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.
Hey, Richard Hamilton was my differential geometry professor. The class was hilarious! Prof. Hamilton would generally just do problems he thought were interesting, and then become entirely consumed by his interest in differentiating certain surfaces on the blackboard for fun. He would also disappear relatively often for conferences without telling us or scheduling a TA to take over. All in all I don't think many of us really rigorously learned differential geometry, but we certainly had an interesting class with a test average of about 12%.
Definitely watch the follow-up video linked on screen at the end.
That whole channel is amazing IMO. It always makes me want to know more and gets me excited on any topic they present.
My understanding is that Wiles's proof of Fermat's Last Theorem is not very accessible, even to experts.
Is Perelman's proof of the Poincaré Conjecture widely understood to geometers?