Well, it is my pleasure to introduce the first plenary speaker, Richard Hamilton. Richard Hamilton had professorships at Cornell University. I am proud to say San Diego, Irvine, and currently he is a professor at the Columbia University. Well, we have already heard and will hear again

that Hamilton's theory of Ricci Flau led to one of the most exciting developments of mathematics. It is not an exaggeration to say that Richard Hamilton is one of the heroes of this Congress. Richard Hamilton.

Thank you very much. Let's see if we can get the screen working again. Great. OK.

The delay was I was trying to get two screens up here. There's some light there. What's that about? Oh, well. Can you see that OK? OK. Well, gee, it was 40 years ago I

was sitting in Jim Eel's seminar at Cornell when he put the Poincare conjecture up on the board and suggested using PDE to prove it. And I said, well, gee, the hypothesis is topology and the conclusion's topology. Why wouldn't you expect the proof to be topology?

And he said, well, the hypothesis is that you can't find any topology so the topologists don't know what to do. And maybe we can help them out. And so that was a great idea. And it goes back at least to Yamabi, who had the idea of trying to put

a nice metric on a three manifold. And he started with a simple case of a metric of constant scalar curvature. And that was the Yamabi conjecture that was brilliantly solved by Rick Shane while we were at San Diego. Anyway, it was 10 years later before I

started thinking seriously about what you could do. And Eel's and Samson had used a parabolic flow to prove existence of harmonic maps, to take a map and straighten it out and make it nice. So I thought, well, maybe you can do the same thing with a metric, start with a metric

and flow it to try to spread the curvature out evenly over the whole manifold. And I couldn't figure out how to do it for quite a while. And then I thought, well, if you just kind of think of what a flow would look like, you want it second order and parabolic and quasi-linear.

And you'd like it to diffuse equally in all directions and all the components of the curvature. And it turns out there's only one flow that has that property, which is the Ricci flow, which is d dt of g, the metric, is minus twice the Ricci curvature. And this turns out to be a very nice, very basic

equation in geometry. The elliptic version is Ricci flat or constant Ricci curvature. And the hyperbolic version is the same thing in Lorentz space, where, of course, Einstein showed that it can be used to describe gravity.

And I don't know if I was the first person to write this equation down. But it said on the Discovery channel that the first person to invent the steam engine wasn't the one who first made something go round and round with steam, but who first figured out

you could power something with it. And so I figured out that you could actually prove theorems in geometry with this. That's kind of fun. However, in order to tackle the general case, you have to allow surgeries, because this equation

forms singularities. And the idea is, before it gets singular, you go in and cut out the bad part and round it off and continue the flow. So let's put up a picture here.

This is the sort of thing that Ricci flow does in two dimensions. If you have an S2 and it has a neck in the middle, well, that neck is just an S1. And S1, from the point of view of Riemannian geometry, doesn't have any curvature.

And the neck opens up, and it turns into a round sphere. And in three dimensions, what I proved is, well, you need to assume that you have positive Ricci curvature. And then this flow actually rounds the metric out to a round one. Of course, while it's doing this, it shrinks down to a point. So you have to re-expand to get a limit.

But I succeeded in proving that with positive Ricci curvature on a three manifold, there's a constant curvature metric, meaning it's either the sphere or a quotient of it.

After this, Yao suggested to me that the,

is there some way to get rid of that white thing there? I don't know what it is. Anyway, Yao suggested that in three dimensions,

the neck that you'd see in a dumbbell shaped thing is actually a two sphere. And that does have intrinsic Riemannian curvature. And that shrinks the two sphere down. It's going to pinch the manifold into two pieces. So this can actually be used to perform a connected sum decomposition.

And here's an example of a typical neck pinch. And the other singularity that can occur is the degenerate neck pinch, which looks like this. So you have, again, a little S2 pinching down.

But the three wall on the other side is also pinching down at the same time. So when it forms a singularity, it's just at an isolated point with nothing left on the other side.

Third Ricci flow is you start with any metric and run the Ricci flow until the curvature gets too big. And then you try to cut the curvature down by metric surgery. And you can look for cylindrical collars and each bounds a ball on one side if you assume that the manifold is prime,

which you can already do by the topology. And see, that's why I like dealing with this case of simply connected and prime because then you don't have any topology. And then you try to cap it off so it looks nice again.

And the claim is that this ends in a finite time on a simply connected manifold after finitely many surgeries. And it ends with seeing that it's actually the three sphere.

So lest you think that there's really any sort of topology going on here, all the surgeries that I'm going to be doing are what I call appendectomy. See, my dad was actually a real surgeon. And when I was a kid, he told me about taking out appendices.

In fact, my parents treated me and my brother so fairly that the only thing he ever got that I didn't was appendicitis. And so here you see all the topologies over here. We're there on the left. And there's this narrow neck pinching off, but it's just got a ball on the other side.

And I started to think of this as the inflammation where the curvature has gotten too big. And you want to cut it out and throw away the appendix, which is trivial, it doesn't matter. And then you sew on a nice cap. And the idea is that you've reduced the inflammation because you got rid of where the curvature was real big

and replaced it with where it isn't nearly as big. And so you see, this is really designed not to fix the topology, which is already fine. It's designed to fix the analysis, to fix the excessive growth of the curvature.

It doesn't work, oh, is this, okay, great. That's fine, thanks. So existence and uniqueness, you start with any metric you like, and then at least for a short time, you get a solution.

And actually I was kind of very interested in the Nash-Moser inverse function theorem then. And I was very proud of the fact that I actually used it because this isn't strictly parabolic, it's only weakly parabolic and you lose derivatives. And I thought, oh boy, now everyone will have to learn Nash-Moser.

But actually two weeks later, Dennis de Terks, they showed how you could actually use a regular theorem in a Bonnock space. But it turned out that was great that it happened because then everybody could read the paper and didn't have to learn Nash-Moser, okay.

I'll come back to that idea later. I proved global derivative estimates and Shih-Won Chong, who was a student of Yao and did excellent work in Ricci flow, managed to do a local version. And it's a standard sort of result

for parabolic equations. It's a little trickier in this case because the metric's changing. But the idea is that if you have a bound on the curvature in a big parabolic cylinder, then you get bounds on all the derivatives in a smaller cylinder.

And it's typical of elliptic and parabolic equations that they smooth everything out. And that really gives you lots and lots of extra control. So for classical geometers, having bounds on all the derivatives of the curvature is something that they think would be way too much.

But as soon as you're into the PDE theory, you actually get it for free. And one of the reasons for liking a parabolic flow instead of just minimizing sequence is that you pick up so much extra regularity. And here's another picture of that cylinder

in the derivative estimate, okay? By the way, the different powers are just according to the scaling. Well, the next basic idea is pinching estimates. And we now have lots of pinching estimates

in parabolic flows. And I've done some Huiskan, Kao Hai Dong did it for the Ricci-Kahler case, Nishikawa, Marjoram, Ivy, and so on. So in three dimensions, the first result was that if Ricci's positive, that's preserved,

or if Riemann's positive, that's preserved. But in three dimensions, you get a very special sort of pinching, which is a pinching towards non-negative curvature, which says that once the curvatures get very, very big, then there could be very, very negative curvature,

but it's not nearly as negative as the curvature is positive in some different direction at the same point. And it's only diminishing by a factor of the logarithm. So just to show you how it is, if the maximum curvature is 10 to 100, you'd expect the minimum to be less

by about log of 10 to the 100, which is about 100 or 10 squared. And so the minimum might be minus 10 to the 98th. So you see it still has a huge amount of negative curvature, but it's very, very small compared to the positive curvature. And that makes this lovely property

that when we do blow up limits, they have non-negative curvature, which is a great thing to have. And the technique for doing this is applying the maximum principle to the diffusion reaction equation satisfied by the curvature. And in order to draw a picture,

this reaction here is actually a sum of two squares. But typically what you do is forget the diffusion and just study the reaction. And since it's homogeneous, you can project curves back to this plane where lambda plus mu plus nu is one.

And then you see that everything wants to flow towards the curvature tensor of S3, except if it hangs up on the cylinders. But if it doesn't actually go into the cylinder, then it skirts around and goes on into the three sphere. And this is kind of a crude picture,

but here's a better one that Muffin drew. Muffin's my computer. And you'll see these three lines here are the lines of the triangle there. And here are the S2 cross S1 curvatures,

and here's the S3 in the middle, and everything is moving in towards the middle, except it tends to hang up on the S2 cross S1, but if it gets beyond it even a little bit, then it comes on into the S3. So the next step in the analysis

depended on recognizing the importance of solitons. And a soliton is something which moves without changing its shape. So in this case, the thing is that the metric actually changes, but the change is affected just by its motion

under a diffeomorphism, that is a re-parametrization. And the condition for that is that the Ricci tensor is the lead derivative of the metric along a vector field. If the vector field's the gradient of a function, then Ricci is the Hessian of the function.

But you can also have expanding solitons, where you add in a constant times the metric, or shrinking solitons with it going the other way. And here's some pictures. Here's steady solitons.

There's this one, which is quite round, opens like a parabola. There's this two-dimensional one I call the cigar, and here the cigar is cross R1 to make it three-dimension. And you can also have expanding solitons, like coming out of a cone,

or a lower-dimensional cone cross R1, or you can have shrinking solitons, like these, S3 shrinking to a point, or S2 cross R1 shrinking down to a line.

So the solitons provide the model for studying various estimates, and typically what you find are Harnack estimates. The whole theory of Harnack estimates started with a very important paper

of Yao-Hsing Tung and Peter Lee, where they studied it for the heat equation and other diffusion equations. And I thought about that in quite a while, and finally realized you could prove this nice inequality that for any complete solution with a non-negative curvature,

you actually need non-negative curvature operator in higher dimensions, you always have this inequality holding point-wise at any point and for any tangent vector V. And this has a nice corollary that if your solution goes all the way back

to minus infinity, you don't have this R over T term, and putting V as zero, you have the RDT positive, and that means R increases point-wise. And this is terribly important for the analysis of these ancient solutions. It also has this interesting corollary that if the curvature is non-negative,

and if some Ricci of V, V is zero, then grad R in the direction V is zero. We'll see that later. The idea of proving a Harnack estimate

is that you need some sort of positivity. For example, if the curvature changed sign, then the positive stuff could cancel the negative stuff. But if R all has one sign, nothing can cancel. So here, the maximum curvature can go away rapidly by cancellation, but here it can only drop down

by diffusion, and that's what keeps the curvature from falling off too fast. Proof involves some intense calculation because you have to define all of these

interesting tensors involving the first and second derivatives of the curvature, and they were found simply by looking at what vanishes on the soliton. And then you get this very nice inequality when you take the infimum of this quadratic overall U,

I call that HAB, it's a matrix, this Harnack matrix, and it satisfies this nice parabolic inequality. And in my original paper, I didn't have this extra term here. I hadn't actually pushed it to the limit, and the calculations are pretty bad,

but I had my computer muffin do it, and my thanks to Dorian Goldfeld for teaching me Mathematica. And muffin found this extra nice quadratic term, and there's even another quadratic term here involving first derivatives that I couldn't use, so you don't see it. But when I want to localize the Harnack estimate later,

that quadratic term's very important. So now we're all very excited by Perelman's work, and the first thing that he gives us

is this absolutely wonderful non-collapsing estimate. And this is something I'd worked on hard quite a while. And so I was totally thrilled to see that he actually had done it, and that it actually looks elegant. So here is Perelman's transport equation,

and this isn't exactly how Perelman wrote it down, but it's completely equivalent. Perelman defines his L function, or reduced length function, this has nothing to do with the real L functions, in terms of minimizing path integral.

But they actually solve this hyperbolic PDE, and now you might think you're gonna see some horrible hyperbolic PDE theory, but it's only first order in a single function. So you can solve it completely by the method of characteristics.

And where would you find this? Well, Perelman had an interesting argument, but the way you might have found it is that, again, you look at all the expressions that vanish on solitons, and if you look at the potential function of the soliton,

then you can make a measure out of it that evolves by the adjoint to the heat equation going backwards. So what you're doing here is you run Ricci flow forwards up to a certain time, and then you stop, and you take another function and solve an equation backwards.

And there's a complete duality between the parabolic equation that he has that gives him his Harnack estimate and this transport equation, which is really nothing but the transport equation for the Li-Yau minimizing path integral for the Harnack estimate.

So there's a whole wonderful theory here, and I could have talked for the whole hour on this, but then you wouldn't see any of the rest of it. So the characteristics of this equation are dP d tau. Now, tau is backward time because we already solved the Ricci flow forward, and now we're going back.

It satisfies dP d tau is grad L, and then you know how L evolves on the characteristics, and you can compute everything out with ODEs. And I take this elliptic operator in L, and it turns out along the characteristics that EL has this very nice evolution

where this term on the right that's squared is exactly the thing that vanishes if it's a soliton with potential function L. And this equation shows if EL is less than or equal to zero to start, then it remains so as tau increases,

that is, if you go back in time, okay? Now, you get this inequality in Perelman's method from the path integral by a rather complicated calculation. I went over it in my seminar in complete detail twice

as calculation's completely correct, but this calculation's a lot easier, actually. Okay, so let me open up the next box.

Good, this slide tried to escape. So then you can introduce the reduced volume

where I take four pi tau to the minus n over two e to the minus L times the Riemannian measure mu, and that gives me a reduced volume form U, and along the characteristics, d d tau of U is a half EL times U, so if EL is less than or equal to zero to start,

U decreases along the characteristics. And if you've read Perelman's paper and seen his entropy, then this minimal reduced volume has the property that the logarithm of it is Perelman's entropy. That is, you do the minimal reduced volume

by minimizing integral of U subject to the condition EL less than or equal to zero. And this gives this lovely non-collapsing theorem, which says that if the metric that you start with is kind of nice at a certain scale or not,

then subsequently you can show it doesn't collapse in a region where you've controlled the curvature. And it'd probably help if I just put up this picture here because it's hard to read the text quick enough.

Where's my picture? Here we go. So we have this region where the curvature is smaller than one over R squared in a ball of radius R, and we're worried that the volume of this ball won't be big enough.

So if it is too small, we want to get a contradiction. So you start by constructing a U where the corresponding L satisfies this elliptic inequality, EL less than or equal to zero. And then you can make U concentrated in this region,

vanishes outside, and you can make the total measure of U very small. Then you propagate L back by the transport equation, which transports the corresponding measure U. And back here, you have nice geometry. So whenever you have this inequality EL less than

or equal to zero, then the integral of U has a lower band. And now by making kappa collapsed enough, you can make this size less than this, but coming back, it decreases, forward, it increases, and you get a contradiction. And that's really all there is to proving this nice,

very, very beautiful, very, very useful result. And I have to say, I'm just so grateful to Grisha for doing this, because in addition to really, really needing it for the Poincare conjecture, I had clawed by my fingernails to get non-collapsing in a few special cases,

and now I never have to worry about it again. Okay, a few miscellaneous facts about solitons that I can refer to later. The first is this cigar, and it has a nice form.

And the second is that if you have a non-compact steady soliton with non-negative curvature, then its asymptotic volume ratio is zero, and its asymptotic curvature ratio is infinity, except in two dimensions where it's zero.

I should define these terms. The asymptotic curvature ratio is the lim sup of the curvature times the distance from an origin squared. The volume is a lim sup of the volume of a ball of radius s over s to the n. And the aperture is the lim sup of the fine diameter

of the sphere of radius s over s. And this fine diameter is, I don't really know the right term, but Tom Elmanan called it the crude Hausdorff measure. But that sounded kind of crude for the talk.

And it's sort of, what it is is that if you're in the non-negative curvature case, the sphere might really have one boundary component or it might have two if it's splitting as a product. And so you take the open covers of the boundary and you look at the infimum of the sums of their diameter.

Okay, come to ancient solutions. This is a very important notion in the whole theory of parabolic equations. That parabolic equations make things better.

And an ancient solution is one which, like me, has been around forever. If something's been around forever and it's been getting better all the time, you can bet it's already pretty good. So it turns out that there aren't many ancient solutions and they're very special and you can say a whole lot about them.

And also they come up a lot in the analysis of singularities when you do blow up limits. And so we want to study the ancient solutions. And what you can show about the ancient solutions is, well, I showed the curvature ratios constant.

And if the curvature goes to zero, s s goes to infinity, the volume ratio v and the aperture alpha are constant also. Now, the next result of Perelman is very, very useful. That he says that on any ancient solution in three dimensions,

that the asymptotic volume ratio is zero. You don't actually need non-collapsing for this because if it collapses, then the volume's zero all the easier.

And he uses this to prove this very important fact that the collection of all ancient solutions, which are non-compact, if you normalize it so that the curvature at your origin is one, then that's compact in the topology of C infinity convergence on compact sets. So let me just try to outline how this goes.

There's another theorem I haven't told you about yet, which is the splitting theorem. And the splitting theorem says that if you have a solution to the Ricci flow with non-negative curvature, and if after a while you actually have some zero in it

in the curvature, this is the curvature operator, then the solution has restricted a holonomy, like for example, being Kahler. And if you know that Ricci has a zero, which is stronger, then you know what the holonomy reduces to,

and it splits the flat factor. And the proof is basically, you go back to the non-negative curvature being preserved and apply the strong maximum principle. So this splitting theorem has the same corollary as Harnack that if the curvature is non-negative everywhere

and Ricci in some direction is zero, then grad R in that direction is zero. And here's a picture of the splitting. And in fact, you see that here when it splits, there are these directions along the flat factor

where Ricci is zero. And in that direction, the grad R in the direction V is zero. Okay, most of Perelman's arguments, you find cones all over the place.

And it's a very, very lovely picture because the idea is that in many cases, you can show that something blows up to a cone. And a cone has the property that you can find directions where Ricci is zero, but where grad R isn't zero,

namely the directions going out away from the cone point. So it contrasts with a product. Well, Perelman uses Alexandrov spaces. And now I was real clever back when and used Nash-Moser to prove existence

and Dennis de Turk showed you didn't need it. So there's a famous story when Alcibiades was a young boy, he found his uncle Pericles in his study hard at work. And he asked his uncle said, what are you doing? And Pericles said, I'm studying how to give an account

of my actions to the Athenian people. Alcibiades replied, you'd do better uncle to study how not to have to give an account. So I thought that I'd study not Alexandrov spaces, but how not to need to learn them.

Now, mind you, Alexandrov spaces are wonderful things. And we really do need a version of weak solutions of the Ricci flow for higher dimensions. But let's see if we can get away without it for now. Well, it turns out that given appropriate derivative estimates of the sort you need

to do the cone argument, that you actually can get a very nice quantitative result. That if you have two points P and Q, that the ratio of the curvature at P to the curvature at Q is bounded by the curvature at Q times the distance P to Q squared to some power.

It's a kind of a power law. And this explicit bound gives you a very quantitative measure of the sort of control that Perelman gets that there must be some control out of the cone argument. And this is a more explicit formula.

So let's see an example of how this might come about. That in proving this lemma that the asymptotic volume ratio is zero, there are three cases.

And I'd originally studied this in my formulation of singularities paper. And I got it in the first two cases and I thought the third case probably couldn't occur but I couldn't show it. And of course, Perelman's cone argument rules it out. But what I managed to show is that if you're in this case where the curvature

times the distance from the origin squared is bounded and the curvature times the time back is bounded, then I showed the asymptotic volume ratios positive and at least away from the origin. The curvature is bounded above and below comparable

to one over the distance squared. And now, if I combine that with some other stuff, I get a contradiction. And the basic motivation behind this is, Perelman would say, well, since it's opening like a cone, you blow it down and you get a cone and then that contradicts splitting.

But I wanna do the contradiction instead from the Harnack. So here, since the curvature is falling off, bounded above and below by one over distance squared, you actually get, you can bound grad R point-wise

by constant R to the three-halves. And since you can bound the dr dt, the Harnack gives you this very nice sort of inequality, which I use a lot. And you also have this volume condition.

So under these assumptions, that's what you need to get this power loss. And the way that you do it, I'll just kind of show you how you can use these estimates, is there's a very nice classical inequality that says you can control integral of Ricci

along a minimal geodesic with this appropriate weighting of the distances by the log of the Jacobian determinant of the exponential map from P to Q. And then you can actually make this Jacobian

have a lower bound in this case because of all the control you have over the curvature and the volume. And then I take this little inequality and I plug it into this integral.

If you integrate grad R over R along the geodesic, you get log of the curvature at one end over the curvature at the other. And then I play Cauchy-Schwartz to get out this thing you can control geometrically and you're left with this other term that isn't too bad.

And then you just have to choose the points X and Y right along the geodesics. Because if I went all the way to the end, this would give an infinite thing. So let me show you the picture. I'm really wishing I had two projectors right about now.

But here's the point where the curvature's big and there's the point where it's fallen off a lot. And here you've got O, and then somewhere around O we have plenty of volume so I can make this Jacobian have a nice lower bound. And I'm gonna pick X close to P and Y close to Q,

but far enough away that I can control everything. And then you pick them all right and you plug it all in. I don't think I have time to really do it in any detail but I just kinda show you that it's not really drawing pictures,

it's a lot of calculating. So Perelman then uses this volume non-collapsing in the following way. The first thing he wants to show is that if you know you have some volume in a ball

in an ancient solution with non-negative curvature, then that actually controls the curvature out to a much greater distance than the radius of the ball. And the proof is that you wanna pick the point P

cleverly so that you can do a blow up around P and get an ancient solution with a positive asymptotic volume ratio which we just saw can't happen. So here's the picture. You look at some radius, like you could scale so R is one

and then W would be the radius. And you've got some volume here and you wanna bound the curvature out to this distance. So I've picked this point where it's about as big as it's gonna be in such a way I can do a blow up limit around it. And when you blow up around here,

then compared to this scale, this little ball here goes off to infinity, but because of its size, it's getting bigger as you scale. So you have to expand to make the curvature control here and then this is getting bigger and further out. But you see that the limit you're gonna get is gonna have this positive asymptotic volume ratio.

And one thing that I've learned from studying Perelman's paper is how much you can get in geometry out of these volumes, especially in the presence of non-negative curvature. Well, the next part of the argument

of proving compactness is then to say that you can actually bound the curvature out to any finite distance in terms of the curvature at a point. And this takes a couple of steps

that you use the previous argument to say, well, you start by picking, you pick a ball here, you know the curvature here and you wanna bound the curvature far away. So you start by picking the ball of a size where you take the biggest radius r, so the maximum of the curvature

over a ball of radius little r is one over r squared. And then I wanna show that that makes the curvature at the origin at least a fraction of that maximum. Well, the idea is that this maximum is attained somewhere here

and then because we have these derivative estimates, because we have volume in here, then we get bound on the curvature out to a bigger radius and Harnack gives us a bound back in time because r doesn't fall off going forward,

so it doesn't grow going back. And then the she's local derivative estimates give controls on the derivatives in here. So you can see if the curvature is the good size at p, it'll be at least a fraction of that if I back up in time.

And then we all know that the Harnack inequality types can be integrated over paths to produce actual Harnack inequalities. So that says the curvature back here at the origin is at least a fraction of q and the curvature at q is at least a fraction of the curvature at p and you get your estimate.

And now the rest is easy because once you know that this ball here of the right size actually is controlled by the curvature at the origin, then the previous estimate says you can bound the curvature out to any finite distance from this ball which is now bounded

by the curvature at the origin. And that gives the first of three steps in the proof of the compactness of the ancient solution. The second step is a pretty easy volume comparison that bounds the curvature out to infinity and the third step in this case is easy

because Harnack controls it backwards. Now in this blow up argument, I've talked about limits. So let me say a little bit about limits of manifolds. There's a definition of a sequence of manifolds converging to a limit in the topology of C infinity

with compact support. And that says that you can find a sequence of, you have to fix origins to take a limit of manifolds and then you find a sequence of diffeomorphism. So it's convergence up to a diffeomorphism where when you pull back,

the metrics are converging to your limit metric in C infinity and compact support which means it looks as close as you like and as many derivatives as you like on as big a set as you like. And this is a very nice notion of convergence for partial differential equations because you have all the smoothness.

And one of the problems with taking a weaker sort of convergence like Alexandrov space sense is that you don't have control of derivatives of curvature. So you have to kind of take a limit in that sense

and then pay a lot of attention to what's going on locally in your curvature bands. This has all been gone over very well by say Bruce Kleiner and John Lott. But if you stay in the smooth category, there are advantages.

Here's an example of limits of manifolds and the sort of blow up procedure we wanna be able to do where you're forming a neck in here and as this neck shrinks, you re-expand to keep it a constant size and then the rest of this manifold gets bigger and bigger

and eventually goes off to infinity. And you see your compact things can converge to a non-compact thing that in this case could be a nice round cylinder, okay? You also notice that here I've picked these base points and if I pick base points over here, I'd get something that didn't look like this but only went out in one direction.

So there's a very nice existence theorem which Cheeger told me he did in his thesis in the C infinity with compact support case. And it says that you get a kind of compactness result that if you have bounds on the curvature

and all the derivatives out to any distance you want going to infinity, and if you can stop the thing from collapsing, then the limit actually exists. And so this is a strong existence theorem

and you can generalize it to Ricci flows. And I showed that actually in the same conditions that you get a limit of Ricci flows. And the advantage in the Ricci flow case is that just having bounds on the curvature in a parabolic cylinder by the derivative estimates

gives you bounds of all the derivatives as well. So the C infinity convergence is natural. And so here you see that Perelman's non-collapsing estimate plays a very important role in being able to take limits because it gives you the other thing you need

in addition to controlling the curvature, you have to control the collapsing. Well, it turns out that another important use of non-collapsing is on surfaces. And if a surface is non-collapsed

and has non-negative curvature and satisfies the derivative estimate but isn't flat, then it actually follows that it's compact. And that's pretty easy to see because if it weren't compact, it would have to stretch out infinitely and then there'd be a region that looked like a S one cross R one

where it would be flat, okay. So in fact, from that, you can do stronger and you can say if you have a non-collapse surface that you can actually bound the diameter in terms of the maximum curvature

and you can bound the ratio of the maximum curvature to the minimum, okay.

The argument coming up, we're gonna wanna be able to bound curvature on three manifolds out to infinity. And there's a very nice result from a volume comparison that lets you do that. And actually, I think since time's short, I'm gonna skip that

and I'm gonna go on to something else, but I'll just mention that there's a very nice volume comparison you could use on annuli. People, I think, are very familiar with volume comparison on balls, but you can get a nice result on the ratio of two concentric annuli also, okay.

Herrmann has three lovely ideas or estimates.

And the first is his non-collapsing result. The second is the compactness of ancient solutions. And the third is the canonical neighborhood theorem, which basically says that if you have a solution to the Ricci flow that's nice to start and the curvature gets big,

then everywhere the curvature is big enough, you actually get it looking very nice. And the way he phrases it is that it looks like a piece of an ancient solution. Now, I went through all the proofs you use in proving canonical neighborhoods and using it.

And it seems to me that there are four essential properties of a canonical neighborhood that are the ones that you need to use. And the first is that you get a bound on the first derivative in space and you have a lower bound on the first derivative in time. Now, notice once you have this,

if you know the curvature at a point, you can control it out some distance in space and then some time backwards. And then you get control of all the higher derivatives by the derivative estimates, okay. But these are the basic ones. And then there's the non-collapsing. That is a nice property of the canonical neighborhoods

and non-collapsing is very important. And finally, there's this thing of having a relatively controlled aperture that this fine diameter of the sphere of radius lambda is really less that if you take a point

where the curvature is one over R squared and you take the sphere of radius lambda R, then it's fine diameter, that is one or two pieces summed up is no more than lambda R. And that keeps things from going off in all directions. It's basically confined to narrow apertures in one or two directions, okay.

So the proof, which is quite clever and a bit elaborate is that say, suppose it doesn't happen

and then you take a sequence of counter examples and then you try to take a limit of them. And remember there were those four conditions and each counter example is the first time when one of them fails. So up until a certain time they all hold and then one fails.

And that was at a certain curvature level. So then you up this curvature level and try again and look at the first time it fails. And if no matter how high you push up this curvature level you still fail to get this nice behavior above it, then you get a sequence of counter examples

and you try to take a limit. And there's three steps to doing the limit. The first is to say you can bound to a finite distance then that you can bound to an infinite distance and then that you can bound all the way back in time. Okay. So in order to do this,

Harrelman uses arguments on limiting cones and I found that you can replace this with the sort of integral of a Harnack estimate that I was showing before. You just have to take care of some error terms. So the first step is to go back to the Harnack estimate

and work a bit harder. And remember I told you there was that extra quadratic term that Muffin found and that's exactly what you need to introduce a barrier and localize the estimate. So you can then show that if you just have a solution in a open set, open parabolic cylinder

and if you have a bound on the curvature above and below where you're thinking the bound below is very small compared to the bound above which is what you get from pinching, then you can bound dr of v squared by well constant k squared Ricci of vb

and a n minus one little k length v squared. So here you see that the negative curvature gives you a little error here but if there's only little negative curvature it's a little error and this is going to be the sort of thing that you can use. And then as before the idea is that

with derivative bounds you can replace these maxima and minima in the whole cylinder by their actual values at the center point. So then what happens is that you try to repeat the previous argument

that I had showed you in the v positive case only now we don't have non-negative curvature we get a little bit of negative curvature and that typically happens if you're forming a neck.

And so now the idea is that in terms of the curvature q and the distance of p from q we wanna control the curvature at p. And the way Perelman does it is to say you take a limit of counter examples and as the curvature blows up the delta goes to zero from pinching

and this limit is a cone and use the splitting theorem to get a contradiction. But if you work a bit harder and use the estimates I showed you from the local Harnack then you actually show that you can control the curvature at p from the curvature at q by this sort of power law

except with the fact that the further away from q you wanna go the more you have to shrink delta down towards zero in order to be able to control the error term. And so here you see very clearly how everything interplays of the error and the curvature

and what happens in the bound. And the idea of this is you repeat the estimate you had before where you're trying to integrate Ricci of TT times this cutoff function over the geodesic

and you get an error that's like delta times r. And in order to handle this I first picked the point p very nice according to maximizing beta and then I picked the point x along that curve to maximize gamma and then I have these two quantities beta and gamma

and it turns out that because of that aperture condition you can compare them. And the gamma satisfies a barrier estimate and the beta is continuous.

And that means that satisfies a barrier estimate which is like this that log gamma is bounded by a constant depending on this proportional distance w wanna go times one plus delta gamma.

And now remember I can make delta as small as I like so no matter what this constant is here depending on the distance w I wanna go if I make delta small enough I get a barrier. And so beta satisfies a barrier also once you show that beta and gamma are comparable

for which you need this aperture condition. And then beta is depending on your choice of this proportional distance w but it depends continuously and you clearly control beta when w is small. So then as w goes to infinity you can keep control but you have to push the delta to zero

as w goes out to infinity. And that gives you very nice control over that part. The control out to infinity is again just the volume comparison but to control the curvature back in time you use the lower bound on dr dt.

So what you do is you assume, there's a nice result that simply says assuming that you have this sort of derivative control that then you can show that the solution

if you have solution defined for t bigger than zero you actually the derivative estimates give you control of r even back at zero which then lets you continue back past that. And this can be done nicely by bound.

times the maximum curvature and you get bounds above and below from the derivative estimates and that lets you pick out a sequence which converges to a type 3 blow up in the terminology. It's a sequence where T times the maximum curvature attains its maximum and I had

showed already in the Harnack that if you get an equality the strong maximum principle makes it a steady soliton and Cao Hui Dong generalized that to the Ricci-Kahler flow and Chen Bin Long

and Zhu Xi Ping showed that if T times R max attains its maximum then it's actually an expanding soliton and now you get a contradiction because the expanding soliton has V bigger than 0 but

all the derivative estimates like we did before make V equal to 0. Okay, now let me say a few words about the surgery that I define an epsilon KL-neck to be one where the metric in space-time

after a dilation and a diffeomorphism is epsilon close in K derivatives to the round metric on a section of length L and back in time L squared and I discussed this sort of surgery in a paper I wrote on Ricci flow on four manifolds with positive isotropic curvature and Perelman,

I was glad to see, quoted and used a number of those results in his surgery arguments. And the surgery procedure that you can use is you set three different levels for the surgery

there's the M hat which is when you reach M hat you have to do a surgery M bar is the level you want to cut it down to and M dip is the level below which you won't have to cut. And then what you show is that by picking these large and inappropriate proportions

you actually get only a finite number of surgeries. Now in his paper Grisha actually lets the curvature go to infinity and form horns and what he says he was interested in is proving the existence of a weak solution continuing through the singularities and he's

given us a lot of ideas for doing that but to just do the surgery part you actually don't need to do anything so elaborate. You can simply do the surgeries according to a fixed procedure like this without the curvature ever becoming infinite. You consider the case where it becomes

infinite to get a limit of counterexamples but in the actual surgery it's staying finite. And then you have to check out a number of things such as that the pinching estimate survives surgery and a number of other estimates survive surgery and the trickiest part is

verifying that the non collapsing survives surgery. Here I'd really like to thank my very good friend for explaining a lot of this to me that you don't have to prove the non collapsing

estimate down to the smallest scales. It suffices to get it down to a scale arbitrarily small compared to your initial scale and you can then fill in the rest of the non collapsing when you repeat the canonical neighborhood argument for the surgery solutions to the Ricci flow.

And the argument for finite surgeries is kind of interesting that you imagine that no matter how big you choose these things that you still reach a point where you can't do the surgery successfully. You can't remove all the excess curvature to cut it down to the level you want

and you say assume that that can't happen no matter how big I push these parameters. So you try over and over again and you keep failing and then you get a limit of counterexamples and you look at the limit of the ratios and then you try to analyze it.

So what you first do is take the limit as it flows up and then you take a blowdown limit going far away from that and that blowdown limit splits as a product because you can blow

down along a minimal geodesic going to infinity and use the splitting theorem. And then you try to extend back in time from that and as you go back in time what you find is that there's a you look at the maximum time you can extend this blowdown of the blow up to and the curvature can't go back to infinity by that little curvature bound I showed you that

follows from the derivative estimates you have and it can't go back into a surgery either because that would violate splitting. So we have positive curvature here and we have a zero here so that can't happen. So actually you can rule out that and the only remaining possibility

is that the this blowdown of the blowup is a product that goes all the way back in time and then the only remaining pieces to show that this cross-section is round. You actually don't need the roundness up until this last point where you want to do the surgery before

then having a bounded obliquity is quite enough and to see that it gets round you can use actually I gave two results and Perelman used the entropy on a surface it's a different

entropy involving the curvature but you can also use the isoparametric ratio where you look at all curves of length L dividing it into areas A1 and A2 and you take the infimum over all curves of L squared times 1 over A1 plus 1 over A2 and then it turns out that this isoparametric ratio increases under the Ricci flow and it's strictly increasing

unless you're on the round sphere. So now if you have an ancient solution to the Ricci flow in two dimensions which is uniformly kappa non collapsed it's been around forever a backwards limit would have this ratio constant and then you see it's actually the

sphere. So this gets you a Ricci flow with finite surgeries in finite time and all you need to see is that in the case where it's simply connected that's all you need. This is my last slide by the way. So now there's a nice theorem by Colting and Minicazzi in

another version of it by Perelman but the Colting Minicazzi theory seems to involve less technical difficulties in the minimal surface theory. You look at a map of degree one into the three manifold which the topologist tell us exists and look at the equatorial

spheres and their corresponding area and now you do a min-max and then it turns out that there's a nice estimate in the Ricci flow which is actually the same one I used in the t goes to infinity case to show the incompressibility of the hyperbolic pieces

where you show this decreases at a fixed rate and that can't go on forever. So now this regularity of the min-max is a bit subtle but Minicazzi told me that it was based on a result that Yao used and I asked Yao and he finally said yeah I looked at it yeah I used

it. So I think I'm pretty happy about this and this way you actually get a proof of the Poincaré conjecture so I think I'm about as surprised as anyone to see this all working and I'm enormously grateful to Grisha Perelman for finishing it off and I I'm real happy.

Okay thanks a lot bye-bye.