Well the next the next activity this afternoon is a special lecture on the Poincare Conjecture by by John Morgan. So Professor Morgan is a topologist who's always had an interest in

the interface between topology and other subjects, algebraic geometry, gauge theory, theoretical physics and so it's no surprise that now he's very interested in Ricci flow and its impact on topology. So as you may well know he and Gang Tian produced an extensive

exposition of the ideas of Perelman in solving the Poincare Conjecture. Today I believe he's going to offer a general view of the Poincare Conjecture for all of us here. So Professor Morgan. Thank you Nigel. It's a great pleasure for me to speak to you today about a stupendous

achievement in mathematics. Gregory Pearlman has solved the Poincare Conjecture. The posing

of this hundred-year-old problem marked in my view the founding of topology as an independent discipline. Since its posing the problem has been studied, generalized, related problems have

been solved. It has been linked in one way or another to most of the progress in topology in the last hundred years. Now while related problems have been solved the original conjecture as posed by Poincare stood untouched resisting all attacks. Before Pearlman's work many viewed

it as a siren song for many boats had foundered on the rocks trying to reach it. There have been innumerable proposed proofs, innumerable proposed counterexamples but none with student

scrutiny. Solving the Poincare Conjecture is a signal achievement for Pearlman but it is also a signal achievement for mathematics. For it is a measure of how far our understanding of the

subject has advanced in the last hundred years. Pearlman has seen far but to do so he stood on the shoulders of giants. One giant in particular stands out. He is Richard Hamilton who addressed this conference on Tuesday about his work and its relationship to Pearlman's. Over a period of 25

years Hamilton painstakingly built the solid and elaborate foundation upon which Pearlman constructed the edifice of his proof. Without Hamilton's work Pearlman's would not have been

possible. Today I will begin by giving you a sense of the importance, centrality, and depth of the Poincare Conjecture. Then I will introduce you to surfaces and three-dimensional spaces we call the manifolds and tell you some of the ways topologists think about these

spaces. Next I will formulate and explain the statement of the Poincare Conjecture. Then I will shift subjects and give you a brief introduction to differential geometry which is the field of mathematics out of which the proof of this topological problem comes. I'll

talk about Hamilton's Ricci flow equation and then briefly describe Pearlman's insights. I will not be able to give you anything but a superficial metaphorical impression of the mathematical arguments. For those interested in more details I will give you links to over a thousand pages

now available on the web where you can learn more in detail. My goal here is to explain the significance of the conjecture and its statement. Let me begin with the role of problems in

mathematics. So from the beginning of the subject there have always been problems and they've always played a significant role. In fact during the 18th and 19th centuries mathematicians

posed each other problems often giving six weeks, two months, half a year for solutions and learned societies often posed mathematical problems to the general public and offered prizes for their solutions. The nature of problems in mathematics I believe changed with Hilbert's

talk at the International Congress of Mathematics in Paris in 1900. There he proposed a list of 23 problems covering a broad range of mathematical fields. These problems were of a different character than the ones that had been posed in earlier times. For Hilbert viewed them as the

central problems of mathematics as he saw it in his day and he thought the study of these problems and their solutions would lead the field forward. And indeed he was quite prescient for much of the progress in 20th century mathematics has revolved around these

questions and to solve one is to be recorded in the Mathematics Hall of Fame. In 2000 following the tradition established by Hilbert the Clay Mathematical Institute identified seven important and central problems in mathematics. Things had changed a little bit in the

intervening hundred years there was no Hilbert so Clay leaned on the support of a committee of leading mathematicians in order to choose these problems and we're a much more materialistic society so they offered a million dollars per problem for the solutions. What makes a good

mathematical problem? Let me quote to you from what Hilbert said in his address to the Congress in 1900. It comes out a little funny in English because I believe it must be a translation from the German but nevertheless I should still more demand for a mathematical problem if it is

to be perfect for what is clear and easily comprehended attracts the complicated repels us. Moreover a mathematical problem should be difficult in order to entice us yet not completely inaccessible lest it mock our efforts. It should be to us a signpost on the tortuous

paths to hidden truths ultimately rewarding us by the pleasure of the successful solution. That's what we're celebrating today the pleasure for all of us of the successful solution. So I think what Hilbert was trying to say was at least these things among others studying a good

problem stimulates a tremendous amount of mathematical research some of its directly on the problem sometimes it simply spawns research in related areas you never know where studying a problem will lead you. I guess you start hoping it will lead you to the solution but wherever it leads you you follow. Problems become more and more famous as they resist more and more

different attempts over time at their solution and the famous and old problems are used as standards for testing new ideas. As a new idea enters mathematics one way to test its power and

relevance is to to try to use it to solve old problems and that's the toughest test and if it passes that test it will be ensconced as important in the subject. So now let's turn to our particular problem Poincare Conjecture. It was formulated in 1904 by Henri Poincaré

so it's a little too young to have made Hilbert's list four years too young. The problem proposed a characterization of the simplest of all three-dimensional shapes that is the three dimensional sphere and I'll come back in a few minutes and try to explain to you how to think about the first of all what the three-dimensional sphere is and how to think about it. It has

been attacked in the intervening 100 years by direct topological means so it was a problem in topology. The hypotheses are topological, the conclusion is topological, it's natural to try

to solve it in the terms in which it's posed by direct topological means and there was no success nobody has managed to do that but there was a lot of effort put into it. Studying the Poincaré Conjecture did however lead to many advances in the study of other three-dimensional

spaces by a variety of workers over a period of about 30 or 40 years. We learned a lot about three-dimensional spaces but none of it was relevant to the Poincaré Conjecture which stood there as a beacon untouched by any of the new methods. In one of the most revolutionary ideas

in all of topology as far as I'm concerned Stephen Smale realized in 1960 that in fact one could formulate the Poincaré Conjecture in all dimensions that wasn't too revolutionary but that in fact it was easier to solve in higher dimensions and he in fact went on to solve it

in dimensions five and higher and for that he won the Fields Medal. I believe that Poincaré's idea was the following simple intuitive naive idea we understand surfaces two-dimensional spaces they were understood by the end of the 19th century clearly the next thing to try to do

is understand three-dimensional spaces. He wrote a long paper about them and at the end of that paper he said well we've achieved a lot but there's this leaves one remaining question and then he posed the Poincaré Conjecture. Surely in the intervening years people thought

well if we can't solve this problem in dimension three there's no hope in dimension four, five, six and all and this problem is just going to get more and more complicated we have less and less intuition about what these higher dimensional spaces look like and Smale's revolutionary idea was to the contrary it becomes much easier in high dimensions. And slightly earlier in

1956 Milner provided counter examples to a closely related problem in dimensions seven and higher. He told me that he when he first discovered this he thought he'd found a counter example to the higher dimensional Poincaré Conjecture but it turned out in the

end he had a counter example to a closely related conjecture and for this work he also won the Fields Medal. In about 1982 Mike Friedman and an incredibly powerful tour de force took the ideas that worked in higher dimensions and managed to squeeze them down so that they

worked in dimension four at the expense of crinkling the space in infinite complexity but he did solve the formulation of the conjecture in dimension four and for that he won a Fields Medal. At almost exactly the same time a young English topologist,

a student of Nigel Hitchen whom he later traded to Michael Attia for future draft choices, Simon Donaldson solved a problem very closely related to what Friedman was doing but not for crinkly spaces but for nice smooth spaces, spaces on which you can do calculus,

write down derivatives and do differential equations and the contrast between what Friedman proved for crinkled spaces and what Donaldson proved for smooth spaces showed us that four dimensions was like no other dimension, the divergence between these two seemingly very close

notions were in fact totally different in dimension four. In fact one had always believed until Milner's result that there was no difference between the two and certainly that's how Poincare thought of things. He was lucky in dimension three there is no difference. Moving the story along in about 1980 Thurston working on hyperbolic three-dimensional

spaces proposed a generalization of the Poincare conjecture to a conjecture about all three dimensional spaces and for that well actually not for that but because he did that he won the Fields Medal. His conjecture is different from Poincare's conjecture in two essential ways.

First of all as I've already said it covers all three-dimensional spaces whereas the Poincare conjecture if it's true only covers one, the simplest and secondly the way Thurston

formulated his conjecture was that the three-dimensional spaces had nice geometries and this suggested the idea that as we'll see later was picked up by Richard Hamilton that one could study these three-dimensional questions using geometric methods rather than purely topological methods, analytic and geometric methods. And here we are today three years ago

Perlman solved the Poincare conjecture and as we now know he has a Fields Medal for it. When I first prepared this talk two weeks ago I had two question marks next to the Fields Medal but I got to erase them for which I'm quite happy. So what is the method of solution

that I've alluded to? Let me say just a couple of more words about it and we will cycle back later to a little more detailed description. Perlman's solution used differential geometry and partial differential equations to attack this problem that in its formulation is purely

topological. He used an equation, an evolution equation called the Ricci flow equation for Riemannian metrics and I'll try to talk both about what a Riemannian metric is and what the Ricci flow equation is in a few minutes. This equation was introduced by Richard Hamilton

who went on to develop a rich theory about the solutions of this equation and to lay out a program for attacking the Poincare conjecture and according to Richard with crucial input from Yao Thurston's more general geometricization conjecture. All right well now it's time to turn

to the topology. I have to tell you a little bit about the topological spaces that we are thinking about and we'll follow Poincare and work by analogy. Two-dimensional spaces are easier to understand. We can see them for the most part and our intuition is guided by this very concrete

present representation we have of them. Three-dimensional spaces are harder and we have to argue by analogy. So I will go back and forth for a few minutes between two-dimensional and three-dimensional spaces. So let's start with the simplest one, the two-sphere. I've given you two representations of the two-sphere. On the right I've written its mathematical

equation and on the left I got a picture of the two-sphere, what I think of as a two-sphere. My wife says it looks nothing like a two-sphere. You can judge for yourself. The one thing I want to emphasize at this point because it's often a point of confusion for non-mathematicians, when we say the two-sphere we mean the surface of the ball. The ball would be the solid

thing and it's a three-dimensional object but its surface which we see sitting here in space, which we see over here, talking about the surface itself, that's a two-dimensional sphere, that is the two-dimensional sphere. Now there are other surfaces and these are

understood to be the surface not the solid object. We have the often called the surface of the doughnut or a torus. We have a two-hole torus and then here I have a picture, I believe this is probably a twelve-hole torus. You see around the sides you see six handles,

each one with a hole, and you see three in front and I believe we're to imagine three more in back. In fact this is what all surfaces look like. They look like some multi-holed torus. I have zero holes as the sphere, here's a one-hole torus, a two-hole torus,

you can imagine the intermeeting ones, here's a twelve-hole torus, keeps going. So that's what all these surfaces look like, that's a complete list of them. Now why are these two-dimensional? They're sitting in three-dimensional space, we look at them, we pick them up, they look three-dimensional. But in fact they are mathematically two-dimensional because

you can describe where you are on one of these surfaces by giving two numbers. I'm sure you're quite used to that in the case of the sphere, if you think of the sphere as being the surface of the earth. If you want to tell somebody where you are you give latitude and longitude. If you were on the donut, you could give two angles, again an

angle around this way and an angle around that way. It would take two angular numbers to specify where you are. It's a little harder to see how you do it on these other surfaces but basically it works the same way. So these are surfaces because it takes two

numbers to describe where we are. Now something that will be important a little later, each of these surfaces as I've drawn them sits in three-dimensional space and in three-dimensional space there are boundary of the inside. So in the case of the surface of the donut,

we could have the donut or bagel itself, that's solid object. That would be a three-dimensional object with boundary, the boundary is the surface. Those objects are called handle bodies or solid handle bodies and we refer to distinguish between them, all we have to say is how many holes it has and the technical term for the number of holes is called the genus

and we apply it both to the surface and the handle body. So the torus is a surface of genus one and it bounds this solid handle body of genus one. Here's a two-holed torus, a surface which bounds a solid two-holed handle body and so on. All right, now we come

to a more difficult part of the talk. Surfaces are easy to see. How do we see the three-sphere? Well, in fact, you can't see the three-sphere, at least I don't know anybody who can see

the three-sphere and the reason is it does not exist as an object in our three-dimensional space. The only thing we have direct visual understanding of are the objects that sit in three-dimensional space and then we can see them. Well, the three-sphere doesn't sit in three-dimensional space just as the two-sphere doesn't sit in the plane or two-dimensional

space. The two-sphere sits as an object in three-dimensional space and the three-sphere sits as an object in four-dimensional space. But we can understand its properties by analogy with what's true for the two-sphere. So how do we think about the two-sphere?

One way to think about the two-sphere is through something called stereographic projection. So I'm imagining the sphere, this red sphere, sitting on this black plane, tangent or touching at one point, which I think of as the south pole. And now I imagine a light at the north pole and I look at the shadows that that light would cast. So if I have

a point on the sphere right here, the light ray coming from the north pole would hit that point and it would have a shadow on the plane. This identifies all the points in the sphere except the north pole with all the points in the plane. And technically this is a topological equivalence between the sphere minus one point and the plane.

Maybe the picture to keep in mind is you've got a ball of cheese that you want to give to somebody as a present. So you have to wrap it up. What do you do? You put it on silver paper. You wrap the silver paper around it and tie a bow. This is a little different. First of all, our paper is no longer a finite extent. It's the infinite

plane. It goes on infinitely in all directions. And secondly, the way we wrap up doesn't leave any left over to tie the bow around. In fact, as you go out to infinity in the plane, you're just converging to this north pole. So going out in any direction

in the plane simply converges up toward the north pole. Another way we can think about the sphere is a union of two disks. So I started here with two copies of a flat disk which I've separated a little bit. Think of bowing the upper one up a little bit and bowing the lower one down a little bit, continuing the bowing until they're

semi-spheres and then simply push them together in glue and you have a copy of the two-sphere. So another way to think of the two-sphere is you can take two disks, two two-dimensional disks, glue their boundaries together entirely but keeping the interior points separate and what you'll end up with is topologically the

sphere. Well, we do the same thing in dimension three. No problem. So the three-sphere can be made by taking two solid three-dimensional balls now. So these are these solid objects whose boundary is the sphere. So I have two solid three-dimensional balls and I simply glue their boundaries together

entirely keeping the interior point separate just like I did for the disk. Easy, right? You can't really see it. The other description you can think of the three-sphere as the usual three-dimensional Euclidean space going on infinitely in all directions plus one more point at infinity that would

be identified with the North Pole of the three-dimensional sphere. Now that's the simplest of all three-dimensional spaces and the others are made there many descriptions of other more complicated three-dimensional spaces but

for me today the representation I want to take is the following. You start with two solid handle bodies. Just a minute ago I was started with two handle bodies of genus zero, the ball, but I could start now with a two solid tori or two solid two-hole tori and glue their entire boundaries

together keeping the interior points separate. Well, if you had trouble visualizing how to glue the boundaries of two balls together this is even more complicated. Nevertheless, it's a completely accurate mathematical description of the way you can make other three-dimensional spaces and in

fact all three-dimensional spaces are made this way by taking some solid multi-hole torus of some genus, two copies of it, and gluing the boundaries together. This is called the handle body decomposition of the three-space and its genus is the genus of the handle bodies you've used in the

construction. So try to indicate this with as good a picture as I can. So I go back to my two-hole torus. I'm now thinking of these solid objects. I have two of them and I glue their boundaries together somehow. Now this time it turns out unlike the case of the sphere there are lots of ways to glue these boundaries together and you can get lots of different three

dimensional spaces by doing it. That's part of the complication of three-dimensional topology. Nevertheless, that's how you make all three dimensional manifolds. So as I've said we've seen the example the three sphere can be obtained by gluing together two three balls but when the

handle bodies are more complicated than balls there are lots of different ways to glue them together and so you can get lots of different three-dimensional spaces from one from two copies of one solid handle body. Most of the attacks on the topological attacks on the Poincare conjecture in the final

analysis involved trying to take this handle body decomposition where we've used solid tori with lots of holes and somehow simplify the decomposition and use smaller fewer solid handle bodies with fewer number of holes. If we get all the way down to no holes that is gluing see the

manifold is obtained by gluing two discs together we would have proved the Poincare conjecture. So you somehow had to simplify this handle body decomposition. That's what's proved to be intractable. So as I indicated earlier

where I think Poincare got his motivation for this problem for his problem was he was thinking by analogy with two dimensions where he understood everything. He was looking for some simple topological property that would characterize the three sphere. So when I say characterize I mean a property that the three sphere has that no other of these three

dimensional shapes has. So if you have a three-dimensional space and you know it has this property then you should be according to the conjecture able to conclude that it was the three-dimensional sphere. So what property should we take? Well back to surfaces. Surfaces and loops. So I've

drawn here the three first three surfaces in our list. The sphere which has no holes the torus that G equals one indicates it has one hole G for genus. Here's a two-hole torus and again you're to imagine this list goes

on and on. Well let's think about a loop on the sphere. So there's a loop on the surface of the sphere. It's pretty clear that you can take this loop and on the surface of the sphere you can drag it and shrink it down to a point. When you do that for example the trace of that shrinking might look like this disc. Just shrink it down over that disc. It's true for

this loop it's not too hard to prove it's true for any loop on the sphere. For example if I had the equator that's another loop on the sphere I could shrink it down to the North Pole just over the top or shrink it down to the South Pole over the bottom. Well the torus has on it a loop that you

cannot shrink to a point. So if I take a loop that goes around this hole no matter how I move it on the torus it'll still always encompass or be caught up in this hole and I'll never be able to continuously shrink it down to a point.

And of course on the two-hole torus I can find lots of curves like that loops like that here are two of them one that goes around this hole one that goes around that hole of course there are others is an interesting that goes around the middle band for example there are these that go around the holes this way and lots of others but the point is that there's at least one on all on this surface on this surface and then on all the other

surfaces there is at least one loop on the surface that cannot be continuously shrunk to a point the sphere does not have such a loop so this property the property that every loop on the surface shrinks to a point

characterizes the two-dimensional sphere among all surfaces now what about on the three sphere well the same argument that works for the two sphere works for the three sphere if you have a loop in the three sphere you can shrink it to a point continuously deform it to a point well

when I was a young graduate student I learned every mathematical talk should have a proof in it so here's your proof this we've seen the three sphere minus a point is our Euclidean three space well this loop is going to miss a point so we can think of it as a loop in three-dimensional space and once it's a loop in three-dimensional space it's obvious we can just shrink it down say

to the origin of three-dimensional space and here I've shown you how to do that here's a loop in three-dimensional space you just pull it by straight lines down to the origin of three-dimensional space and that's a shrinking of the loop to a point and by our identification of three-dimensional space minus a point with the sphere we can use this to

shrink all loops on the three sphere as well so the three sphere is a three-dimensional space with the same property that the two sphere has namely every loop on the three sphere shrinks to a point what about

the converse the converse statement is the Poincare conjecture the converse says if you have a three-dimensional space and every loop on it shrinks to a point then your space is the three sphere that's the Poincare conjecture

intuitively if your three-dimensional space doesn't have any holes that you can wrap a loop around like the doughnut has a hole that you can wrap a loop around if you don't have any holes like that then your space must be the three sphere so technically a three-dimensional space with the property that every loop in the space shrinks to a point is topologically

equivalent to the three sphere this is almost verbatim what Poincare said at least in the language of his day why should we believe it's true why did Poincare believe it's true well it's not completely clear he did believe it's true he didn't actually state it as a conjecture he stated it

as a question and then he said studying this question would take us too far afield well one reason to believe it's true Poincare's motivation was the as we've just seen the analogous statement is true for two spheres distinguishing the two sphere from all other surfaces as far as I'm

concerned a much more important and powerful reason it's true came 80 years later with Thurston 75 years later when Thurston formulated his more general conjecture about three-dimensional spaces the Poincare conjecture was a very special case of that conjecture and then he went on to establish his

more general conjecture in certain special cases unfortunately not a case that included the Poincare conjecture I think if you'd taken a poll of math of topologists when I first entered the subject in 68 or 70 my sense was they were split as many would have voted for the Poincare conjecture being true

as would have voted for it to be false if you asked the same question in 1985 I think you would have gotten 10 to 1 odds that it was true this is a psychological phenomenon because in the intervening 20 years 18 years there had been no advance on the Poincare conjecture but there had been advances

on this more general conjecture putting the Poincare conjecture in a more general framework and seeing that that framework whole sometimes gave a lot of credence to the fact that it was true and three-dimensional topologist since about that time have been working on three-dimensional spaces attacking many

other questions assuming that the Poincare conjecture is true this does not come as a surprise to them the fact that somebody's proved it and the way it was proved probably comes as a surprise the fact that it's true doesn't shock anyone all right that's your

lesson in topology now we switch to another field of mathematics geometry differential geometry and in particular romanian geometry this is a slightly different subject so bear with me so any three-dimensional space in fact there's nothing special about three

here on any space of any dimension one can put or impose a notion of structure let's call it a way to measure angles and links so if you have a curve in your space think of it as parameterized by time a particle moving in your space there's a notion of what the velocity

vector or tangent vector is at each point and there's a notion then of how long the trajectory is and if you have two of them and they meet at a point there's a notion of the angle between them that structure is called a romanian metric after Riemann who introduced it or at studied it the problem with this idea maybe it's not a problem some see it as a gift is that there

are huge numbers of ways to put on these structures and there's no obvious way to construct one that has some properties that you would like it to have you decide ahead of time I wanted to have this this and this property there's no obvious way to make one and since

you're working in technically what's an infinite dimensional space of possibilities finding one is that the proper with the properties you want isn't going to be easy either so now I can tell you a little bit more about Thurston's more general conjecture it says

that any three-dimensional space admits an especially nice one of these romanian metrics now in the brackets there I've put for all the mathematicians in the audience the technically correct statement you have to cut the thing in pieces in some canonical way but let's not worry about that this conjecture in and of itself as I indicated earlier suggests a completely

different approach to the Poincare conjecture because here the conclusion is not topological anymore the conclusion is geometric the conclusion is that there's a nice romanian

metric so we have a geometric conclusion instead of a topological conclusion we might try to establish that conclusion by analytic differential geometric methods and I believe that that insight was crucial to Hamilton as he was studying the Ricci flow so Hamilton proposed

a program to deal to establish these conjectures and if I can paraphrase it in this slide in the following way you start with any metric remember we had no way to pick out a good one

to begin with and there were so many possibilities it wasn't going to be reasonable to expect we could just pick a good one that we wanted but heat it up heat the manifold up and let it cool this is a euphemism but let's go on as it cools it the metric should distribute itself

over the manifold homogeneously just as heat in a metal bar will distribute itself homogeneously as it cools and if it really distributes itself homogeneously over the manifold in the end we should find this nice metric the one that we are looking for so in a nutshell that was Hamilton's

idea notice if it works it will establish not only the Poincare conjecture but also Thurston's more general conjecture because it's going to produce these nice metrics in great generality well cooling was a euphemism so let me try to say a little bit more about what's really

involved in Hamilton's equation evolution equation and to do that I have to tell you about curvature and let me try to illustrate curvature by the following phenomenon I think we're all familiar with if you have the top of an orange peel about that much of it and you put it down

on the table of course it will be bubbled up like this if you squash it flat it will tear the reason that it has to tear is that there's not enough of the orange to cover the region in the plane there's not enough stuff in the orange more technically if you look at the very

top point of the orange and you look at the amount of stuff within radius one within one unit of the top that's our cap you ask how much area do I have there and compare that with the area of the unit disk in the plane you will find you have less area in the orange

peel than you do in the disk in the plane so that's in the sense in which there isn't enough of it this property that there's less area in the orange peel than there is in the corresponding region in the plane is an intrinsic property of the metric the distance on the orange peel and it's a reflection of the fact that the metric has positive curvature this point

actually this curvature goes back to Gauss and it's called the Gaussian curvature so that's the simplest form of curvature in higher dimensions curvature is quite a bit more complicated you have

a higher dimensional space every two-dimensional direction has a curvature of the type I was just trying to describe so you get a whole bunch of curvatures at each point in different directions and they fit together into a complicated mathematical object to go has the name of a

associated to the metric and it's called the remaining and curvature tensor after Riemann who introduced it a late 19th century middle 19th century remind went on to show that this is the only local property of your metric that's intrinsic to the metric so for example Riemann

showed if you have a small piece of an indimensional space and you can flatten it out onto a piece of the plane then it has zero remaining in curvature and conversely if it has zero remaining in curvature you can flatten it out onto the plane so curvature captures is the intrinsic fundamental notion about the way metric space metrics on spaces curve

now that's not the curvature that directly enters into Hamilton's Ricci flow equation there's a related simplified curvature called the Ricci curvature it's exactly the same curvature that comes up in Einstein's equation and it comes up for the same reason so Hamilton's

cooling process what I called it before is really the idea of letting the metric evolve by requiring it to change in time so that it's time derivative is proportional to this curvature this Ricci curvature if you write this equation out in local coordinates on your space you'll

see that it looks very much like the heat equation though it's nonlinear and it's a tensor equation instead of a scalar equation nonlinearity is very important because nonlinear equations as is well known in the theory of partial differential equations often almost always develop

singularities and in fact it was trying to understand these singularities and deal with them that was the hang up in Hamilton's program for establishing these nice metrics if you let your metric flow under Ricci flow you think it's you hope it's going to go to a nice metric indeed Hamilton proved under certain circumstances that really happens but if you

get hung up on these singularities you're not going to be able to prove anything but Hamilton did study this equation for 20-25 years he established innumerable basic analytic properties of the equation in the solution and as I just mentioned he showed that in certain circumstances

you could use it to produce the nice metric for example the metric you're looking for to produce that to prove the Planck rate conjecture but we do face the issue of singularities to really use Hamilton's program to give the topological result you have to deal with the

singularities you have to analyze them and figure out how to continue the flow in spite of them and that's what Perlman did so in a series of three preprints in 2002-2003 sorry I misspelled his name here Grigory Perlman gave arguments showing how to deal with the

singularities arising from Ricci flow and how to continue the process in something he called or is called Ricci flow with surgery he then used this augmented flow this Ricci flow with surgery to prove the Planck rate conjecture and he's indicated he indicated in the second

of his three preprints how to use the same technique to prove the general Thurston geometrization conjecture if you're interested here's a link www.claymath.org that will take you to Perlman's three preprints and also the three long manuscripts the one that

Tian and I wrote that weighs in at 473 pages I hope you'll see that next summer on the summer reading beach table at your local library there's an article weighing in at over 300 pages by cow and Jew and an article on the archive by Kleiner and lot a lightweight at

200 pages but what what Perlman's preprints do and what these articles explain in much greater detail is a complete comprehensive correct argument for the Poincare conjecture and it's

now been thoroughly checked he has proved the Poincare conjecture and to me what's as exciting as the fact that the Poincare conjecture is true is been proved is that he did it using ideas and results from differential geometry and partial differential equations to solve this

purely topological problem anytime one sees this sort of interaction between disparate fields of mathematics it certainly makes me sit up and take notice and be very excited what's its significance well the obvious one it brings to a conclusion the classification

of three-dimensional spaces we've now solved this hundred-year-old problem and that's satisfying and is a great victory for Perlman and also for mathematics and as I just said the reaction the interaction between the disparate fields makes the argument all the more interesting

and important but I think that in the significant impact in the future will be not on three-dimensional spaces but rather applications of Ricci flow in other mathematical context for example it's obvious to try to apply this in four dimensions and also to apply to technically

what are known as Taylor manifolds or algebraic varieties these should be the first applications to come they're the closest to the three-dimensional questions but then there's singularity development and other geometric evolution equations that are similar to the Ricci

flow for example the mean curvature equation has many properties in common with this one so other geometric flow equations the way they develop singularities and how you treat these singularities I think we'll see lots of advances there and finally and probably more speculatively because I know nothing about applications of mathematics to physical processes

it is nevertheless true that many many physical processes are described by the same kind of evolution equation technically a parabolic differential equation that the Ricci flow is and understanding singularity development in those equations can have tremendous applications

we may look back in a hundred years and see Perlman's treatment of the singularity of Ricci flows is the beginning of an understanding much deeper understanding of how singularities develop in these equations how to handle them and treat them having untold consequences for us

in everyday life but as I say I'm not an applied mathematician so take that with a grain of salt thank you