The Extraordinary Theorems of John Nash – with Cédric Villani


[APPLAUSE] John Nash was a most peculiar
man– an extraordinary man. And it will be the
purpose of this talk to pay due attribute
to this man. And recall, as suggested by
this title, that what he did was amazing or, to put it in
the words of the great geometer Mischa Gromov, speaking of
an accomplishment of Nash, “It could not be true,
and it was true.” Let’s explain what
Gromov meant by that. Here’s a short biography
of John Forbes Nash. Born in Virginia, he
starts as an engineer at heart, studies chemistry, but
eventually goes to mathematics. In 1948, he becomes a
student in Princeton. 1949, 1950, he defends his
PhD, including in particular the famous Nash equilibria. It’s a two-page paper. From 1951 to 1958, he does
a first-order world-class mathematical career in MIT,
Courant Institute, in New York, and Princeton. And then he proved
some theorems which are extraordinarily famous
among analysts and geometers– embedding and continuity. 1959, history– a sad history
of acute paranoid schizophrenia. And, around 1990, spontaneous,
progressive healing. 1994, Nobel Prize in Economics,
while lived in Princeton. This slide was written
a few years ago, as I was for the first
time lecturing on Nash. And later on I will update
this, at the end of this talk. Here is a paradox. Maybe 95%, I know it’s
subjective measurement of Nash’s fame is due to
game theory and his work on the so-called
Nash equilibria– famous in economics,
famous in biology. But is that not, to use
his own words, the least of his accomplishments? Should not Nash be
thought of as one of the greatest analysts
in this century? As Gromov said, he was combining
extreme analytical power with geometric intuition. And the goal tonight is to focus
on the less-known story– less known than the game-theory
stuff– to ask ourselves, what makes a value of
a mathematical result, and to get behind the scene
of a famous and dramatic paper of mathematical research. Let’s get 60 years in the past,
at the moment when young Nash arrives in Courant Institute. He’s 28 years old. His a, you know,
strongly built guy. He arrives in New York. And in New York, he arrives in
this famous– already famous– institute for advanced,
uh– pride mathematics, in some sense– Courant
Institute, a place that, after the Second
World War has reason to extreme mathematical fame
and which is part of the world history of mathematics. As he arrives in
Courant Institute, he’s young but already
preceded by his legend. And Louis Nirenberg, one of
the most important people in Courant Institute,
has a problem for him. Hey, young Nash is coming. I know what I will ask. What was it that had
made Nash famous already? Let’s do a little bit of
flashback and talk about that. What had made Nash famous
was the so-called isometric embedding theorem. To motivate this, let
us say that geometry can be done in several ways. Historically, geometry
is measuring the Earth or making a map of the world. “Map of the world,”
we think of something which is two-dimensional. You know, on a
sheet of paper, let us draw the shape of the
continents and the rivers and whatever. But, as we understood,
as ancient Greeks already understood, the
Earth is not flat. The Earth is spherical. And, as was first observed at
the time of space exploration– you know, the ’60s– the
Earth is indeed round. And somehow that
second view is simpler. The flat view is
more complicated. Why more complicated? Well, you have all kinds
of problems and paradoxes when you want to represent
the world on a flat space. Problems of the shapes
and, you know, the areas. If you look at the usual map
of the world, in a rectangle, like, Antarctica
looks like it’s huge! And the countries
in center of Africa, they look small, when
in fact they are huge and Antarctica is not so big. It’s distortion, due to
the flat representation. There are other problems
which are interesting. This is a picture which
I took some years ago, in a hotel in Palestine. And, as people who
traveled in some places in the world in which
dominating region is Islam, when you arrive in
a hotel you always have this. A qibla. That’s an arrow indicating
the direction of Mecca. Because, if you’re
a good Muslim, that’s the direction in
which you should pray. Now, my friends, what does it
mean, “the direction of Mecca”? Well, viewed from here, it’s OK. South, east. But the answer is not
so OK, not so easy, if you are in any
part of the world. If you are in North
Pole, what does it mean, “direction of Mecca”? It actually was a problem. It was one of the motivations
for development of geometry in the Arab world– at a time
in which European mathematicians were complete amateurs, compared
to the Arabic and Persian mathematicians. And here is a
problem, for instance, that really was a problem
when mosques started to be built in the United States. Viewed from, say, New York,
which is direction of Mecca? And the answer is easy, if you
think of the Earth as a sphere. Put it on your globe. Say you take some
elastic string, and you put one end in Mecca,
the other end in New York, and you see what is
the trajectory, there, of the string. But if you look on
the planisphere, it’s not obvious at all. And the answer, if
you are in New York and you want to go in
direction of Mecca, you have to face,
somehow, northeast. This is a thing that
people, some people, could never understand. Actually, there
were some incidents with this in which some
embassy was involved and, you know,
saying, you are crazy. Put it direction of south,
because it’s south– No, no, no! But it’s geodesic– it’s
shortest path– and so on. I don’t understand your thing. It’s south– it should
be south– whatever. Now you know, with the
development of aviation, it’s obvious. You look at the
trajectories to go, say, from Paris to New York. Has to be this way, you know? It’s a curved trajectory. And we know it’s the best way. It looks as a curve, it
looks as a path which is not economic on the
plane, in two dimensions. But if you think of it as
a sphere, it makes sense. It’s obvious that’s
the way it should be. So spherical
geometry, in a sense, is more simple than plane
geometry, for that reason. On the other hand,
plane geometry is more economical, because
you just use two dimensions. While spherical geometry,
you need the three dimensions to put your sphere in
something, in our space, which is three-dimensional. So it seems that it demands
more information, the sphere. So this is the kind of
dilemma that you have. Either work on a
simple geometry that requires “many” dimensions–
in this case, many three– or work with a
complicated geometry that requires few dimensions– and,
in this case, “few” is two. This is a problem that can
get much more complicated than this, but this is the
essence of this problem, which geometers called “embedding.” Embedding is like the
spherical representation, because we think of
the sphere as part of the three-dimensional
space, as embedded in the three-dimensional space. And, on the contrary,
the plane representation, geometers would
call it “intrinsic,” because it only uses
the two dimensions which are intrinsic to the
surface of the sphere. So, intrinsic, or embedded. What is geometry? Should it be intrinsic,
or should it be embedded? It takes us back to
the problem of what we want to do in geometry. At first, geometry
developed, you know, with the ancient Greeks
as a Euclidean geometry, drawing figures that
you can draw in the sand or on paper or on the
flat stone or whatever. But then people understood,
when you travel long distance, and so on, you need to
understand spherical geometry. And then some people
say, after all, you could do geometry on any
surface, not specially a sphere. Could be some weird thing. And then people said, let’s
work on any possible geometry and devise a theory that
can encompass all of them. And let’s try,
also, to do things which withstand
deformation, things which are as intrinsic as possible. You know, if you take
a sheet of paper, and you make a drawing
on your sheet of paper– this is my sheet of paper. I do some drawing on it. If I bend the sheet of paper,
from an embedding point of view it will be very different,
but the figure, here, will remain the same. So, if I look at
it intrinsically, it should be the same object. And geometry made
huge progress when it started to think
in intrinsic terms, even for curved geometries. This is related
to the development of the non-Euclidean
geometries in the 19th century and, in particular, the
so-called hyperbolic geometry, which was devised by people
like Gauss and Lobachevsky. Hyperbolic geometry, like
the Euclidean geometry, was a geometry in which
all points were equivalent. No privileged point. A sphere, also, is such
a geometry, you know? On a sphere, from any point
the sphere looks just the same. There are very
few geometry which have this property
of looking the same. And the hyperbolic
geometry is one like this. But somehow it’s the
inverse of the sphere. While the sphere is
always positively curved, the hyperbolic plane is
always negatively curved. And it has some
weird properties. Here is an example of a picture,
in the hyperbolic plane, of these lines which
are all geodesic lines, meaning “shortest path.” And we are making
it rotate, like we would do to rotate a figure
by rotation in the plane. See how unusual it looks. Here is a famous rendition of
the hyperbolic geometry by MC Escher, the famous artist. And, you see,
hyperbolic geometry is a geometry in
which units of length will change from place to place
in a plane representation. I told you, intrinsically
every point is the same. But when we represent
it on the plane we are forced to make it
having dimensions that change from place to place,
so that, in this, each fish has the same length
as any other fish. This is hyperbolic geometry. And you may ask,
OK, but now, can I represent this hyperbolic
geometry in such a way that the distances
are not distorted, like I would do on the sphere? Can I embed this
hyperbolic geometry in our three-dimensional space? And this is a problem
which already tormented Gauss, the early 19th century. And he thought, well,
there’s a problem in there. There’s a problem. And it was a huge progress when
Riemann, the student of Gauss, said, let’s not worry whether
it’s embeddable or not. Let’s go on and work with
the intrinsic properties. And we’ll make
progress from this. And he defined the so-called
curvature which, to this day, has remained the most
important tool used by geometers to study
non-Euclidean geometries– the Riemann curvature. Without Riemann curvature,
general relativity would never have existed, and
an enormous amount of everything that we
see in image processing could not exist, either. And Riemann set up the axiom
that we still use, to this day, to describe a
non-Euclidean geometry– the concept of
Riemannian manifold, which is a geometry
in which each point is in a neighborhood resembling
a distorted Euclidean space. This was a marvelous
achievement and one in which you say, let’s forget
whether it can be embedded. So we can treat much
more general situations. OK. Looks much more general. Is it, really? Is it really more general
than embedded geometry? Hmm! Can one embed hyperbolic
space, for instance? Can we have a representation
of the geometry that I showed you
which would be set in our three-dimensional
space without distortion? Answer is, no. We can have approximations. We can have some
partial geometries with some so-called
singularities, like crests, like on these representations. This one, on the right top,
is called the “pseudosphere.” Below it, it’s called a Coons surface. And, on the left
bottom corner is called a “hyperbolic crochet.” [LAUGHTER] I have one here. Let’s hand it over to you. And you may pass it from
people to people, OK? [LAUGHTER] That’s a hyperbolic crochet. You can distort it, and so
on, and reflect on the fact that every point of this
has the same geometry properties as the others. And that they should
try to continue this for knitting and knitting. There are recipes from this. You’ll find this online, how
to do hyperbolic crochet. If you try to make a very
large one, you will fail. Such a knitting, such a crochet,
has to be rather limited. And it was proven by some
of the famous mathematicians of 20th century. Hilbert, for
instance, proved it’s impossible to have a very
large hyperbolic crochet. So, OK, this seems like,
yes, the embedded geometry is restricted. And there are some geometries
like the hyperbolic space that we cannot embed. No, no, no. And the mathematician
will tell you, OK, we cannot embed it in our
three-dimensional space. But who said we need
three dimensions? Let’s dream! Let’s embed it in a space
of four dimensions, maybe, or five, or six– whatever. It was proven by Blanuša that you can embed hyperbolic
geometry, infinitely large, in a six-dimensional space. So, in the end, it is embedded
geometry, all the same. But we just needed to
enlarge the point of view. And now you may say, is it
the same for any geometry? So here’s the rule. I give you an abstract
geometry– an abstract surface, for instance. Can you find a Euclidean
space, a straight space, but possibly with 10 or 20
or 100 dimensions, in which I can embed my
geometry such that it would be a part
of that geometry, as the sphere is part of the
three-dimensional geometry. And this was a problem
that stood open, ever since the time of Riemann. An old, respectable problem. Now, let’s go back to Nash. You have to understand that
Nash was not very humble, as a young man. [LAUGHTER] Nash was rather,
you know, annoying. And when he arrived in MIT,
one of his colleague, Ambrose– a quite good mathematician–
became so annoyed with Nash’s arrogance– like,
you know, I am a genius. I’m the best in here. OK, Wiener is good, but
I think I’m even better. And so on. And once he was so
angry, he told him, well, if you’re so good, why don’t you
solve the isometric embedding problem? Nash’s reaction was, what? What is this embedding problem? What is this about? And Nash was, like, thrilled. Oh, is this a difficult program? Maybe I can become
famous by solving it. OK. So he checks, you
know, asking people whether this is really a problem
that can make him famous. [LAUGHTER] Starts working on it. Spends more than
two years on it. Wow, he said, I will solve it,
I will solve it, et cetera. Here is my idea, here
is this, and so on. [EXHALE] Ambrose laughs at him. We have a letter
of Ambrose in which he says to one of his
colleagues, well, there’s Nash. “We’ve got him, and
we saved ourselves the possibility of having
gotten a real mathematician. He’s a bright guy but
conceited as hell, childish as Wiener, hasty
as X, obstreperous as Y, for arbitrary X and Y.” [LAUGHTER] This is mathematician
humor, you know? [LAUGHTER] OK. But now the thing is, Nash
did solve the problem. Not only one proof, but two
amazing proofs– two amazing theorems that he got from it. Nonsmooth embedding,
smooth embedding. People were not even aware
there was an interest in nonsmooth embedding, here. He proved both. Let me explain a little
bit what it was about. First amazement was
the method of proof. Here was an abstract
geometry question, you see– general question. When you have a
general question, in math, it’s natural
to think it will be solved by general reasoning. Abstract question will be
solved by abstract proof. Not at all. He solved it by
concrete analysis, getting his dirty hands
into big calculations. And we’ll get back to that. It was to anticipate a
little bit the same amazement as 50 years later, when Russian
genius Grigori Perelman solved the most famous
Poincare conjecture, about all possible shapes of
the three-dimensional universe– a very general question–
by some very hands-on and technical calculations
and reasonings. Of Nash’s proof,
Gromov said, it’s “one of the first works which
made Riemannian geometry simple– an incredible
change of attitude on how to think of manifolds. You could manipulate them
with your bare hands.” This leads us to the
question, what is an analyst? We asked the question,
what is geometry? Now, what is an analyst? What is analysis? You know, mathematicians
is not a single species. You have several [INAUDIBLE]. You have several subspecies
and subsubspecies and so on. Analysis can be compared
to fine cuisine. In Japanese it’s the
same word, by the way. It needs fine tuning,
precise control. Analysts pride themselves on
the strength and sharpness of attack with simple
and powerful tools. We like to study,
in great detail, functions– signals– often
unknowns, because they are solutions of problems. And they ask how
fast they change. is it a small variation, a fast
variation, a big variation? Like stock exchange, will it
fluctuate very much, and so on? Et cetera. Here are some
examples of functions. Some of them are
smooth– slowly varying. Others are wild, and so on. Analysts spend
their life on this. Well, the main– the
first tool of analysis are the derivatives–
the differentials. This is the peak of variation. This is a slope. And inventors of
this famously were Newton and Leibniz, who were
both geniuses and engaged in a horrible
battle, for the shame of mathematical
community, in a sense. But it was brilliant
what they did. You know, look at
this graph, there. If I drew the tangent
to the graph, which is the line which
touches the graph, and I look at the
slope of this line, it will tell me
instantly what is the variation– if it’s growing
fast, or decreasing, and so on. And when I have
this slope, I can, for any value of the variable,
plot the value of the slope and then look at the
slope of the slope. Which is a variation
of the variation, the second
derivative, and so on. And I can continue. And the study of these
successive derivatives gives me strong information
about the way these functions change. This is what people
call the “regularity.” Derivative is easy
to understand. You know, 1% interest rate–
we understand this easily. Second derivative, not so easy. Starting from third
derivative, it’s really difficult to get it. Well, actually there is a famous
example of third derivative in political speech. This was Nixon, 1972,
when he announced publicly that the rate of
increase of inflation had started to decrease. [LAUGHTER] Which, as you may guess,
was not such great news. You know? And which, as you see, was
a good way to say things are improving in a way
that nobody can understand. [LAUGHTER] But for analysts,
it’s no problem. We’re used to this. We use arbitrary numbers of
derivatives– or dimensions, for that matter. Even fractional number of
derivatives is no problem. And the more
derivatives there are, the more the function is smooth. This is our daily
bread, and it can change the conclusion of a problem. For instance, if you’re
looking at a problem about fluid mechanics,
whether you’re looking for smooth solutions
or nonsmooth solutions, it will change not only
the mathematical attack but the physical conclusion
of the problem, and so on. And, as a true
analyst, Nash revealed that, in that geometry
problem, the regularity was very important. And, depending whether you’re
looking for smooth embedding or nonsmooth
embedding, the answer could be completely different. Geometers had no
idea about this. The proofs are incredible. For instance, to construct
his nonsmooth embedding, Nash started by grossly
reducing distances, you know, in a way that was
certainly not an embedding and then increasing them
back, progressively, by some progressive
process, by spiraling. Looked like crazy idea. And from the smooth
embedding, he attacked an incredibly
difficult system of equations– solutions, in
a sense, loses derivatives. That may not say
anything about this, except that it was
identified as a nightmare. And he understood that you
can counteract this nightmare by a numerical
method which had been devised by Newton to solve
equations in a very, very fast way. In one problem, it’s
a regularity issue. In the other, it’s like
a numerical problem– finding solution. But he understood he could
play one against the other. Nobody had a clue about this. The methods were founding. And the tools that he
introduced gave rise to new, powerful theories
which later would be taught in mathematical classes. Even without referring
to the geometry problems that they had been
introduced to. It was not only solving problems
but finding new techniques to solve these problems. And the conclusions were
powerful and amazing. Here is one of the things
that Gromov referred to as “impossible.” You may take a
sphere and crunch it, without altering its geometry–
its intrinsic geometry– neither getting bumps, you know? Not like taking a hammer
and– boc, boc, boc, boc. There will be no bumps–
still, it will be crunched. This is contradicting
our experience. We know in some sense
that the sphere is rigid. But that transform
defies experience, because it is hardly smooth. Here is how it looks like. This is the way that
Nash– or, rather, this is how, a few years ago, a
team of mathematicians in Lyon represented what Nash had
proved to exist– a way to embed the flat torus. What is a flat torus? It’s a geometry
that is well known to people of a certain age
who used to play Pac-Man. [LAUGHTER] You know Pac-Man–
it’s like a square, and there is some
labyrinth, whatever, and there is your
Pac-Man, your body. And it goes, goes–
and when it goes out in some direction it enters
back through the other side, you know. And when it goes this direction
and gets over the top, it comes back from the bottom. This is known to mathematicians
as the “flat torus” geometry, a geometry in which you identify
left side and right side and up with the down. And that geometry,
if you try to do it on a sheet of
paper– hmm-hmm, OK– let’s identify this with this. Easy. I do like this. OK. And I glue this. And indeed, if my little
guy will go there, it will enter the
other side, et cetera. But now you try to
glue this to that. Mm-hmm! [LAUGHTER] Does not work. And you can prove
it doesn’t work. Well, Nash
[INAUDIBLE], my friend, it doesn’t work if you’ll
go for a smooth embedding. But if you do it
in a clever way, and that is the
clever way, it works. So that thing,
here, which nowadays is called a “smooth
fractal,” looks like this. You see how it has tiny
structures and micro structures and micro micro
structures and so on. But still it’s not
irregular as a fractal. There is a derivative
at every place in there. And it is flat. If you are a tiny,
tiny, microscopic being, living on the
surface of this, you would not distinguish– you
will not feel that it is curved. For you, it will
be perfectly flat. It took years before
geometers digested these new things of Nash. And then came, two years later,
the great embedding theorem. If you take an
abstract geometry, you may embed it in a very
smooth manner, much smoother than this, but provided
you put enough dimensions. In this, you only
needed three dimensions. If you get it smooth, you cannot
make it in three dimensions, but you can do it in
a large-enough number of dimensions. This was great achievement. And there, it solved
a problem which had been asked
something like 80 years before– whether the point
of view of Gauss, embedded, and the point of
view of Riemann, intrinsic, were
actually equivalent. Answer? They are. That was good. That was good, and,
as a result of this, Nash had demonstrated
that he mastered regularity better than anyone. And that’s the reason
why Louis Nirenberg, when he saw Nash arriving in
Courant Institute, thought, that’s the guy to
solve my problem. It was for a problem of
regularity– regularity of partial
differential equations. What are partial
differential equations? Partial differential
equations are equations about derivatives– tangents. But you know, in real
life a function doesn’t depend on one variable. Depends on many variable. Take temperature, for instance. If you’re interested
in meteorology, temperature depends on
the time– you know, this morning it was quite cold,
today it’s better, and so on. It depends on the latitude and
the longitude and the altitude. Depends on four parameters. So you may compute the
derivative with respect to any of these parameters. Is it getting warmer
by the minute, or colder by the minute? Is it getting warmer when I
get to the south– et cetera. Partial derivatives capture
these, these tendencies with respect to the parameters. And this is a big
discovery of people, starting from the 18th century,
that almost any phenomenon you can think of
eventually is modeled by these partial differential
equations– related to tendencies of the functions. For instance,
temperature in this room is a problem of partial
differential equations. Electric potential in our brain? It’s a problem of partial
differential equations. Whatever. You know? Control, understanding
the motion of fluids? It’s partial differential
equations, et cetera. Here are some of the most famous
partial differential equations. I don’t want you
to understand them, but first you may appreciate
how beautiful they are. [LAUGHTER] This here is Boltzmann equation. Gosh– I spent 10 years
of my life on this! [LAUGHTER] This describes the
evolution of a gas. Was first devised by James Clark
Maxwell and Ludwig Boltzmann and changed the face
of theoretical physics. This here is the
Vlasov equations. Tells us about the evolution
of a galaxy, for instance, over billions of years. This one, on the
contrary, is so, so small. Schrodinger equation, the
basis of quantum mechanics. This one, actually,
when you go to Paris you may visit a public place
in which it is engraved. It’s in the Paris subway. The sculpture, in station
Chatelet-Les Halles– I like it, because every day
probably hundreds of thousands of people pass near this
without noticing whatsoever that there is the Schrodinger
equation on there. [LAUGHTER] You know? Like a metaphor of the
fact that we are surrounded by these marvelous
equations around us without us noticing this. Here is some other couple ones. These are the equations
of fluid mechanics. Euler equation,
Navier-Stokes equations. They changed everything in our
technology, or many things. They are solved every day
to predict the weather. They are used at enormous
length by the Hollywood industry to make all kinds of special
effects in the movie theaters. Go to see Titanic
or whatever– it’s full of resolutions of
these partial differential equations of fluid mechanics. OK. And here’s another
one, which was solved by other mathematical
genius, Alan Turing, to understand the problems
of pattern formation on the skins of animals. And here is another one, which
was used by Joseph Fourier, in the 19th century,
to understand the evolution of temperature–
say, in a block of iron. Here it is. Heat equation. It’s a very famous partial
differential equations and, when you take courses
in partial differential equations, one of the
first that you study. It’s about temperature in a
block of metal, conducting. On the left is the
time derivative– the tendency of the temperature
with respect to time. Is it getting warmer, or colder? And on the right, there
is the space derivative– and, actually, two
space derivatives. The tendency of the
tendency, in respect to the spatial variable. And there is a coefficiency. It’s the conductivity. Or, to be more rigorous,
thermal diffusivity. They are in relation
of each other. Because, you know,
some materials– in some materials, the
heat is easy to transmit, in some others
difficult to transmit. OK. When you have a
conductivity that changes from place to place– say,
in a mixture of two metals– the distribution of heat can
become quite complicated. The equation is more complicated
than the solution, though. You cannot compute it exactly,
then, but you can study it. Let me show you some examples. Here’s an example
in which you have a heat distribution in a metal
bar– hot, cold, hot, cold. You see this is x. This is, like, the
distance to the origin. And initially you heat
some places and see how it evolves with time. You let it cool down. Look carefully. This is the evolution of
temperature, as time goes. At the beginning, I
had several bumps. Now I only have one. It’s getting a bit boring,
because the evolution at first was very first and
now it’s very slow. That’s one of the first
things that we learn when we study heat equation. Starts first and
then becomes slow. Other things that we learn is
that, even if globally it’s cooling down, some places are
cooling down– others are not, initially. Look at this hot spot
and this cold spot. After just a moment, the
hot spot is becoming colder, but the cold spot
is becoming warmer– until there is some
kind of equilibration, and then they start
to get decreasing. OK. Another thing that we
learn in these courses is that heat equation
regularizes things. It makes you smoother. Look here. Is it smooth? Well, not much. You see these strong
variations, wild variations, of the slope–
changes completely. But look after a
fraction of a second. It has a smooth. You know, as a
mountain gets eroded, heat equation does some
kind of erosion on the data. This is the regularizing
effect of the heat equation. OK. This simulation, here, is for
a constant– the homogeneous metal bar. Let’s now look at another
one, in which it would be a mixture of various metals. And let’s also start with
initial data of temperature that is crazy– you know,
hot, cold, hot, cold– completely crazy. Look what happens
after a few seconds. Ah, not so smooth, but better. And, you know, here it
was quite discontinuous. But here, continuous. And better and better. We see it’s not as
regularized as it was with the homogeneous metal
bar, but still not so bad. So you have some
regularization effect. At least, that was
the conjecture. And that is exactly
what Nirenberg wanted Nash to prove with mathematics. Take any alloy– you
know, any mixture of metals– in any
geometry, in any dimensions, and any distribution
of heat, initially. And let it act
for a few seconds. Will it become smooth? It may seem like a
very specific problem, but this was an
important problem because it was a key problem
to understand a whole class of related problems, you know? And so Nirenberg explained
this to Nash, and Nash– ha! How to do this? OK. This is what I was
just explaining to you. And this is the
mathematical statement. So, if temperature of time
and x is a heat distribution in a medium with discontinuity
conductivity– you know, any mixture– and a
discontinuous initial distribution of temperature,
after one second, will the temperature
be continuous? This is the problem. And Nash is very
interested– also check that he can
became famous for this. [LAUGHTER] And starts working on this. Starts working on
this, gets started, and works on the problem. It’s fascinating, because we
have accounts and testimonies on how he worked on it. Going and meeting
the people, you know, betting back to
Nirenberg– tell me more about it! I want to know this and this. And is it true that
this and that and that? Nash was not a specialist
at all of that– of mathematical physics–
diffusion equation. Had no idea. And then he went to see other
people– people in Princeton, people in New York. Hey, I heard that you
are specialist of this. Can you explain me
this, that, and that? And at first his questions
were quite stupid, you know? Like, he was an outsider,
not knowing about this. And Nirenberg was
starting to wonder, hey, is this guy as
smart as they said? And, little by little, questions
were becoming more and more to the point. And he was putting everybody
through contribution as a conductor, you know? Hey, my friend, I need you
to prove me this and this. I think that you are the expert,
and you can give me this. I can use it to prove
something more– and so on. As a conductor who would
give assignments– you know, here, you’re the violin player–
will play this and this. You are the trumpet. You will play this and this. Each one does their part. Nobody understands
the great plan, except when the
orchestra starts to play. And Nash had the
overall plan for this. And everybody was amazed
when, after six months, the problem was solved. Putting all people
to contributions. And, again, the
solution was amazing. Let’s examine this
famous paper that he wrote from there– one
of the most famous papers in the 20th century– partial
differential equations. “Continuity of Solutions
of Parabolic and Elliptic Equations,” by John Nash. And now you understand
what it was about. 1958, 24 pages– rather short. By current standards,
very short. It’s interesting to read
what he thought about this. “The open problems in the
area of nonlinear partial differential equations are very
relevant to applied mathematics and science as a
whole, perhaps more so than the open problems in any
other area of mathematics, and this field seems poised
for rapid development. It seems clear, however, that
fresh methods must be employed. We hope this paper contributes
significantly in this way and also that the new methods
used in our previous paper will be of value.” That was absolutely true. “Little is known about
the existence, uniqueness, and smoothness of solutions of
the general equations of flow of a viscous, compressible,
and heat-conducting fluid.” It’s still true. Almost 50 years
after what he wrote, we are still in the dark about
some basic features about this. The style was informal and
informative– very interesting. You know, he doesn’t suggest
this and that and that. He also says about key results. Also tells you what he
had to work hard on. And he also says things
like “This is dimensionally the only possible form for
a bound”– reasoning a bit like a physicist. He speaks of “dynamic
inequalities.” What does it mean, a
“dynamic inequality”? Well, you know, it’s to
convey some impression. It’s precious. Or “powerful
inequality”– et cetera. “The methods here were
used by physical intuition, but the ritual of
mathematical expression tends to hide this
natural basis.” He doesn’t want just people
to see right through. He wants them to understand
how he got into this. Very generously, actually. The notions were unexpected. He was thinking of the solution
of the heat equation in terms of statistical mechanics,
with temperature being like density of matter. As if there was such a
thing as atoms of heat. Doesn’t exist, the atom of heat. But let’s think
like it’s this way. And he used the entropy
solution of disorder used by Boltzmann,
and later by Shannon, to measure the disorder
or microscopic uncertainty of that temperature. From physical point
of view, this quantity minus integral of T
log T makes no sense. It would make sense, you
know, if T was, like, a density of matter. But there’s no such
thing, as I said, as matter– as a particular
associated to temperature. But it works. In a radically unusual context. And he used this
notion in a context which is very different from
the one it was introduced for. And he demonstrated,
with his proof, the power of differential
inequalities, that these inequalities between
slopes of various quantities, involving simple quantities
containing information, in some sense. It was, a bit, the same
amazement as the one that Grigori Perelman, again, would
generate when he showed that, to solve the Poincare
problem of geometry, it was useful to
introduce some entropy. . This was a masterpiece
in several parts. Let me not go into details
but very rapidly say that it was like first act
was about understanding displacement of what
would be something like an atomic sort of heat. Like a Brownian motion, like
a particle of temperature. And, through them,
the temperature was neither too
low nor too strong. That contributions of
sources of temperature would overlap, in some sense. That, if two point
sources are closed, then the resulting
heat distributions are closed too, and
the decontinuity. And each of these steps has
precise mathematical statement. You know, in the
ideal of mathematics the Holy Grail is to have
a nice, beautiful proof. And we get this concept as a
legacy of the Ancient Greeks, and geometry in particular. This implies this
from that, et cetera. They like playing with this,
but the– like, the bricks are not triangles and lines. These are, like,
qualitative properties of the heat distribution. Let me skip this,
even though these were some of the inequalities. And let’s focus on
one thing, here. You see, what is written
here is Nash’s inequality, in the middle of this slide. Everybody in analysis knows
this is Nash’s inequality. Truth is– and
it’s clear, if you read the paper by Nash– Nash
did not prove this inequality. He asked one of his
colleagues, named Stein, to prove the inequality. Stein was an expert in
this kind of things. You want this inequality? Yeah. Let me prove it for you. Here is how you do it. Thank you. And Nash showed how to use
it in that problem of heat distribution. He was genius in this kind of
integrating the various parts. And of on. Let’s skip this– also this. He got from another
guy, actually, this one, from his colleague
Carlson, who introduced him to the concept of entropy
after he had studied that from his colleague [INAUDIBLE],
who had been specialist of Boltzmann equation. That was the style of Nash,
taking this idea here, that idea there. Mmm– I think they are linked. I will use them. And so on. This was brilliant. [INAUDIBLE] Hollywood
and movie, after that would be the big celebration. I don’t know– Nash would have
married a beautiful heroine or whatever. It was not this way. The celebration was lost. In 1957, Nash heard that
a young, unknown Italian mathematician, Ennio de
Giorgi, proved the same result by a different method. To all mathematicians, this
would become as the, you know, schoolbook example,
textbook case, of simultaneous discovery–
by different people, at the same time, with
different methods. Di Giorgi was completely
unknown at that time. He was eccentric, he was
monachal, he was genius. He would become a living legend. His proofs set the standard
for generations of experts. Nash, in spite of
being so bright, was in pathological
need of recognition and attributed to
this coincident his failure to get
the 1958 Fields Medal, which went to French
mathematician Rene Thom. The epilogue was sinister. Nash’s paper was accepted in
Acta Mathematica, arguably the best journal in the
world, in those days– maybe nowadays, also. And after it was
accepted he withdrew it. Which is unheard of,
you know, if your paper is accepted in such a
journal, to withdraw it. And he sent it to the American
Journal of Mathematics– maybe the most famous paper in the
history of American Journal of Mathematics– in the hope
of getting the Bocher prize, 1959, a prize which had to go
to a paper published in there. OK. In vain. It was Nirenberg who
got the prize that year! For some other work. Hmm! Nash actually is already
undergoing paranoid delirium in there– the start
of a long tragedy and more than two decades
of going mental hospital, at times, having some relapse
and some clear thoughts, at other times being
completely miserable– haunting the corridors of Princeton,
talking nonsense, whatever. Meanwhile his papers will
make their revolution. To Louis Nirenberg, in
1980, somebody asked, do you know mathematicians
that you can consider geniuses? And Nirenberg answered
“I can think only of one, and that’s John Nash.” In 2011, to a young, brilliant
Princeton mathematician, John Pardon, somebody
in an interview asked, who is your favorite
Princetonian, living or dead, and he answered
“Probably John Nash.” In 1994 as he had been out
of his mental condition, Nash was awarded a Nobel prize
in Economics for his PhD, for that two-page paper in which
he defined the Nash equilibria. Here is the situation. OK, you know it’s not
really the Nobel prize. It’s the Sveriges Riksbank
Prize in Economic Sciences in memory of Alfred Nobel,
awarded jointly to– et cetera, et cetera– including John Nash
“for their pioneering analysis of equilibria in the theory
of noncooperative games.” Looked like Nash’s genius
had been recognized at last. 2001, it was the
movie Beautiful Mind. So this is Russell Crowe,
playing the role of Nash. I’m not sure he understands
anything of what’s written on the blackboard. [LAUGHTER] I can recognize these
are the equations of isometric embedding. The film beautifully manages
to massacre everything in the life of Nash. [LAUGHTER] The chronology’s wrong, the
nature of the mental condition is wrong, the nature of the
science contributions is wrong. It’s amazing how
the film manages to get it wrong on all points. Brilliant. Nash hated every single
bit of the movie. But interestingly, his
wife– his wife Alicia– found it was lovely to be played
onscreen by Hollywood beauty Jennifer Connelly. [LAUGHTER] Now, for
mathematicians, it still remained as a problem, you
know, that Nash’s genius had been recognized in
economics but not for his truly beautiful
mathematical work related to the geometry and analysis. And little by little came
this public recognition. 2009 was a big splash, when
Nash’s ideas led Camillo De Lellis and Laszlo Szekelyhidi,
bright young mathematicians from Italy and
Hungary, respectively, to construct some impossible
solutions of Euler’s equation– crazy solutions. Imagine a fluid that would
be at rest initially, then start to
agitate like crazy, then be at rest again, without
any force acting on it. Something that made us
rethink the definition of what is a solution to
a fluid equation. In 2012, I depicted my
emotional encounter with Nash in “Birth of a Theorem.” This was, like, autobiographical
book, or at least talking about how it is to
prove a theorem. All the ups and downs,
difficulties, mistakes, traveling, meeting
with people– whatever. And one chapter is
devoted to my encounter with Nash in Princeton,
at the T. And I was so, you know– he represented
so much to me. I was so, how to say,
impressed that I did not even dare to speak with him
on the first encounter. And 2015, Nash at last was
awarded the Abel Prize. Abel Prize is, together
with Fields Medal, for sure the most
prestigious prize. Abel Prize is certainly
more difficult to get than Fields Medal. Abel Prize is younger–
started, like, 15 years ago. Abel Prize typically
goes to the living, old legends in their 70s
who made contributions that everybody knows
and is amazed of. And I sat on the committee on
the Abel Prize, in those days. It was very emotional
for me, also, as being on that
committee the person who was scientifically
closest to Nash– you know, the one in charge
of defending the work of Nash. I will not say more
about the discussions, because, of course, it’s secret. But this was very interesting
discussions we had about this. 19 May 2015, in
Oslo, the Abel Prize was awarded jointly to John
Nash and Louis Nirenberg– you know, these guys, they
have a long history together– for striking and similar
contributions to the theory of nonlinear
partial differential equations and its applications
to geometric analysis. Very emotional moment. You have to imagine the
ceremony, also– King of Norway was there, Nash gave
speech, and so on. There was public speech of Nash,
in which Nash recollected some of his older work,
some other work that he discussed with Einstein,
related to general relativity. I had the honor to
be his chairman. This was our fourth
encounter, actually. And for all the community,
everybody thought, at last– Nash got this reward. And it was quite due. So it was the mood–
after this prize, he will at least– he
can relax– recognize, money, whatever. Not so. 23 May 2015, back
from the ceremony, en route to Princeton, he
died, together with his wife, in a taxi crash. We met again with
Nirenberg, later on, trying to make sense
of it, whatever. Nirenberg [INAUDIBLE] Nash
ever did anything like anybody. Extra-terrestrian. Always gives these proofs that
no one can understand at first. He himself was unable to
explain them– always saw things that people think is impossible. Gets Nobel Prize
for his PhD work, after having becoming
insane from an illness that you’re supposed
never to recover. And, in the end, dies
like nobody else. Anyway, this is tragic. But the beauty of it, of
course, is the methods. And many mathematicians
feel that Nash is part of their family, in
the sense that he brought so many ideas and techniques. If I think of my own
work, I can see clearly relations with Nash’s work. The taste
statistical-mechanics problems, the taste for
entropy, the key role of regularity– which I
uncovered with my collaborators in similar problems. We, for instance,
in the problem that was one of my– our–
most noticed papers, about behavior of
plasmas– we uncovered how critical the role of
regularity was, in a way that had not been understood
before by physicists or mathematicians and this. And, also like Nash,
I admire his talent take pleasure in this sharp,
massive attack of given simple problem, in some
sense– with simple and well-calibrating
tools, always trying to uncovering new connections. This idea, that idea, let’s
put it together, and so on. And so, in a way, the Nash
legacy still lives on. And I will conclude with this. Putting on, flashing,
just this bibliography, recommending to you, by
the way, the Beautiful Mind book by Nasar– well,
especially chapters 20, 30, 31, in which you will find
the stories behind the scenes for the mathematical
achievements– which, for us, is even much more
important than the mental in the story. Nash wrote four big
papers, you know. That’s nothing, compared
to a big list of papers. But, of these four
papers, three, probably, in retrospect, would
have deserved the Fields Medal. On my web page, on my blog,
on date of 26 December 2015, you will also find, in
French, a short article– or not so short– in
memoriam John Nash, entitled “Breve rencontre”–
“Brief Encounter.” And, with this, I will
conclude this discourse. Thank you. [APPLAUSE]

Comments 100

Leave a Reply

Your email address will not be published. Required fields are marked *