[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]

Listen to him, his looks or attire are unimportant! He is actually quite brilliant!

intentionally not showing the transformations = your a git!!

amazing speech. period.

John Nash was born in Bluefield, West Virginia

comme disent les italiens Monsieur Villani grazie di esistere

So lovely Accent ! and Theme

Thieving shtable boy

I can't do it

A beautiful lecture, communicating with kindness and humility, so much of the spirit of mathematics

Great presentation.

This was great! So fascinating and his accent is killer.

I would have loved to watch this but the speakers accent is way too much.

I am not kidding, the speaker sounds a bit too much like :

https://www.youtube.com/watch?v=BqnFnWSYzaQ

Spelling error in "Unexpected Notions"……an example of partial differential precision.

at first, i thought this guy is a freak, but then i started to listen and he's in fact good in talking. i am not a mathematician, so i can not judge on whether he's making any point, really, but it was nice to listen 🙂

Why are the slides so full of broken patches?

I was here for Nash and got overdose of geometry instead

Plano= 180° , Esfera >180° (earth goes here), Hyperbólica <180°

x*(v*a)^n. where x=1. x=v. x=a. x=(a*v)=1=Q. Q*v=I. Q*a=O. Q*a*v=m. P=ion=Element=compound etc… to urban growth decay rejuvenation to city or country life from mathematics expanding polynomials also showing m*a*v theory faster than light superimpose pulsed on constant Q +- time. magnetic ns astral planes

1956: Young Nash arrives in the Big Apple as a budgie-smuggler.

Win the Fields and you can be as eccentric as you want thereafter.

Both mathematics and this lecture are a piece of art.

The Fields Medal Winner is harder than Nobel Prize of Mathematics (awarded only every 4 years). Congratulations, Monsieur Villani.

His head voice when he begins a sentence

I like how @ 41:49 he mentions conductor because a few years ago I was watching a completely unrelated video to math about a classical music awards ceremony in France and a pianist that I admire (Cyprien Katsaris) was getting an award. Well I spotted Dr. Villani in the the crowd. Of course I emailed the video to him for confirmation. He confirmed and laughed because I recognized him.

Dr. Villani used to be a serious piano student when he was younger.

Why did they stop showing his slides around 37 minutes? You can't even see what he's trying to demonstrate!

I just love this guy!

WATCH OUT! WATCH OUT! VILLANI'S GOT A ~SPODER~ ON HIS SUIT.

I bet "the spider man" was bullied at school lol.

This Oscar Wilde knows more about Nash than I do.

… It#s Catweazel!

OMG, it’s so weird. We move in 3 dimensions witch is much more space then 2 dimensions. But we think of it as 2 dimensions! We are the ants on the basketball 🏀. That mathematical thinking made me realize how Physics work. Damn, why did I study law instead of it. Is it too late? Where do I start?

This talk was amazing. The first 12 minutes make you realize by yourself the following minutes of the video and Cédric keeps up giving us more and more. Amazing explanation. Glad I could watch it on Youtube.

From 37:30 – 39:00 the punchline is missing.

thanks nash we have subtitles. sounds just too harsh and annoying for a female voice

37:00 why on earth would the camera be focused on him when he describes an EXAMPLE !!!!!! I WANT TO SEE WHAT HE IS TALKING ABOUT GOD DAM IT

So which is the highest award a mathematician can rcv? The Abel or, 6 sentences later, the Fields? How hard IS IT to have some level of consistency? They probably have 2 different values for Pi also.

well, the style of every person can be from their way of thinking, who cares, just pay attention what he is saying…Jhon Nash and Cedric Villani are brilliant minds

at least the are making history, they are not using their time judge others, instead they are very smart with their time….

37:40, who had the idea to show Cédric clicking instead of the heat equation in action?.. What a tease.

This boi in the phantom troupe

Evidence a people in history possessed knowledge isn't evidence they came up with it.

this guy's voice ruined my life

This a great presentation. Unfortunately, photographer forget to switch to the slides when Cédric demonstrates a crucial example. What a putty.

I like this guy.

5:10 – TRIGGER WARNING

I really wanted this guy to do magic during this lecture…

He is brilliant

37:04 the cameran now expects us to picture the graph of the temp on his face. Nice cinematography… Very creative.

Besides outlandish approach to mathematical formulation and interpretation, Nash seems to have learned the essential structure of Ramanujan's Theta Function and Tau Function. But he seems to have had a unique approach to view problems.

This is a man who knows how to dress. We have much to learn from him.

405 comments with this one, only 12 about his spider, took me 20 minutes to notice, 5 min before that i was wondering about his cool outfit, so far rly great talk

I'd like a room, please..A rume?, yes a roooome!

Holy shit. This man is genius he said Palestine

"What are you wearing tonight?"

"Oh I don't know, probably a tux with a big-ass spider as a measure of good taste"

Always great to have physics explained by a James Bond villain

37:30 It'd have been nice to see the slides he is discussing in the video.

Cédric is a national treasure ! Brillant exposé

Fantastic lecture. No qdos for no animation however..

I don't know enough about the world of mathematicians to know who this gentleman is (to my loss, I'm sure), but I like him. He seems to have a nice since of humor, is earnest & engaging, & is passionate about math (always a good thing!) I had no problem understanding his French accent – it was quite pleasant, in fact. I'm not bothered by showing equations on the screen – apparently unlike most people [I don't REALLY believe that it's "death" to show equations in a book or at a talk, anyway.] So, I very much enjoyed his talk, and would like to know more about John Nash. I would like to see more talks by Mssr. Villani. tavi.

God I wish this guy had more lectures. His voice, to a middle-of-the-road american English speaker such as myself, is so very well adapted to getting both the technical AND aesthetic nuances across, to me at least, that I could listen all day to stuff I don't even begin to understand, and still enjoy it immensely… 🙂

also in den ersten 10 minuten redet er über flache erde und gibt ne sehr kurze stichpunktartige biographie wider. für mich is das ne zeitverschwendung. da vergeht mir die lust das anzusehen. kommt mir vor wie einer dieser verkäufer auf kaffeefahrten.

39:24 his desktop is amazing 🙂

If Floki was a mathematician.

Great lecture overall but sorry you lost me when you said that the Greeks started geometry. There's an intellectual wall around Europe and its derivatives that only a few can escape out of.

His voice is simply impossible …

You could say Nash's work was kind of … derivative.

Very interesting, entertaining and inspiring. The outfit worn by the speaker is really peculiar as well.

This is stupid.

Not sure if that's supposed to be a tie.

What happened to the slides at 37 mins?

Does he live in a castle?

Clicked for the outfit, stayed for the accent.

6:20 <3

I understood it perfectly well up to the opening applause

What's with the spider

Hyperbolic crochet is my new reason to live.

A man who has a large spider pin always has something interesting to say.

Cédric is a brilliant communicator.

Listening to this man makes me feel tiny like a sand corn in the universe!

Very interesting lecture – but really poor camera work and/or editing around 37 minutes –

veryfrustrating.He has difficulty speaking English

"the methods used here were inspired by intuition, but the Ritual of mathematical exposition tends to hide this natural basis", exactly, haha…you'd think they started defining things and "setting it all up"…there's something much more dry about the finished product than, I would guess, the process used to attain it…

He's not speaking English…he's speaking French but with English words.

I so needed to see that temp. evolution in a metal bar…. I machine them every day🙈

Fan-boy with spider broach?

French accent sounds like a mouse squeaking

Do you need some mathematical justification to introduce certain notions to your mathematical model? Why was it possible for Nash to view temperature as 'particles'? Do you need rigour to introduce such concepts?

Excellent video! That truly unmasks John Nash. What an evil man!

THIS IS NOT A WIZARD HARRY

Biri türkçe altyazı yapaydı eyiydi

I really didn't know how mathematics have fun if this man teaches.i hate math.This guy is a great speaker…. i am not a mathematician yet i find this lecture so interesting….

so french.

This guy looks like if John Wick had a PHD in mathematics

why does one freak always draw more freaks of rising levels of freakism? This guy is a moron.

Hi, I have schizophrenia, thanx for the video.

I learned basic complex numbers before my illness began.

Now, I dream of quaternions

!to Z and at the point Kif you go without medication, then it is an override !

Sadly the cameras are not well positioned nor utilized appropriately to represent the speaker's specific examples for non-attendees. Maybe a function of style fascination vs a vs in depth understanding.

I wanted to see how the Heat Equation works. Cameraman showed him pressing button repeatedly. Where is the animation?

Great lecture and storytelling!

What century is he from? This guy is so creepy I couldn't pay attention to what he was saying.Mathematics is a form of French poetry.

the strong french accent is buggering me.

You talk about so called 'arabic' mathematicians but in fact you ignore there where no one arab among them : only Kurds, Assyrians, Berbers, Perses, Jews, etc.