So today I want to talk about John Nash
and John Nash is slightly unusual as a mathematician
because he's a mathematician
that is better known
to the public in general ; mathematicians
don't tend to be known in the public eye.
He was the subject of a film called
A Beautiful Mind, the film had Russell Crowe
in it starring as John Nash.
There's also a book a beautiful mind
that it was based on by Sylvia Nasar.
I enjoyed the book more than the film.
The film is a little bit fictionalized.
I have to confess I haven't seen the film yet,
I know I'm such a big film fan
I'm such a massive film nerd.
I imagine it's absolutely accurate
in all respects
because that's how Hollywood works.
You may have heard of him because of his
work in economics he won a Nobel Prize
for economics and that was for game
theory so he analyzed what are called
non zero sum games and he's known as
like a father of game theory one of the
pioneers of game theory.
For a pure mathematician that was quite a simple
application of those things but the
really big deal was that he brought
those mathematical ideas into economics.
But he's really well-known among
mathematicians as well for some
serious pure math work that he did in a
subject called "Differential geometry".
Mathematicians do think that
actually his better work is his work in
geometry and partial differential
equations and so it was that work which
was then finally recognized when he got
the Abel Prize in 2015.
Well the mathematician Mikhail Gromov
said that he thought that Nash's work in
differential geometry and on elliptic
partial differential equations was
orders of magnitude more important than
his work in game theory but of course
that's the point of view of a pure mathematician.
So I want to talk about
why he won the Abel prize for his work in geometry.
So if we cut off a bit of
square brown paper - and this is something
that is possibly well-known to some of you -
but this square of brown
paper, another way of thinking about this
square, if we add some rules, it's called a
flat torus. What are the rules ?
The rules are : if I'm traveling along this
paper here and I go off the right-hand side like that
I then reappear on the
left-hand side. If instead I was
traveling up the paper here and go off
the top, then I will reappear at
the bottom of the square piece of paper
(Brady) - It's like a game of asteroids.
It's exactly like that, it's like a game of asteroids. The old computer games
used to get, particularly the game of
asteroids, where you'd go off the edge of
the universe and come out the other side.
So this is a flat torus it's called.
Now there's something you can do with this
you can embed this - they call this -
you can embed this in three dimensions.
This is two dimensions, this is a two-dimensional flat surface
but we can embed it in three dimensions by doing this.
So if I go off the right hand side
and reappear on the left then the
right and the left is the same
- we're calling that the same -
so let's glue them together and we make a cylinder.
Can you see me there ? So we make a tube.
In the same way if I go off the top of this torus
I reappear at the bottom so the
top and the bottom are the same we can
join them together which is going to be
difficult to do with this piece of paper.
If we join them together they would make
a doughnut shape like a classic
Homer Simpson style doughnut. It is going to be
difficult to do with this piece of
paper to show you then I've got
something I can use.
We have the torus shape and for my
next trick here's a poodle so there's
our torus shape and that's what the
shape you want to make so if you play in
the game of asteroids this is actually
the universe that you're living on
you're actually living on the surface of
this universe.
If in my universe I travel
let's say from top to bottom like that
or I compare that with traveling from
left to right like that
then if I wrap this up...
That's not quite perfect there - so if I wrap this up
this is me going round my cylinder
universe and if I can join the ends
of this tube I would make my torus shape.
And on the actual torus itself the green
line would be me traveling around like
this through the hole and then back out
here joining up like that the red line
however would be traveling around this
universe like this.
The squeaks are extra.
So what I've done is I've turned this flat two-dimensional
shape into sit in three dimensions.
So this is occupying a three dimensional space.
There is one problem with this.
You see in my square torus
this red line and this green line
they're the same length, right, because
it's square they're the same length.
When I've stretched it into this shape the
green line and the red line are no
longer the same length so there's the
green line going around you see this red
line is much longer now so I had to
stretch it to stretch around the tube.
Is there a way to embed a square torus
like that into our 3-dimensional world
- maybe it will be a different shape, it
might not look like this but can we do it
in such a way so that we preserve
distances so these lengths don't get
stretched and they all get the same
distances ? That's the question and it
looked like you couldn't do that.
And why can't you do that ? Because this is
flat and the curvature
- let's talk about curvature - it's a flat
object, it has zero curvature.
Something like this is a curved object, it has
"positive curvature" we call it.
So it looks like it's impossible, like
we can't embed something because it
would have a positive curvature and we
want it to keep its flat curvature.
Problem. John Nash found a way to do it.
Maybe he's slightly breaking the rules,
but he found a way to do it.
And the method he found was, he was able to embed
the object and deform the object in such
a way that some points curvature had no meaning.
Here's an example of the sort of
thing I mean. So imagine you're traveling
along the motorway at 100 miles per hour
You deaccelerate and then you
continue your journey at 30 miles per hour.
So, that might be something that could happen.
So this is speed, this is your speed
against time.
Imagine if I drew that graph in a slightly
different way, very similar graph but I'm
going to draw it slightly differently.
I still want the speed up here,
This is me traveling at 100 miles per hour
and the end result is
I'm going to be ending on 30 miles per hour
and then I'm going to join up these.
So they're very similar graphs.
There is a difference between them.
If we look at our acceleration,
here there's zero acceleration
you're travelling at a constant
speed 100 miles per hour
there's zero acceleration or the
gradient of the graph is zero.
Here, the acceleration might be let's call it
minus 10 and then here it's it's zero again
You're at a constant speed.
Now with that graph at this point here and
at this point here acceleration isn't defined,
acceleration has no meaning here.
What you've done - you've gone from zero
acceleration suddenly, at an instant
to minus ten and then suddenly again to
zero
And at that point here and at
that point there there is no meaning to
what the acceleration is.
You've gone from 0 to -10 to 0.
Now speed is defined, we know what the speed is here
that's no problem, the speed is
100 miles per hour there and 30 miles
per hour there. No problem with the speed
but the acceleration isn't defined.
With this curve - very similar
that's not a problem here :
acceleration is defined for every point on this curve.
(Brady) - This one's the real world one isn't it ?
- Yeah exactly that's real world , I agree
And so this is maybe... you're taking some
more abstract look at what you're doing.
It works the same in geometry :
if you have something like this then
curvature is defined everywhere here.
If we instead made that sort of shape
like a hook, if you made that
shape with a straight edge followed by a
semicircle, like that, then here the
curvature is flat flat flat flat
suddenly here it's positive and at this
point where you join on, curvature has no meaning
There's no definition for curvature there.
What Nash was able to do was :
he was able to find a way to embed
your square flat torus into the 3d world
and there are some points like that
where the curvature isn't defined.
You can take your flat torus and then you
can embed it in the 3d world and you
embed it in such a way so maybe
something like that you can embed it in
such a way that the distances are a
bit short so that your map,
your deformation, your distances
have become a bit shortened.
All the distances have become a bit shortened.
Then we're going to
lengthen a little bit, we're going to hit
this object with waves
this is going to ripple across the surface of
this object, waves, and those waves
will ripple across - they're called corrugations -
and they will lengthen all those
distances slightly. And you hit it with
another set of waves, those waves are a
bit smaller, so smaller ripples.
But then we're getting closer to the
lengths again we're getting a bit closer
to what the correct lengths are.
And then you hit them with another set of waves
and we get a little bit closer again to
what the correct distances should be and
you keep doing that and then you get an
object which is full of these
corrugations but all the lengths are the
correct lengths as they were in the
original flat torus.
(Brady) - Are you saying by
making the surface bumpy this green line
has to travel further closer to what the
red line is traveling ?
- Yes a shortened distance like this green line
would get lengthened so it becomes
the same as this red line around the
outside of the torus so they become the
same length again like they where in the
original flat torus, like they should be.
What Nash was doing, he was studying
this kind of thing and he was actually
doing it more abstractly more general
terms and he was do it in higher
dimensions so much more abstract
mathematics but this would be an
application of what Nash was doing,
this work in geometry.
When he was doing this work,
his solutions involved these
things called partial differential equations.
That's about the rate of change of things :
the rate of change of curvature
the rate of change of speed...
Using these equations... His methods for solving
these equations were so innovative that
not only are they useful in geometry
they turned out to be useful
in the topic of partial differential
equations itself as its own subject.
That is why Nash got the Abel Prize for
his work in geometry and partial
differential equations.
That's right of course his life story is
interesting to all of us because of his
of his mental illness and the fact the
remarkable fact that he recovered from it
to become an active mathematician again
but really his mathematics is
what's going to last.