The search for answers and the romance of math - Cédric Villani at TEDxObserver

Cédric Villani is a world-renowned mathematician, one of the best specialists of the equations of the kinetic theory of gases and plasmas, and optimal transport. Former student of the École Normale Superieure in Paris, in 2010 he received the Fields Medal, the most prestigious award of the mathematical world. Outstanding scientific popularizer, he loves sharing his passion with enthusiasm and humor.

Hello everybody.

So, let me introduce me.
(Laughter)

I am a French mathematician, as was said.

French, probably you have ideas
about what this means:

good food, great literature, cute girls,

of course, all that is true.

I believe that I need my crispy baguette,

good cheese, raw meal to be happy
in my daily life.

But it is less known that France is also
a great country for science

and in particular mathematics,
and has been so for hundreds of years.

As I was a high school student,
shy, kind of nerdy,

I went through this very elitist
French system of preparatory classes

and Grandes écoles
inherited from Napoleon.

I attended courses in
École Normale Supérieure in Paris.

It is the institution that claims
the most Fields Medalists

among its former students,

more than
any other institution in the world.

Last year my name was added to the list,

and that was it.

When you receive such an award

of course you are full with joy and pride

but also with a bit of terror

like all of a sudden you become
a spokesperson for your field

and you will meet many people

who will like to hear you talk
about what you do.

So you meet everybody,
you meet CEOs, you meet politicians,

you meet kids in high schools,

you meet musicians,

you go in public newspapers,

everybody sees you in the street.

I even made my way to fashion magazines.
(Laughter)

You know, these French people,
they like fashion. Ok.

So I met everybody, from garbage men
to President of Republic

and, all of them, they wanted me to talk
about science and mathematicians.

It is maybe a surprise for some people
in the audience

that there is a job called mathematician,

what the hell is this?

Do they exist? What do they do? Compute?

No, mathematics is not about figures.

It is a job, a damned good job,
(Laughter)

and in fact when the very serious
Wall Street Journal

had the weird idea to rank
all jobs in the world,

guess what came first?

Was it princess? Or lawyer? Or trader?

No, my friends, it was mathematician.

(Laughter)

Well, I hate rankings but, well, knowing
they are in the good direction, it's OK.

Now, mathematician,

these people define as somebody who
applies mathematical theories and formulas

to teach or solve problems

in the business, educational
or industrial climate.

To some extents it is fine

but these guys forgot what for me
is the most important:

to create mathematics,
to invent mathematics,

to discover new mathematics.

And that's my job.

The world is full of mathematical problems

that are still not solved.

Problems with numbers
are the simplest to explain,

there are problem about numbers
that you can explain

to eight years old kids
in a couple of minutes

and though the smartest mathematician
in the world

would not be able to solve them.

My inspiration is not in numbers,
it comes rather from physics.

The world around us.

It's so full of great problems,

whenever you boil water,
this is a problem, a mathematical problem

that nobody has solved yet.

Think about... in this room,

there are all the particles
of gas going around,

the speech that I am making,
waves of sound,

the heat going on.

Close your eyes for a second, maybe,
and try to imagine

the complexity of the gas around

with billions and billions of molecules
going and bumping into each other

and exchanges of energy
all around your body and around you

billions and billions of molecules.

This is crazy.
The world is so difficult to understand.

And still it is a marvel that Einstein
was very surprised of, for instance,

that the world in some sense
can be explained and studied,

thanks to mathematical formulas
and physics.

And it is our job, understanding the world

with the help of mathematical formulas

and with only our logic.

Here are some of the objects we work with,

take them as artistic objects if you want.

Each of them for mathematicians' eyes
stands for some natural feature.

Here in blue
you have the equation of fluids,

incompressible fluids, all are equations.

Here the equation for gas,
here the equation for plasmas and so on.

And the first one on top, here,
this is the Boltzmann equation,

I made my PhD on this equation.

When my adviser first wrote down
the equation for me

and told me
"You know, OK, we'll do this and this"

and after a couple of years
when you start to master the equation

you even can do this:

"Wait a minute!

Am I going to spend
two or three years of my life

on this one-line damned equation?

Aren't there better things to do?"

And I did it and I loved it.

(Laughter)

Not only did I do it, but I spent ten years
on the damned equation.

(Laughter)

And little by little I appreciated
how rich it is,

what beauty there is in it,

what complexity of natural phenomena
are hidden.

Many papers, many things that I still
don't understand about this equation

but some things I understood and this is
part of what awarded me the Fields Medal.

Boltzmann was a great scientist,
this equation is from 140 years ago.

He understood a concept that is universal
deep and associated with the equation,

the concept of Entropy.

Entropy is associated
with uncertainty, disorder.

When we look at some system
and we can make some experiments

like feeling the pressure with our hand,
or something

it always involves
a great number of particles

so we only have access to it
in statistical sense,

not the exact position
of all the molecules.

So there is a great deal of uncertainty,

and Entropy captures
this uncertainty disorder.

Here is the formula of Entropy: S=k.log W

This is me near the grave
of Boltzmann in Vienna.

I love this picture.

It shows me like a heir of Boltzmann,
thinking about his ideas

you know there is communion
throughout the centuries

I worked at the same problem as he.

And this formula is magical, I swear;

when I met a stranger
in this huge cemetery

and I asked him if he knew
about Boltzmann's grave

the guy looked at me
and said something like "S = k log W ..."

like a secret, a secret password, you know

for those who know.

(Laughter)

So I love this picture,
there is so much in it.

The concept of entropy
and uncertainty is everywhere

and was re-discovered 80 years
after Boltzmann by the great Shannon.

Shannon, a mathematician and engineer,
is the one who put

the Theory of Communication,

of transmission between computers,
of coding.

Whenever we use computers or cellphones
or whatever, there is Shannon inside,

there is entropy without us knowing it.

In all our inventions of daily life there
are some great minds who have contributed.

Now, Boltzmann had this great idea
that entropy always goes up

can only increase,
disorder should only increase

under some circumstances.

And that is a great idea, an idea
that in physics is fundamental,

but from the mathematical point of view
still lacks a rigorous understanding,

there are some mysteries
in this Boltzmann's idea.

It would be like an architect who
has given the global plan

but maybe in his building

there are some panels missing,
there are some windows non sealed there,

and mathematician
wants to understand it all,

and make the whole building stand,
and with only his logic.

And then it is such a rich problem that
it decomposes in a number of sub-problems

that have esoteric names like
"Regularization by dressing collision",

like "Entropy production inequality",
whatever.

You learn, when you work on it,
to work on these problems

and it is very emotional

it is like a love affair in some sense.

You get on one problem and then,
as I said, it's very emotional;

when you arrive in a new problem first
in this you are in total obscurity

you don't understand anything,
what's going on?

I can't understand.

Like darkness everywhere, like Bilbo,

the Hobbit, in the Gollum cave,
for those who know. (Laughter)

Everywhere so dark.

And then at some point you feel
some tiny breeze of wind

something that shows you that will open it
and you will see the light

that's the moment I prefere, excitement.

At first stage you understand it all,
light comes and it comes all of a sudden

and it is so bright,

and you understand how
the various mathematical concepts

can be put together to solve your problem.

It is a great moment like
if you put together

pieces of a symphony;

believe it or not, André Weil,
the legendary mathematician from France

compared it to sexual pleasure, orgasm,

only lasting longer, he said
(Laughter).

And to some extent that is correct.
(Laughter)

So you go and go and, so excited, you tell
the whole world about your discoveries

and then after some time
you start to become bored

no excited any longer, is time for
a new mathematical romance.

I've been through it a number of times
finding new problems, new affairs,

and trying to solve them,
some of them I solved, some not.

And that is my duty,
to solve these problems

and then to explain it.

Mathematics is not
just about solving things for yourself

it is about sharing: it's science,
it's art, it's also social activity.

Let me show for you,
as an impressionistic picture,

something that was part of my life;

a book which took years for me to write.

This is a book on Optimal Transport,

it is a problem involving mathematics,
engineering, probability and geometry

everything that was said.

I will not try to desribe it for you

even though it corresponds to things
in your daily experience

and started as an engineering problem.

On this I spent years, literally,
trying to solve the mysteries

that are behind this
optimal transport problem.

The book is about a thousand pages,

(Laughter)

it was concluded exactly on the day
of the fifth anniversary of my daughter,

and it is also a baby of mine
in some sense.

So when you have this discovery
then you go and share and tell the people

and that is what I have done,

going around the world,
meeting fellow scientists

regardless of boundaries and cultures,

I have been through Europe
and Africa and America.

I have been in Israel, in Palestine,

I've been in India, in China, in Japan,
in Australia. Everywhere.

Everywhere you feel solidarity
between scientists

all together fighting against the unknown,

fighting to increase the knowledge
and the understanding

of the world by mankind.

Always with their rigor, with tenacity,
with hard work

but also with the imagination and passion.

That's it. Thank you.

(Applause)