ERIN SODERBERG:
Welcome, everyone, and thank you for coming. My name is Erin Soderberg. And I'm very excited
today to welcome Cedric Vallini, professor
at Lyon University and the head of the
Henri Poincare Institute. Cedric is widely regarded
as one of the most talented mathematicians of
his generation. In addition to winning the
Fermat and Henri Poincare prizes in 2009, he was also
awarded the prestigious Fields Medal for his work on optimal
transport and kinetic theory in 2010. For those who aren't
familiar, the Fields Medal is often referred to as the
Nobel Prize of Mathematics and is only awarded
once every four years. Today, he is going to speak
to us about his latest book, the "Birth of a Theorem,"
which was released last week. Let's welcome Cedric. [APPLAUSE] CEDRIC VILLANI:
Thank you very much. It's a great pleasure to
be here and to continue this tour of presentation of
the US version of my book. The book appeared three years
ago, two years and a half ago, in France, but somehow
the English translation took quite some time. And in the end, everything
needed to be perfect. In the meantime, has been
translated into something like 12 languages. And the book was an opportunity
for a lot of outreach about our field and what
we do as mathematicians at a time in which many
countries consider it the single biggest
problem about science is to motivate people
to do sciences. Of course, if you are
in the United States or in other countries which
are extremely good at welcoming people from all the
world, you can just import the mathematicians. But in most other countries,
it doesn't work this way, and you need to train
mathematicians from young age. And people see the numbers
of aspiring mathematicians in the universities going
down and down and down and wonder how to prevent
this from occurring. In spite of these
trends, and in spite of the bad reputation
of mathematics, the time could hardly be
better for mathematicians. A few years ago-- some of
you probably have seen this. When "Wall Street
Journal" made its ranking of all the jobs
in the world, they ranked mathematician as number
1 best job in the world. This was a ranking based
on a number of parameters, including salary, freedom,
possibility of promotion, possibility of hiring,
many, many things. Last year, in 2014,
a company named CareerCast, which
started as a spin-off of this kind of ranking,
also ranked mathematician as number 1 job again. In a time where nobody knows
what the future jobs will look like and where a
mathematician, by definition, is a job in which you have
to adapt to any situation, in the same sense as mathematics
is the science of working on anything, and as a
computer, by definition, is a machine able to perform any
task, any mathematical task. So as computers continue to
invade the world in algorithms, mathematical tools
have such resonance and are more and more used
in every kind of business, and applications of mathematics
have never been so large. In this ranking, by
the way, the definition which was taken
for mathematician is somebody who applies
mathematical theories and formulas to teach
or solve problems in a business, educational,
or industrial climate, which is not bad in the
sense that it insists on solving problems, which is
what mathematics was invented for. However, it misses
a crucial point, that there are also people
who not only teach or apply mathematics, but there's also
people who do mathematics, who create it, who change it. And mathematics is changing
at a very fast pace nowadays. It is often a
surprise to audiences when we talk about the numbers. Here probably, in this room,
people are more familiar. And if I ask you, what's
the number of researchers in mathematics
around the world, you will probably get the
right order of magnitude. What do you think? AUDIENCE: [INAUDIBLE]. CEDRIC VILLANI: That's not bad. That's not bad. It's a bit more than that. It's in the hundreds
of thousands. But maybe it's somewhere
between 100,000 and 200,000, something like that. Increasing rapidly, in
particular with India and China producing, if I may say,
many more mathematicians. It's exponential growth. If you go around, if you look
around at a large time scale, in particular it's as usual. If you take the number
of dead mathematicians, it's smaller than the number
of living mathematicians, as is always the
case when you have fast enough exponential growth. So mathematics is a field that
is being renewed constantly. At the same time,
it is also, almost by definition, the
science in which things last for a very, very long. There is no other subject,
of course, in which knowledge can last thousands of years,
and the theorems of Euclid are still accurate nowadays. Without going that
far, if you look at modern applications of
computer and mathematics, you will see every day
around the world everywhere people are applying
contributions of Laplace, like the law of
central limit theorem, the normal errors, contributions
by Shannon about sampling, contributions by Fourier about
signal analysis processing, and so on. And all this will be mixed
with the modern technology. So at the same time, it is
very contemporary and very old at the same time. In the past, International
Congress of Mathematicians, so this is the big rendezvous
of mathematicians every four years. The past one was in
Seoul, summer of 2014. And as a sign of the times,
one of the plenary talks was from a
mathematician expressing the pride of mathematics, and
more precisely in big data at large. This was Emmanuel Candes
from Stanford University. He was a member of the
French delegation, let's say. By the way, in this
International Congress of Mathematicians, the highest
number of invited researchers were from French. It was, even in absolute
numbers, the most represented country in this conference. And so Emmanuel Candes
explained, for instance, to the audience how he had
collaborated with people from hospitals,
medical doctors and so on, with a combination of
old and new mathematics-- Fourier transform
analysis, data processing, so- called sparse
methods, and so on, to reduce drastically the time
that you need to get a scan. This machine, getting to this. Machine makes basically a
Fourier transform of your body. And in particular, when
you are handling children, it is at stake how
long this should last, and you don't want
this to last too long. And they managed to reduce
by a factor of eight the amount of time
which is needed by using the fact that if
you want to reconstruct a signal with sufficient mixing
from its Fourier transform, you need to know only a very
small portion of the Fourier transform. In experiments, they showed
that to reconstruct images, sometimes you can do it
with the best accuracy that you need having only
2% of the Fourier transform. And so this was very
significant of the evolution. At the same time good,
old Fourier analysis, and at the same time,
very modern methods of big data related--
here in this audience, I guess, for most people
it will be familiar if I say Netflix problem of
reconstructing the missing information, the same
problem reconstructing the preferences of
users for movies-- and reconstructed the image. If you have a Fourier transform
which has holes in it, there is common
mathematical structure. And in all this, again, we
see, as usual, the strength of mathematics. The same mathematical
structure can be here in many different problems. However, the life
of the mathematician is not that full of
adventures at first time. This is the typical state-- [LAUGHTER] --you know, not knowing
what is going to happen, how you are going to
get out of the problem, and waiting for good,
old Henri Poincare to whisper some
hint in your ear. Most of our time as a
researcher is spent in failure. That is objective fact. My colleague from Berkeley
University, Craig Evans, who is a world-famous specialist
of partial differential equations, likes to say,
"Every day is a failure. This is our life." And however, when you look
back on the amounts of time everything works and progress
over a 10-year period maybe is always enormous,
partly because, of course, of the collaborative
nature of science, which makes it that many things
that you did not understand, through the work, the help
of other contributions and so on, you understand
little by little. And every year, the number of
new theorems in mathematics is also counted in the
hundreds of thousands, number of [INAUDIBLE],
the number of new theorems per year. So there is this
apparent discrepancy. But also, at the
individual level, if you look at the process
of proving a theorem, it is not what
one could believe. And it, on a wider
scale, it doesn't look like this, neither like
the usual view that we may have. This is a nice drawing I took
from the Electron Cafe blog contrasting the public
perception of science with the actual science. This is like what we in
school, we learn sometimes science is theory, experiment,
conclusions, theory, experiment, conclusion. It's not the way it works. And sometimes you may think,
OK, when you have a problem, you read hard, you think
hard, you do science as told, and then you get a big reward. In practice, it's
more like this. So do the science,
not quite as expected. Amazing result turn
out to be crap. What the hell is going on? Wait, think, no, it doesn't. Funny. This makes sense. And then you figure out
somebody did it 50 years ago, and then you go
on and on and on. And sometimes in
the process, here is something that is
worthy of publication. A very chaotic process. This picture,
still however we're missing one so important
point in our life, which is collaboration. So this is at the scale
of one individual. Take this as the unit
loop and put this as a string with many,
many, many other loops, which are all together,
interconnected, and you have a better picture. And that's only one
part of the story, of course, which is
when you have something that is ready to publish. After you're right in
this step of publishing, you still have the
task to convince people that your theorem is good,
or that your idea is good, or that your theory is good. This can be very
tricky, and sometimes it takes long before people
recognize it as interesting. There are episodes
in science in which this problem of
having folks recognize the importance of your discovery
was critical or even completely dramatic, as is the
case with the sad story of this gentleman,
Ignaz Semmelweis. Any of you heard of him? Yes, you did. So Semmelweis, as those
who heard about him know, is, in some sense, the
discoverer of hygiene, the fact that it's good to wash your
hands in certain situations. And in particular, if
you're a medical doctor, and you work in a
hospital and you train by dissecting
corpses, and after that you have babies delivered,
it is good in the meantime to wash hands. This he discovered, and
this people did not believe. They were thinking he was crazy. Look at my hands. They are clean. What the hell are you
talking about soap? What will it change? Of course, in those
days, there was no model for microbes
and bacteria or whatever, so it looked like nonsense. And he had, with
statistics, he had how there were good indications
that his discovery was important but did not
manage to make his point. He ended up in an
asylum, by the way. AUDIENCE: Ooh. So this is a dramatic,
extreme example of how difficult it may
be to make your point, even when you have
succeeded in your research. Another thing which is
tricky is when people ask you to predict what will occur. And it's becoming more and more
so in research institutions, as governmental
agencies and so on want to give you money
only if you tell them what you will discover. [LAUGHTER] Of course, if you are engaged in
a big industrial project and so on, and which we think is
kind on trail, on track, then you can set up
the lines and so on. Even in that way, it may turn
out to be not as planned. But when you are in fundamental
research, by definition, the interesting stuff is
what you do not predict, and that's a big problem. Here we see in this
journal article from 1900 how people in those
days, already of course, were trying to have these
mathematicians predict the future of their discoveries. And there was this question
asked of Henri Poincare, who was, in his day, arguably
the most famous mathematician in the world, and
people asking him about science of
the 20th century. This was 1900. His answer was, "Sir, your
letter is coming to me today after having been around. If, in 1800, one had
asked any scientist what would science in
the 19th century be, how much nonsense
he would have said. Good heavens. This thought prevents
me from answering. I believe that surprising
results shall be obtained. And that's precisely the reason
why I cannot tell you anything about them. For if I could predict
them, how could they still remain surprising? So please excuse my silence." So this was the
answer of Poincare. I used this quote of Poincare
a couple of times on television when journalist is
asking questions about, can you predict the future
of mathematics in 2050 or something like this? If you think about it,
Poincare is making an answer which is deeper than it seems. It's not just a joke. "I believe that surprising
results shall be obtained" is asserting the
belief in the fact that there will be renewed
innovation always and always, and that you will never exhaust
the amount of discoveries. In those days, 1900s,
some physicists believed that physics
was essentially complete. Everything was explained,
except for a few anomalies. And from the few anomalies,
a few years later arose quantum
mechanics, relativity, and so on, a bunch
of revolutions. And so this is a good
example of the fact that new discoveries
indeed come. And if you ask
people, by the way, now with the
theoretical physicists, there are many in
sharing the belief that some new revolution
is in preparation, and they have no idea
which it will be. But somehow, they
feel unsatisfied by the present state
of fundamental physics and some of the anomalies
and things that bother them. Here are a few people with
examples of what we said first, the fact that the fundamental
ideas that say the new ones are impossible to predict. Also the fact that their
impact can be huge. The fact also that when they
are improved and relayed by many, many people, the
ideas of one individual can make a tremendous
difference. And the fact that these
new ideas contributed to change, not just
the history of ideas, but the history of
mankind in general. Of course, we start to the right
with Alan Turing, who has now been turned into
a Hollywood star with a horrible Hollywood movie. And Alan Turing, as we know,
was instrumental in cracking the secret codes used
by the German Army during the Second World War. By the way, and this is
forgotten by a bunch of people but should be reminded,
Turing could never have done anything
if it had not been for the preliminary work of
three Polish mathematicians who worked in the '30s
and who participated in literally saving the world. So Turing anyway had
these remarkable ideas, and it was
mathematical new ideas, about how to handle this code. And it is generally considered
without the contribution of Turing and
cracking of the code, the war would have lasted
at least two more years. In particular, in
1944, would have been impossible to do
the Normandy operation. So this was for Turing. And here we see it's
fascinating to think that just one idea in
one brain can have such an impact on the whole stuff. Well, these ideas,
what motivates them? I'd say in the case of
Turing, his main motivation clearly was solving a riddle. He had never strong
political ideas. The person on the
left of Turing is also one of the symbols of these
people whose only goal in life is to solve problems. They just lived for it. This is Paul Erdos, 20th century
most productive mathematician. Erdos is part of this
extremely important event in the history of ideas. When a bunch of scores of
Jews immigrated from Europe, in particular Central
Europe and Eastern Europe, to this continent, America,
because of the persecutions of the war and so
on, and here was Erdos with all the
remaining of his life. No home, no car, no bank
account, no salary, nothing, with just one
suitcase and ideas, ideas, ideas, working
for in the theorem, theorem, theorem, theorem. Whole life was like this. Whenever he got some money
from some reward or something, he would always use part of it
to put a reward on the theorems that he liked best
or give part of it to people he thought needed
the money more than him. Anyway, here was the most
productive mathematician of 20th century. No need to motivate these
kind of people particularly. Interesting also to know
that he was representing the culture of Hungary,
which was, at some point, the most innovative,
most productive ecosystem for ideas in mathematics
and theoretical physics in the world. They literally changed
the world, the Hungarians, the Hungarian scientists. And interestingly enough, also
the amount of idea production, let's say, from this part
of the world collapsed at the same time as
communism collapsed. This is one of the big
problems that stay contemporary in the ecosystem of ideas. Let's continue with another
of these Hungarian Jews. Here, uh uh. Anybody recognize him? AUDIENCE: Pauli? CEDRIC VILLANI: No, not Pauli. This is Leo Szilard,
Leo Szilard, the first human to have
understood the concept of chain reaction back in 1933. At the time where
it was even not known whether uranium was
fissile, in those days he had understood the concept. The story is a good one. So this guy also was wandering,
going everywhere and so on. So he had read in the newspaper
the account of a public lecture by Rutherford, the father
of nuclear physics. And in that lecture,
Rutherford once had a question, had said that, yes, there
is some energy in the atom, but it's so tiny that you would
never do anything about it. The exact words was,
"It would be moonshine talking if you are willing to
extract energy from the atom." And Szilard liked contradiction. And so he was very annoyed
with this, and he thought, I will prove that guy wrong. And he thought
for days and days. And at some point, aha, while
crossing a street in London, he had this illumination
of how to do what will be the principle
of chain reaction and exponential growth
of this energy and so on. This is the patent
for the atomic reactor after Fermi and Szilard. And so Szilard is the starter
of the Manhattan Project. There is a famous letter which
Einstein sent to Roosevelt, which was the start,
but the letter actually was written-- the draft
was written by Szilard. When Szilard and Teller came
to visit Einstein in Princeton, and they told him
chain reaction, atomic bomb, his answer was, ah,
I had never thought about this. This is ironic because
nowadays many people associate Einstein with the atomic bomb. The real guy behind
the atomic bomb is this guy that
essentially nobody knows. Szilard spent all
the final years of his life fighting for
pacifism in the world. And also, at some point, was
cured for cancer by radiation, and changed himself the
protocol of his cure so that he would be cured
with higher doses of radiation that he was sure were the
correct answer in his case. So fascinating,
fascinating character. So with all these, we see we
have here examples of people who changed the world with
just the right idea who were motivated mainly by
their appetite for ideas. And here comes the problem. And here I put a picture
of some famous university. Here comes the problem,
which this arises naturally. How to make a good
research institution, and what is a good
research institution? A good research institution,
a good university, a great university
is not a university in which you are taught the
most up-to-date science. It is a university in
which students will later do science which is better than
the one that they were told. So that's the big problem
of master and student. How should the student
be taught in such a way that he will do better
than the master? Nobody has a real
satisfactory answer to this. Still, that's the most important
problem that you want to solve. Henri Poincare
gives us some hints about how the ideas came
to him, were coming to him. Poincare was considered the
symbol of the genius type with illuminations,
you know, flashes. In one of these famous
takes, he gives us some diary, some
account, of some of these ideas that
made him so famous and revolutionized
mathematics in those days. And he tried to analyze this. It's also very unclear
what you can get from this. So in one of these
most famous passages, he talks about the
time when he was working like crazy on some
problem and can never make it. And goes and goes, and, pfft,
doesn't work, and at some point goes on a trip with his friends. And when as he's coming
back from the trip and sets his foot again on the
bus-- it was organized trip, whatever-- at this
precise moment came the idea about the
solution of the problem. Here is a longer passage
of the same text. "Disgusted from
my failure, I went to spend a few days near the
sea thinking anything else. And one day, while walking on
the cliff, the idea came to me. And as before, it was
very brief, sudden, with immediate certainty,
that arithmetic transforms of indefinite ternary
quadratic forms are identical to those of
non-Euclidean geometry." So this is the
comment of Poincare. Let's comment this text. First, under which circumstance
does the big idea come? In popular culture, there
are these representations of episodes, like Newton
sees the apple falling down, and eureka, I get it. In real life, it's not
the way it happens. You do have the eureka
moment, illumination, but usually there is nothing
in the environment that relates to the problem
that you are working on. It's like it's a separated,
independent event. And here is the same, the
[INAUDIBLE], of course, has nothing to do
with quadratic forms, even if you forgot what
quadratic forms are and even more, arithmetic
transforms of blah, blah, blah. The other noticeable
comment is the process that Poincare describes is
neither hard work continuously, nor just waiting for the flash. It's a combination of both. Our talents work
very hard, then there is something of the relaxation,
and then the illumination, and then continue
work very hard, and then there is
something again. And if the brain has not been
prepared by the hard work part, the illumination will not come. And it's the same in all these
stories I told, no [INAUDIBLE], Szilard, or Turing, or whatever. It's always first
the hard preparation, and then something
has to be arranged by some unconscious process
that nobody really knows about. Last comment. One of the reasons why
this text by Poincare became famous and known by all
people working in committee sciences is that he used the
big words without explanations. It seems paradoxical that the
fact of not explaining things adds to the success. But indeed, assume that Poincare
wanted to share with the reader the science of this. OK, my friends, let's explain to
you what the quadratic form is and the ternary
quadratic form when it is indefinite and so on. Even if Poincare had been the
best pedagogue in the world, it would have taken
pages and pages. Most of the
audience, if not all, would have been
lost while reading. But here he gives the big
words with zero explanations, so you don't even try and
understand what it's about, and it's just the
circumstances which matter. And in this, this is one
of the testimonies of, from the inside,
what happens when you are working on this idea. So here we are
arriving at the book. The book which I did
is like an outgrowth of the Poincare testimony. By the way, even though
I knew the Poincare text, I was not aware at
all that I was here applying the same kind
of process than Poincare. But is, as usual everything that
is in your mind you don't know. So the book is like the
diary of the discovery of the big theorem. I say big because it took more
than two years of my life, big because it played a major
role in my Fields Medal, and because it was
solving a problem of more than 50 years old about the
behavior of plasma physics. Let's take just a few moments
to explain what it was about. Here you see a
picture of Lev Landau, one of the most [INAUDIBLE]
physicists of the 20th century. He doesn't look happy,
and that's normal. That's the mug shot
when he was taken to prison for no good reason. And these are images of plasmas. Plasma is any state
of matter in which electrons have been torn apart
and separated from the nuclei. And it's an element. It's a state of matter as fluid,
or gas, or solid, or crystal. It's a very common. Even though it was
discovered quite late, it's very common
in the universe. And the main equation describing
it is, for the electron motion, is a statistical variant
of the Newton equations, which is called [INAUDIBLE]
Poisson equation, whatever. This equation tells you
about the statistics of the electrons, of the
electron distribution, and how it evolves with time. People, by the way,
in galactic dynamics use the same equation
with the idea that at the scale
of a galaxy, a star is something like an electron
at the scale of a plasma. Of course, it's not
true, but still a star is so tiny compared
to the galaxy that may contain hundreds of
thousands of billions of stars. So that it's OK to have a
statistical treatment of stars in a galaxy. So you have this equation. And it's a problem that
people, for instance, working on nuclear fusion, for
instance, in working in-- well, there are many applications
of plasmas, including screens, as we know. And it's a problem to
understand how it reacts to electric stimuli and so on. And Landau suggested
that there would be equilibration processes,
that is, spontaneous damping of electric perturbations
in the plasma, even though the plasma is like a
perfect world with no friction, no entropy increase, as the
second law of thermodynamics would say. So it's like you
would have stability but nothing irreversible,
no friction or whatever, which is very paradoxical. At our scale, the main
factor of stability, where there are two-- why
do things stop thrusting is that we are all
attracted by the Earth. And we see in outer space
how things are unstable, like when "Philae" wants
to land on the asteroid and so on, it's such a big mess,
it goes up it's so unstable. So first, we're all
attracted by the Earth. Second, there's
friction everywhere, otherwise nothing would work. In plasma, there's no
friction, and there's no common attractor. However, there's Landau,
still it's a stable media. And if you make a
small perturbation, it will spontaneously damp. This is the paradoxical
Landau damping effect, which, more or less, appears in
one paper of plasma physics out of three, so common
is it in its use. So what we did, the
two of us-- that is me together with Clement
Mouhot, a former collaborator-- was establish this
Landau damping in the mathematical world
under nonlinear equations, so to speak, the true equation
of plasma, so to speak, while it had always been
treated under approximate linear equation. There was a huge
jump in difficulty. The linearized treatment
is like three pages. The nonlinear treatment took
up a whole paper of 180 pages. And this came at the same
time with new insight, in particular to the fact that
this problem was connected to the stability problem,
which was devised in the '50s for the solar
system, and to a paradoxical phenomena of
spontaneous response by the plasma of a
couple of impulses, known as the plasma echo. So new insights, new
theorem, and so on. But in the end,
how did it arrive? In many cases, I do lecture
about this Landau damping. And I give these
lectures around the world to physicists, to
mathematicians, and so on, but as if it's something
simple because a posteriori. It's always simple
when you understand. When you are in the middle
of it, it's always a mess, and it's not simple. And in the book, it's
precisely the complementary of the usual talk
that there would be. I don't explain what
the science is about. I only explain about
the discovery process. And like in the Poincare,
using the big words without any explanation,
but that's the way we talk with each other,
showing all sorts of formulas. And by the way, this is
the first [INAUDIBLE] book that I know of
which has been entirely typeset with the word
processor that we use for mathematical formulas. That is the TeX software
from Newton Stanford. Also showing the moments
where you work, like Poincare, the moments of hard
work, like when you are at 3:00 AM
in your apartment working like crazy to
make the computation work, the moments
of illumination, like, ah, that's the way
it has to do, the moments of interaction, and so on. The book is like a pregnancy. "Birth of the Theorem"
is like-- in fact, it's the last chapter is the
birth because that's publication for a paper. Publication is when the idea
is out and going in the world. And all the what comes
before is like the pregnancy. The first chapter
is the fecundation. There is a discussion
between Clement and myself. And we are discussing,
and this is interaction. And like, we wanted to work on
something completely different. We discuss on the blackboard,
tak, tak, tak, tak. And oh, this reminds
me of a conversation that they had two years ago
in Princeton with a postdoc, and that was the question
he had, et cetera, and then we think of it. And then said but
it also rings a bell with some other conversation
three months ago in Providence with some other guy,
and we put together. And then, ah, but
then this should have to do with the Landau damping. What? It was not the intention
to work on this. But from the interaction
started the project. And then took two
years to-- first took several months
to understand what we wanted to prove. And then took more than one
year to put the proof together with many mistakes, with
false announcements. Now we know we can do it. Now we cannot do it, et cetera. With conversations, discussions,
and also a lot of environment, like listening to
this music, doing this, whatever, and that keeps
you going, keeps you going. So by the way, a
word about the title. Original French title was,
[FRENCH], "Birth of a Theorem." And that's really what it is. Then the French editor was
not so happy about the title. Why can't you make it a bit more
ambiguous, more romantic maybe? And I thought, OK, we
give various tries. And then "Theoreme
vivant," "Living Theorem," yes, that's more ambiguous. What does it mean,
"Theoreme vivant"? Well, maybe that's a guy
who's a living theorem. Maybe it's about the
fact that mathematics is living, et cetera. Also it rang a bell
with a book which I read as a child, Disney
photograph book named "Desert Leafs," [FRENCH]. Maybe some of you saw it. No? Maybe. And you know the desert,
when you are not there, you think it's some
boring, dry, dead place. But you look at the pictures. You see when you know where
to look and when to look, it's full of color, full
of life, fascinating. It's the same with mathematics. For people out there, it's
this desert, dry, dead subject. And when you know
where to look, it's this fascinating,
colorful, living subject. So the analogy was good for me. So "Theoreme vivant" it was. But then the English
translators, the publisher, OK, you know, "Theoreme
vivant" is a bit too ambiguous. Why don't we use something
more to the point? Why not "Birth of a Theorem"? And I said, OK, good. OK, if you wish. That's one of the great
things when you publish a book and it's being translated,
you have all the variants. And by the way, all
the cliches are true. That's what you learn
with this kind of process. To finish, what do
you find in the book? You find all the
ingredients which you need to make your
idea stand up on its feet. We don't know how
the idea occurs, how the theorem occurs in the end. But we know what you
need, all the ingredients. Then the chemical
reaction is a mystery. The first ingredient
is documentation, because you never do
research out of the blue. It always builds on previous
research and insights. Documentation can be things
several centuries ago, can be things very
recent and so on. Of course, it was transformed
by computers and by internet. In one example that
I quote in the book, at some point I need to have a
certain formula, like whatever, iterated derivatives or
compositions of functions. Eh, I have no idea what it is. But I remember, it seems to
me that 15 years ago as I was a student, our professor
was telling of something that looks like this. Ah, never mind if you
don't remember the formula. Just a few keywords,
and you get the formula. You get the history. You get the conditions
application. You get everything you
want to know about it. So the problem becomes not,
what is the formula, but just, ah, that's the formula. I need a formula of that type. And still, of course,
it's quite a big deal, but it helps you a lot to
have all this knowledge which is dispersed and accessible. Second ingredient is motivation. Motivation is maybe the most
important of the ingredients, but it's the most mysterious. Nobody knows what
motivates people. And it's not rational. Sometimes you lose
the motivation, and there's nothing
one can do about it. It's like when you are going
into a nervous breakdown. There's no rational thing
taking you back in business, and there's no rational argument
keeping you motivated again. Actually, when
you read the book, you have the impression
it's completely insane, the amount of
motivation and faith in the final result
that we display. Like, you think, we
already worked for one year on this 100 pages. We still don't know if it works. Do you really
think it will work? Yes, yes, we'll do it. We have to do it and so on. It's just a gut feeling. So why do people
become motivated? Why do people
become researchers? By the way, a creative process. Some people say childhood
is so important for this. By the way, both Szilard
and Turing all their life were influenced,
strongly influenced, by a book they read when
they were 10 years old. In the case of Turing, it's a
book called "Natural Wonders Every Child Should Know." And all his work on
artificial intelligence, whatever, you can find
it in German in this book he read when he was 10. I don't know what I
read when I was 10, but I was watching this movie,
"Donald in Mathmagic Land." I don't tell you it's
high mathematics, but I thought it
was fascinating. Maybe this played a role in
me becoming a mathematician. Who knows? Here is an interesting
experiment about motivation, by the way. It was made in the UK. It was all these [INAUDIBLE]
here as school children, and there's one
researcher there. So they all worked on the
experiment and so on, kids from 8 to 10-year-old. And they worked on how do
bees get where to forage from and whether they are
sensitive to patterns, and it was published in
a scientific journal. "We discovered that bumblebees
can use a combination of color and spatial relationships in
deciding which color of flower to forage from. We also discovered that
science is cool and fun because you get to do stuff that
no one has ever done before." So this is one example
of experiment devised to trigger strong motivation. OK. Third ingredient is ecosystem. The innovation never
arrives on its own. The inventor is never
alone, and there's always a whole ecosystem
which is around. Often it comes in
cities, for instance, or regions of the world. There was one point in
history in which Persepolis was the most innovative city in
the world, one point in history in which was Paris, one point
in which it was Budapest. Of course, everybody is aware
of Silicon Valley innovation. And as an ecosystem, it
[INAUDIBLE] and so on. If you are a director
of a laboratory, as is my case with
Poincare, your work is to make in the lab an
atmosphere so that people could be creative and profit,
meet each other, whatever. So that the fact that
there are many people together will make
them, as a whole, more productive, more efficient,
more creative than if it's just individual, and this other,
and this other, and so on. In this case, in the
case of the book, the fact that it originated
in Lyon was so important. My colleagues in Lyon, in
particular, [INAUDIBLE] meet the colleagues
at some point. Put me in the right direction
for wrong reasons, by the way. But it was in discussion and
so on that the project started. And later, at some point in
the book, I moved to Princeton. And half of the hard
work, or more than that, is done while I am in
Princeton, and my collaborator is in Paris. And in fact, at some point,
I am in this very particular atmosphere of the Institute for
Advanced Studies in Princeton, in which you can concentrate
like crazy for days and days on a certain problem
with no obligations and so on, also was enormously
important part of the work. The next ingredient
is the communication. I told you when I
was in Princeton and Clement was in Paris,
how we would communicate? Well, email, lots of emails. Every day emails, hundred
of emails to work on this. And of course, it makes like
your kind of delocalized brain with this. In all my experience
of researcher, all the time, projects
were originated by face-to-face
contact and deepened by email collaboration. And these are very different
phases of the project. The start, the initiation
of the project, and the deepening of the project
requires different environment, different skills, and so on. Tim Gowers in the
UK, by the way, who got Fields Medal
in the late '90s, made experiments of
massive collaboration in mathematics, like hundreds of
mathematicians working together on a single theorem. A bit like, of course, is done
in free software industry, but at a very theoretical level. The next ingredient that
you need for the ideas are the constraints. If you have no constraints,
then you have no creativity. And this is not
particular to science. In mathematics, we do
feel the constraints so strongly because it
needs to be all rigorous. It's crazy constraint. You know, famous
problem in mathematics is the Riemann hypothesis. It was checked on thousands
of billions of experiments. But that is not the
proof for mathematicians. The proof will only be
a set of logical rules at such a crazy constraint. And so it's no wonder
that mathematics can lose lots of
creativity because you have to overcome the constraint. It's the same in every field
of human culture and so on. For instance, poetry, at the
same time, the most constrained and the most creative
of literary genres. Here I put as
illustration of the power of constraint two
of my favorite, very innovative pieces of art. One is the study by Gyorgy
Ligeti consisting of only A's. You know, like A, B, C,
D. It's a [INAUDIBLE]. Here there's only A. And
still it's not boring. That's the surprising
stuff, because you have to work on all the rest. And the other is the
famous thick novel by Georges Perec, which doesn't
use the single letter e. So in each case, you
have the impression it's kind of a new kind of
art because of the constraint. The constraint in itself
puts the creativity. In mathematics, one example,
depending on whether you are imposing yourself
to be constructive or non-constructive. Being constructive is an
the additional constraint, and it requires
much more creativity to find how to estimate
and construct the things. I belong to this
strain in some sense. For instance, I
always want theorems to come with a kind
of recipe, algorithm. The next ingredient
is the intuition. Intuition is this thing
that we don't understand, which comes together with the
hard work, the illumination, and so on. One story, which I
tell in the book. At some point in the process,
after months and months of work, I am here
in my apartment working late at night
trying to fix a proof. You have to imagine
the situation. I already announced that I can
prove the theorem in public. And I discover there's a
big gap here, still there. Well, it should not be so
big, but still there is a gap. And so it's a big
problem, and so I'm working hard to fix the
gap, trying all my tricks, years of experience. And 1:00 AM, 2:00 AM, 3:00
AM, 4:00 AM, go to bed. And doesn't work, doesn't work. And then I wake
up in the morning, order the kids to take
to school, or what. And then there is this
voice in my head-- take the second term on
the other side and Fourier transform. And that was the start
of the solution indeed. And I tell you,
having this sensation of the voice in your
head is something-- it's an experience
that-- it's exhilarating. I don't know. And showing how much the brain
is still working and working after this preparation
phase, and so that there can be something like a light. The last ingredient you
need is chance, is luck. Because you will
try and try and try, and it will always be a failure,
except from sometime where, by luck, you have
some good feeling. And the most important is
recognize when you are lucky. So this is so tremendously
important in the research. So that's it, seven
ingredients that are there. And mix together a reaction,
whatever, luck, whatever, after some time get the idea. In this case, get the theorem. And after, it's another process. You will go and publishing
it and share it and so on. While we are there, let us put
a nice quote of Thomas Jefferson about ideas here for sharing. "If nature has made any one
thing less susceptible than all others of exclusive property,
it is the action of the thinking power called an idea, which
an individual may exclusively possess as long as he
keeps it to himself; but the moment it is
divulged, it forces itself into the possession
of every one, and the receiver cannot
dispossess himself of it. Its peculiar character, too, is
that no one possesses the less, because every other
possesses the whole of it. He who receives an idea from
me receives instruction himself without lessening
mine; as he who lights his taper at
the mine, receives light without darkening me." In those days, politicians did
speak quite well, you know. And this is from Poincare. "Thought is like lightning
of striking the long night. But it is that lightning
which is everything." Because, of course, it may
be a big idea, difficulty, years of work, and whatever. Whatever you have as
an idea, it will only be a tiny, tiny,
tiny part of what you would like to prove and
to know in the general nature. So if you ask, for
instance, mathematicians what we understand about
the world around us, they will tell you,
pah, so little. It's like tiny islands
in ocean of unknown. And it's exactly
like Poincare said. It's like this
lightning strikes, so thin in the long night. So we know almost nothing,
but the almost nothing is so precious because that
something really came out of the culture and the
richness of the thought of many, many people
and the efforts. OK. Thank you. [APPLAUSE] ERIN SODERBERG:
Thank you, Cedric. I think we have time for
one question, maybe two. AUDIENCE: So some of us here are
software engineers, not quite the number 1 profession. But we're OK on the list. And when we write
180-page program, we're pretty sure there's
some problems in it, and we have some
tools for fixing that. How does that
compare to the tours you have to make sure
your 180 pages are right? CEDRIC VILLANI: It's
pretty much the same. We know there are
always mistakes. And big problem is to
understand it sufficiently well that you are confident
it will work, even if there are
these small mistakes. From time to time, you fail. And there are famous
cases in which a famous theorem, for instance,
becomes a conjecture when the big mistake is discovered. And there are some--
in the history of 20th-century
mathematics, there are some famous cases like this,
in particular, for some reason, in the field of
dynamical systems. And so we are never really sure
if some important criteria is later, when someone simplifies
a proof, when someone gets a different proof of the same
result, when the theorem is taught, when you get lectures,
et cetera and so forth. But it's so difficult to
take the mistakes out. So right now, there
is all this trend about automatic checking of
proofs, with, in particular, research institutions like
[INAUDIBLE] in France. And for instance, one
of the leaders in this, Georges recently worked
out this challenge. He took a famous
proof in group theory, I think like a 200-page
proof, a big thing, and-or translated it
in this Coq language, which supports
automatic verification. And showed, indeed, that
even though in the proof there were some minor mistakes,
the whole thing was correct. Certainly this will
develop and develop. In the same group--
because, of course, it's a mixture of logic,
computer science, mathematics, and so on-- in the same
group there is Xavier Leroy.. Some of you may know him. Yes. So he worked on the
certified C compiler, and he explains every
single of the C compilers which are out there
has some bugs. A bug meaning that sometimes
you do the program, and the executive
file resulting from it is not the one
that you would want and in some locations
gives the wrong answer. Except his one,
which he developed specifically using
this machinery of automatic checking. So I think it's quite
the same problem. With long computer softwares
and long mathematical proofs, there are always bugs in there,
except if you can, as you said, check them specifically
and systematically. ERIN SODERBERG: I thank
you all for coming. And let's thank
Cedric one more time. [APPLAUSE] CEDRIC VILLANI: Thank you. [APPLAUSE]
