Ricci flow is a mathematical tool, analogous to curve shortening and mean curvature flows, used to deform geometric shapes by evolving their curvature. Its significance was amplified by Grigori Perelman's proof of the Poincaré and Geometrization conjectures, which involved understanding and managing singularities that arise during the flow.
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Ricci flow when it was invented by
Hamilton in the early 80s nobody would
have paid much attention to it except
mathematicians. What happened is 20 years
later, you know, Hamilton and others,
including me, were working on this and
trying to prove this so-called Poincaré
conjecture and also something a bit
more inclusive called the
Geometrization conjecture. It was going okay but
there was some big roadblocks. What got
people's attention is in in the year
2002 this fellow called Perelman, who is
famous as much for his eccentricities as
as his math, was able to actually prove
these important theorems using the Ricci flow.
It's hard to describe exactly how
Ricci flow works, so I thought what I'd do
is talk about something which is very
closely related to Ricci flow. We'll
start with something called curve
shortening flow. Think about the two
dimensional space and let's just draw a
smooth path, closed path, on that surface
and what we want what we want to do is
see if we can move it, deform it, change
it into a new path. Let's take this this point
here and I'm going to draw the tangent
line here; and so we're going to move it
in a direction which is perpendicular to
that, so either this way or that way. And
remember this is happening at each point
along this path, the amount you
move is is proportional to how much
curvature you have. Curvature is sort of-
think of it as how tightly- you know, if
you were driving along there how much
how much you'd have to be turning your
wheel on the car, it's also sort of the
circle that's closest to matching it
there. Here there's a small circle, here
there's a very large circle and the
curvature is sort of 1 over the radius
of that circle. Here the circle is
very small and so the curvature is very
large so the amount- the speed that this
point is going to move is going to be
very fast and it's going to be moving in
this direction quite fast. Here, because
the curvature is much smaller you're
going to be moving in this direction but
it's going to be only a little bit. So if
you think about this
path, after a certain amount of time - and
this time is artificial it's not real
time it's just something you're putting
on artificially on this - this this point
will move just a little bit in there,
this point will move quite a bit more.
Here it's going to basically not- so the
circle is going to be huge, here it's
going to basically not move at all. Qhen
the curvature is sort of going backwards
so the circle is not inside but outside
then you're going to be moving in that
direction. - (Brady: What's happening? Is this is)
(this thing shrinking or is this thing)
(expanding? When this thing flows what's)
(going to happen to it?) - Well it's going to-
actually if you take a- if you start with
a big circle and you move it according
to the so called curve shortening flow
it's going to be moving uniformly in at
each point; the same amount because the
curvature - lousy drawing - but the
curvature's supposed to be around the
same. So you're going to get: this is what
it is at time T is equal to now, at time
T is equal to say five seconds later
it'll be smaller like that and then it's
just going to get smaller and smaller.
But sometimes, because we don't like to
have these things run away from us
we'll change the flow a little bit and
say let's flow in such a way that the
total area inside doesn't change. So if
that were the case, if you started with a
big circle it would just stay the same.
But if you start with something like
that or something that has a lot of
curvature this bit is going to move out
like- so this is going to move out, this
is going to move in, this isn't going to
move very much, this will move out, this
will move in. This will become rounder as
you go on in time so it'll become more
like that and then eventually as time
goes on it will become essentially round.
This is a very simple model, there's not
a lot of interesting things going on
here but it's a nice first picture of of
what what I'll ultimately relate to,
Ricci flow. - (This is like an artificial)
(thing then? You- this is just like a game)
(mathematicians play? Like this isn't- )
(you've just said let's draw a shape and)
(let's- ) - Yeah it's- yeah it's it's a little
yeah it's it's a game and you can you
can ask questions like; say you don't
keep the area fixed, how long does it take
for it to- for to go down to a tiny point?
And when it when it gets to be
very very small the curvature remember
is sort of 1 over the radius of the
circle, so here the curvature is getting
arbitrarily big, so it forms what's
called a singularity and it stops- it
stops changing. But you could look at
something like this: say you have
something which has lots and lots of
really screwy little things like that
and you can say maybe this'll- maybe
some of these things will join up, what
will ultimately happen? In this simple
case what will happen is it will always
become a circle after enough time. But if
we look at it sort of the next I think
simple model for Ricci flow, which is
mean curvature flow, there it gets a lot
more interesting. So now instead of
thinking of this as a two-dimensional
surface with one-dimensional closed
paths in there, now let's think of this
as a three dimensional surface. So I'll
think of a axis there, I don't want to
always draw that, but think of this as
three dimensions and we could be
thinking about say a sphere, a beach ball.
You know you can think of a beach ball
which, if it's all blown up, is is nice
and round but if it's if it's not you
know if you've left it you know get to
the middle of the winter it's going to
be maybe instead of being a nice round
one it's going to be you know sort of
something like this, and think of this
always as two dimensions and maybe it
has a little, you know sort of a- this is
a two dimensional surface in three
dimensions. So we're in a three
dimensional space but we're thinking of
a two-dimensional surface, just like the
surface of the Earth. The so-called mean
curvature flow is similar to the
curve shortening flow except now we're
moving this two-dimensional surface. And
because- and we're going to be mov- the
the notion of curvature is is is quite a
bit more complicated when you have a
higher dimensional- a higher dimensional
surface, but it's moved according to
what's called the mean curvature which is
sort of the average of the curvature in
this direction and that direction. The
flow - and remember we have this word flow
which is telling how is this
two-dimensional surface, this this sort
of half deflated beach ball, how is it
going to move in space? And before
when we were talking about a path in
space we talked about a tangent line, now
we have a tangent plane. Think of the
surface of this ball
and you sort of take a two dimensional surface and fit
it at that point, that'll be the tangent
plane, and this point is going to move in
the direction which is 90 degrees to
every- to that plane so it's in the- again
the sort of orthogonal normal direction.
(So our wonky beach ball is going to get bigger)
(or smaller or?) - Well actually - and I
shouldn't have drawn it this way because
this curvature it would tend to go in
this direction - our beach ball, unless we
tell it to make sure that the that the
volume contained inside is is constant,
you can do that. You can change the flow
of it, if you don't it's going to become
smaller and smaller and rounder and
rounder. Unless- but the interesting thing
that can happen here that's a little bit
different from that simple curve curve
shortening I was talking about before is
say you have one that starts out like
this, I think of this as a sphere with a
very tight corset. People don't wear
corsets anymore I don't think but anyway-
(That's 3D though?) - This is 3D, so think of this as, so this
is 3D, think of this the surface as as
taking that that beach ball- - (It's like an hourglass?)
Like an hourglass, very good, like an hourglass. So
what's going to happen here for reasons
that are I think a little tricky to
explain; but right here where you have a
lot of curvature in certain directions,
this thing instead of becoming round
this whole thing is going to shrink off
and it's going to - as we go in time - this
is going to basically go down to
the radius around here is going to go
down to zero. The curvature now blows up;
when I say blow up I mean it becomes
infinite. What happens here now is you
have to stop the flow because the flow
can't move anymore. The flow is saying,
move a given point in a direction which
is orthogonal to the surface, orthogonal to
the tangent plane, and move it in an amount
that's proportional to the curvature. If
the curvature is infinite you can't do that anymore.
(Jim isn't it only that point that can't move?)
(Can't all the other points on)
(our hourglass continue?) - You can. You can
do that if you say I'm going to- so what
people- you sort of do a quarantine. You
say okay we're not going to work with
these guys anymore let's- in fact you
could try to throw it out and look we can
continue to work with these. The problem
with just continuing
everywhere else, the disease that's happened
here if you will is going to spread very
quickly and very quickly the whole thing
is not going to make any sense.
So what one would like to do, and this
was a very important thing in the Ricci
flow, in Perelman's use of Ricci flow to
prove the Poincaré conjecture, is
figure out how do I quarantine these
singularities for me? How do I cut this
thing off so to put a little bandage
around it? They call it surgery and
surgery off the bad place and continue
the flow elsewhere. So can we get rid of that stuff?
And that's- you can think of that as a
fairly arbitrary thing but you have to
do it very cleverly if you want to learn
things about the nature of the
relationship of topology which is kinds
of surfaces and curvature which is about
how how round and how curved- you know how
pointy things are. So here we're dealing
with 2-dimensional surfaces in
3-dimensional space;
mathematicians like to abstract things
so we could think about
3-dimensional surfaces in 4
dimensional space. Now I'm not going to
try to draw a picture, even here this is
a bit of a lie because I'm drawing on a
2-dimensional surface. But one can do
that. Now you have more curvatures and
more interesting things you can do.
That's still not Ricci flow. Ricci flow
is a big step away from mean curvature
flow or inverse mean curvature flow in
the following abstract way: in the in
this kind of thing where we're first
talking about 2-dimensional surfaces
in 3-space or even 3-dimensional
surfaces in 4-space; you can still
picture, picture that big empty space and
you have this this like the the deflated
beach ball you have that thing sitting
in the space. And it's easy for our minds
to picture going to a 4-dimensional
space I think relatively easy,
maybe I'm fooling myself; but to picture
a 3-dimensional thing or even a
2-dimensional thing in 4
dimensional space. Ricci flow is very
different. Now the object you're looking
at is not sitting inside a space, you're
not moving the object. So maybe I'll try
to explain now.
With the curve shortening flow and the
mean curvature flow we were taking an
object, be it a path or the surface of a
ball, I mean surface of a deformed ball, and
we're moving that in the empty space.
Ricci flow you're doing something very
different. You're dealing with something
called Riemannian geometry. Just think,
so I'm going to have to go back - how do I
do this? Mathematics is about
abstracting. Some abstractions, like going
from 2-dimensional objects in 3
space to 3-dimensional objects in
4-space, you can sort of trick your
mind into thinking about that. But when
you do the abstraction in going to
Riemannian geometry you have to think
about- think about the surface of
that ball but forget the inside, forget
the outside, it's not moving. So I'm going
to draw the same old picture of this
this- think of this 2-dimensional
surface. Except, don't think of it sitting
inside a big space, think of it as
something where you specify a function
and the function is going to assign a
matrix. So think of a matrix, well usually
it's called G, and at each point - let's
say the point is labelled by x - there's a
component G11 in this corner,
function of x G12 and maybe- it's a
square matrix that goes. So you have a
square array of numbers and these
numbers, and this is called the metric,
these numbers tell in an abstract way
what kind of information would you like
to know. One thing you would like to know
is if you're starting at a point here
and you're going to a point there what's
the shortest possible path between them?
You can take any point on Earth and then
another point and find the great circle
route between them. And they tend to be
just sort of almost straight paths. But
say you're in a mountain range and you
want to get from the top of one mountain
to another, this is a very curvy place
and if you try to find the shortest
possible path between them it can be
very complicated. You can think about
this kind of thing for a surface in
space or what you can do is say I'm not
thinking about this this this surface in
space, I'm just going to assign it
something called the metric and using a
little bit of mathematics you can use
that metric to decide what shortest
possible paths are going to be, what
volume- what the area of different
regions is going to be and what the
curvature of this thing is going to be.
You can think of Ricci flow as, let's take this
metric, the metric changes in accordance
with something that involves the
curvature. So the basic idea is the same
as the surface moving in space but we
don't move the surface in space we move
this thing, we change this thing which is
describing the geometry. A very
interesting and important feature of
Ricci flow, like that mean curvature flow,
is if you have something and I'm going
to- this is not fair but if I draw that
same picture where somewhere there's a
lot of curvature there, maybe it has some
interesting holes in a topology. But if
you have- now this wouldn't work in 2
dimensions, in 2 dimensions it's always
going to become as round as possible. But
if you go to higher dimensions, when you
have a lot of curvature in some place
it's going to tend to- the flow is going
to tend to hit a singularity where the
curvature gets arbitrarily big.
Understanding how to work with the
curvature was very important in
Perelman's proof of the Poincaré
conjecture and the geometrization
conjecture using Ricci flow.
If you'd like to see and hear a bit more about the
Poincaré conjecture there will be links
on the screen and in the video
description; we've got sort of a fairly
simple video about it. And there's also a
lot more of my interview with Jim about
the conjecture and also how the Ricci
flow sort of contributed to it finally
been cracked. If you haven't subscribed
to Numberphile please do, we'd love to
have you on board. And if there are other
subjects you're into like chemistry and
physics, astronomy have a look in the
video description, I'll have links to my
other channels there.
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