up with much more[br]sophisticated algorithms,

0:09:51.781,0:09:54.061
but already this was beautiful.

0:09:54.981,0:09:56.457
And still,

0:09:56.481,0:09:58.092
just one problem in a million.

0:09:58.734,0:10:01.004
With the advent of digital area,

0:10:01.028,0:10:06.044
more and more problems lend[br]themselves to mathematical analysis,

0:10:06.068,0:10:10.433
making the job of mathematician[br]a more and more useful one,

0:10:11.166,0:10:13.888
to the extent that a few years ago,

0:10:13.912,0:10:17.691
it was ranked number one[br]among hundreds of jobs

0:10:17.715,0:10:21.683
in a study about the best and worst jobs

0:10:21.707,0:10:24.682
published by the Wall Street[br]Journal in 2009.

0:10:25.445,0:10:27.297
Mathematician --

0:10:27.321,0:10:28.754
best job in the world.

0:10:29.646,0:10:32.714
That's because of the applications:

0:10:32.738,0:10:34.877
communication theory,

0:10:34.901,0:10:36.721
information theory,

0:10:36.745,0:10:38.005
game theory,

0:10:38.029,0:10:39.475
compressed sensing,

0:10:39.499,0:10:41.061
machine learning,

0:10:41.085,0:10:42.652
graph analysis,

0:10:42.676,0:10:44.418
harmonic analysis.

0:10:44.442,0:10:47.082
And why not stochastic processes,

0:10:47.106,0:10:48.736
linear programming,

0:10:48.760,0:10:50.788
or fluid simulation?

0:10:51.292,0:10:55.187
Each of these fields have[br]monster industrial applications.

0:10:55.211,0:10:56.362
And through them,

0:10:56.386,0:10:58.385
there is big money in mathematics.

0:10:59.400,0:11:01.440
And let me concede

0:11:01.464,0:11:03.941
that when it comes to making[br]money from the math,

0:11:03.965,0:11:07.789
the Americans are by a long shot[br]the world champions,

0:11:07.813,0:11:12.432
with clever, emblematic billionaires[br]and amazing, giant companies,

0:11:12.456,0:11:15.736
all resting, ultimately,[br]on good algorithm.

0:11:17.091,0:11:21.063
Now with all this beauty,[br]usefulness and wealth,

0:11:21.087,0:11:23.371
mathematics does look more sexy.

0:11:24.399,0:11:26.016
But don't you think

0:11:26.040,0:11:30.160
that the life a mathematical[br]researcher is an easy one.

0:11:30.959,0:11:33.700
It is filled with perplexity,

0:11:34.347,0:11:35.497
frustration,

0:11:36.172,0:11:38.617
a desperate fight for understanding.

0:11:39.955,0:11:42.095
Let me evoke for you

0:11:42.119,0:11:46.499
one of the most striking days[br]in my mathematician's life.

0:11:46.523,0:11:47.674
Or should I say,

0:11:47.698,0:11:49.435
one of the most striking nights.

0:11:50.713,0:11:51.864
At that time,

0:11:51.888,0:11:55.039
I was staying at the Institute[br]for Advanced Studies in Princeton --

0:11:55.063,0:11:57.202
for many years, the home[br]of Albert Einstein

0:11:57.226,0:12:01.654
and arguably the most holy place[br]for mathematical research in the world.

0:12:02.878,0:12:06.722
And that night I was working[br]and working on an elusive proof,

0:12:06.746,0:12:08.124
which was incomplete.

0:12:09.304,0:12:11.512
It was all about understanding

0:12:11.536,0:12:15.359
the paradoxical stability[br]property of plasmas,

0:12:15.383,0:12:17.341
which are a crowd of electrons.

0:12:18.423,0:12:21.159
In the perfect world of plasma,

0:12:21.183,0:12:22.961
there are no collisions

0:12:22.985,0:12:26.643
and no friction to provide[br]the stability like we are used to.

0:12:27.392,0:12:28.543
But still,

0:12:28.567,0:12:31.600
if you slightly perturb[br]a plasma equilibrium,

0:12:31.624,0:12:34.312
you will find that the[br]resulting electric field

0:12:34.336,0:12:36.675
spontaneously vanishes,

0:12:36.699,0:12:38.674
or damps out,

0:12:38.698,0:12:41.992
as if by some mysterious friction force.

0:12:42.728,0:12:44.563
This paradoxical effect,

0:12:44.587,0:12:46.064
called the Landau damping,

0:12:46.088,0:12:49.077
is one of the most important[br]in plasma physics,

0:12:49.101,0:12:52.103
and it was discovered[br]through mathematical ideas.

0:12:52.970,0:12:54.121
But still,

0:12:54.145,0:12:58.375
a full mathematical understanding[br]of this phenomenon was missing.

0:12:58.399,0:13:03.185
And together with my former student[br]and main collaborator Clément Mouhot,

0:13:03.209,0:13:04.701
in Paris at the time,

0:13:04.725,0:13:08.811
we had been working for months[br]and months on such a proof.

0:13:09.832,0:13:11.167
Actually,

0:13:11.191,0:13:15.937
I had already announced by mistake[br]that we could solve it.

0:13:15.961,0:13:17.686
But the truth is,

0:13:17.710,0:13:19.857
the proof was just not working.

0:13:20.196,0:13:24.545
In spite of more than 100 pages[br]of complicated, mathematical arguments,

0:13:24.569,0:13:26.259
and a bunch discoveries,

0:13:26.283,0:13:27.550
and huge calculation,

0:13:27.574,0:13:28.743
it was not working.

0:13:29.290,0:13:30.971
And that night in Princeton,

0:13:30.995,0:13:35.296
a certain gap in the chain of arguments[br]was driving me crazy.

0:13:35.658,0:13:40.251
I was putting in there all my energy[br]and experience and tricks,

0:13:40.275,0:13:42.017
and still nothing was working.

0:13:42.553,0:13:46.435
1 a.m., 2 a.m., 3 a.m.,

0:13:46.459,0:13:47.767
not working.

0:13:48.545,0:13:52.866
Around 4 a.m., I go to bed in low spirits.

0:13:53.915,0:13:56.375
Then a few hours later,

0:13:56.399,0:13:57.550
waking up and go,

0:13:57.574,0:14:00.931
"Ah, it's time to get[br]the kids to school --"

0:14:00.955,0:14:02.106
What is this?

0:14:02.130,0:14:04.272
There was this voice in my head, I swear.

0:14:04.894,0:14:06.807
"Take the second term to the other side,

0:14:06.831,0:14:08.750
Fourier transform and invert in L2."

0:14:09.257,0:14:10.408
(Laughter)

0:14:10.432,0:14:12.134
Damn it,

0:14:12.158,0:14:14.271
that was the start of the solution!

0:14:15.519,0:14:16.670
You see,

0:14:16.694,0:14:18.977
I thought I had taken some rest,

0:14:19.001,0:14:22.389
but really my brain had[br]continued to work on it.

0:14:23.008,0:14:24.605
In those moments,

0:14:24.629,0:14:27.230
you don't think of your career[br]or your colleagues,

0:14:27.254,0:14:30.944
it's just a complete battle[br]between the problem and you.

0:14:32.098,0:14:33.426
That being said,

0:14:33.450,0:14:37.399
it does not harm when you do get[br]a promotion in reward for your hard work.

0:14:37.808,0:14:42.968
And after we completed our huge[br]analysis of the Landau damping,

0:14:42.992,0:14:44.607
I was lucky enough

0:14:44.631,0:14:47.661
to get the most coveted Fields Medal

0:14:47.685,0:14:50.552
from the hands of the President of India,

0:14:50.576,0:14:54.496
in Hyderabad on 19 August, 2010 --

0:14:55.453,0:14:58.704
an honor that mathematicians[br]never dare to dream,

0:14:58.728,0:15:01.127
a day that I will remember until I live.

0:15:02.366,0:15:03.813
What do you think,

0:15:03.837,0:15:05.978
on such an occasion?

0:15:06.002,0:15:07.152
Pride, yes?

0:15:07.791,0:15:11.431
And gratitude to the many collaborators[br]who made this possible.

0:15:12.304,0:15:14.516
And because it was a collective adventure,

0:15:14.540,0:15:18.682
you need to share it,[br]not just with your collaborators.

0:15:19.548,0:15:25.240
I believe that everybody can appreciate[br]the thrill of mathematical research,

0:15:25.264,0:15:29.582
and share the passionate stories[br]of humans and ideas behind it.

0:15:30.494,0:15:35.268
And I've been working with my staff[br]at Institut Henri Poincaré,

0:15:35.292,0:15:40.473
together with partners and artists[br]of mathematical communication worldwide,

0:15:40.497,0:15:45.084
so that we can found our own,[br]very special museum of mathematics there.

0:15:46.537,0:15:48.314
So in a few years,

0:15:48.885,0:15:50.462
when you come to Paris,

0:15:50.486,0:15:56.144
after tasting the great, crispy[br]baguette and macaroon,

0:15:56.168,0:15:59.831
please come and visit us[br]at Institut Henri Poincaré,

0:15:59.856,0:16:02.371
and share the mathematical dream with us.

0:16:02.395,0:16:03.546
Thank you.

0:16:03.570,0:16:10.570
(Applause)