It started with the fact that I got my first job at Cornell and Jim Eells was there. He and Joe Sampson had just finished this beautiful paper on harmonic maps where they used a parabolic flow to deform a map into a surface of negative Riemannian curvature to a harmonic map. And I had mentioned briefly with Jim: "Well, maybe you can deform other structures?" Then I came back to that later, but meanwhile, I did a bunch of other stuff. And how did I get onto the Ricci flow? Well, that has to do with Jimmy Carter and the oil crisis. I had bought a nice water ski boat on Lake Cayuga, but it was taking all this money to fill the gas tank. I had to get something going for my NSF proposal and I started thinking about the Ricci flow, and I thought about it for a while and it sounded like it might go somewhere. So, I sent that in for my NSF proposal and this was the first time I got rejected and I think I know why - because after Yau talked to me, he said: "Oh, when I first heard your idea about the Ricci flow, I thought you were a madman." I thought that was the nicest compliment!

I didn't know much geometry at the time, and I was trying to imitate Eells-Sampson where they have a Dirichlet energy. So, I wanted to take an integral of the first derivative of the metric squared and minimise that. And that wasn't working because I finally found out that in any sort of invariant sense, the first derivative of the metric was zero (the covariant derivative). But then one day I had this bright idea - what if there were such a thing? What would I do next? And I figured I'd integrate by parts and get the d/dt of the metric is something that would be two derivatives of the metric. And I say, "Oh, the only thing that's intrinsic about two derivatives in metric is the curvature." And then I thought. "Well, which curvature?" You got scalar curvature, Ricci curvature, Riemannian curvature ... And the scalar curvature didn't have enough indices, the Riemannian curvature had too many, and the Ricci flow was just right. It looked like the metric. So, I wrote down d/dt of the metric is the Ricci curvature. And I computed out the evolution of the Riemannian curvature, and I realised it was a backwards parabolic equation. So, I thought, "Okay, I'll just put in d/dt of the metric is minus the Ricci curvature." And then I put a two in to get rid of the unpleasant one half and started working on that.

And then I knew that the Riemannian curvature was evolving via a nice parabolic type equation and the Ricci curvature, and I made a curious decision to start working on three dimensions instead of two. And one person was to say, well, you should start on two and if you can't do that, you should give up. But see, I was always quite vain, and I thought, well, I should do something better than just reprove the 100-year old geometrisation or 100-year-old Ricci manifold thing. So, I had read this thing in Eisenhart that said that in three dimensions you can capture all the curvature from the Ricci curvature. And I thought, "Oh, well, that sounds like a good place to start." Which turned out to be a lucky guess, because positive Ricci curvature in three dimensions is in some ways stronger than positive scalar in two dimensions. It kind of has more constraints to it. So, then I started working on it and the real breakthrough came one day when I had a girlfriend who was teaching at Gettysburg College, and the only thing to do in Gettysburg was to walk around the cemeteries - and I think it was raining that day and you couldn't even do that. So, I just cranked on and I got two good estimates and came up with that good theorem and I kind of got it started. I mean, back then it seemed nearly impossible that you could actually do Poincaré with it. But, you know, a lot of success in maths is being lucky, being in the right place at the right time, and trying the right thing.

Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds). The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the centre).

Since we cannot directly visualise objects in n-dimensional space, Poincaré asked whether there is a test for recognising when a shape is the three-sphere by performing measurements and other operations inside the shape. The goal was to recognise all three-spheres even though they may be highly distorted. Poincaré found the right test (simple connectivity, see below). However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere.

In the last century, there were many attempts to prove, and also to disprove, the Poincaré conjecture using the methods of topology. Around 1982, however, a new line of attack was opened. This was the Ricci flow method pioneered and developed by Richard Hamilton. It was based on a differential equation related to the one introduced by Joseph Fourier 160 years earlier to study the conduction of heat. With the Ricci flow equation, Hamilton obtained a series of spectacular results in geometry. However, progress in applying it to the conjecture eventually came to a standstill, largely because formation of singularities, akin to formation of black holes in the evolution of the cosmos, defied mathematical understanding.

Perelman's breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. He achieved a complete understanding of singularity formation in Ricci flow, as well as the way parts of the shape collapse onto lower-dimensional spaces. He introduced a new quantity, the entropy, which instead of measuring disorder at the atomic level, as in the classical theory of heat exchange, measures disorder in the global geometry of the space. This new entropy, like the thermodynamic quantity, increases as time passes. Perelman also introduced a related local quantity, the $L$-functional, and he used the theories originated by Cheeger and Aleksandrov to understand limits of spaces changing under Ricci flow. He showed that the time between formation of singularities could not become smaller and smaller, with singularities becoming spaced so closely - infinitesimally close - that the Ricci flow method would no longer apply. Perelman deployed his new ideas and methods with great technical mastery and described the results he obtained with elegant brevity. Mathematics has been deeply enriched.

The differential equation that was to play a key role in solving the Poincaré conjecture is the Ricci flow equation. It was discovered two times, independently. In physics, the equation originated with the thesis of Friedan, although it was perhaps implicit in the work of Honerkamp. In mathematics it originated with the 1982 paper of Richard Hamilton. The physicists were working on the renormalisation group of quantum field theory, while Hamilton was interested in geometric applications of the Ricci flow equation itself. Hamilton, now at Columbia University, was then at Cornell University.

On the left-hand side of the Ricci flow equation is a quantity that expresses how the geometry changes with time - the derivative of the metric tensor, as the mathematicians like to say. On the right-hand side is the Ricci tensor, a measure of the extent to which the shape is curved. The Ricci tensor, based on Riemann's theory of geometry (1854), also appears in Einstein's equations for general relativity (1915). Those equations govern the interaction of matter, energy, curvature of space, and the motion of material bodies.

The Ricci flow equation is the analogue, in the geometric context, of Fourier's heat equation. The idea, grosso modo, for its application to geometry is that, just as Fourier's heat equation disperses temperature, the Ricci flow equation disperses curvature. Thus, even if a shape was irregular and distorted, Ricci flow would gradually remove these anomalies, resulting in a very regular shape whose topological nature was evident. Indeed, in 1982 Hamilton showed that for positively curved, simply connected shapes of dimension three (compact three-manifolds) the Ricci flow transforms the shape into one that is ever more like the round three-sphere. In the long run, it becomes almost indistinguishable from this perfect, ideal shape. When the curvature is not strictly positive, however, solutions of the Ricci flow equation behave in a much more complicated way. This is because the equation is nonlinear. While parts of the shape may evolve towards a smoother, more regular state, other parts might develop singularities. This richer behaviour posed serious difficulties. But it also held promise: it was conceivable that the formation of singularities could reveal Thurston's decomposition of a shape into its constituent geometric atoms.

Hamilton was the driving force in developing the theory of Ricci flow in mathematics, both conceptually and technically. Among his many notable results is his 1999 paper, which showed that in a Ricci flow, the curvature is pushed towards the positive near a singularity. In that paper Hamilton also made use of the collapsing theory mentioned earlier. Another result, which played a crucial role in Perelman's proof, was the Hamilton Harnack inequality, which generalised to positive Ricci flows a result of Peter Li and Shing-Tung Yau for positive solutions of Fourier's heat equation.

Hamilton had established the Ricci flow equation as a tool with the potential to resolve both conjectures as well as other geometric problems. Nevertheless, serious obstacles barred the way to a proof of the Poincaré conjecture. Notable among these obstacles was lack of an adequate understanding of the formation of singularities in Ricci flow, akin to the formation of black holes in the evolution of the cosmos. Indeed, it was not at all clear how or if formation of singularities could be understood. Despite the new front opened by Hamilton, and despite continued work by others using traditional topological tools for either a proof or a disproof, progress on the conjectures came to a standstill.

Such was the state of affairs in 2000, when John Milnor wrote an article describing the Poincaré conjecture and the many attempts to solve it. At that writing, it was not clear whether the conjecture was true or false, and it was not clear which method might decide the issue.

It was thus a huge surprise when Grigori Perelman announced, in a series of preprints posted on ArXiv.org in 2002 and 2003, a solution not only of the Poincaré conjecture, but also of Thurston's geometrisation conjecture. The core of Perelman's method of proof is the theory of Ricci flow.

In this paper the author surveys some of the basic geometrical properties of the Ricci flow with a view to considering what kind of singularities might form. The Ricci flow was introduced by the author in his celebrated work on deformation of metrics the by Ricci flow on compact 3-manifolds with positive Ricci curvature [J. Differential Geom. 17 (2) (1982), 255-306]. It has been found to be a useful tool in the study of Riemannian geometry, both for compact and for open manifolds. The idea is to study the evolution of a Riemannian metric along its Ricci curvature. Many interesting results have been obtained using the Ricci flow, and it is most desirable for the author to have written a survey in the field where he initiated and continues to make contributions. This article is a beautiful example of how to use techniques in differential equations and Riemannian geometry to study manifolds.

The paper starts with some examples of how singularities might be formed under the Ricci flow. Short-time existence, derivative estimates and long-time existence are reviewed. Then the author discusses the convergence of the Ricci flow on dimension two, three and four, as well as on Kähler manifolds. It is known that the Ricci flow is invariant under the full diffeomorphism group and any isometries in the initial metric will persist as isometries in each subsequent Riemannian metric. The author illustrates the idea by considering a Riemannian metric on a 3-manifold where the torus group $T^{2} = S^{1} \times S^{1}$ acts freely as a subgroup of the isometry group. It is shown that the Ricci flow of such a metric exists for all time and converges to a flat metric. The author discusses an estimate on the derivatives of curvature from local conditions and the Harnack inequality for the Ricci flow. He gives a discussion on the limits of solutions to the Ricci flow. To do so, a lower bound on the injectivity radius at a point is obtained in terms of a local bound on the curvature. The author also gives a control on the distance between two fixed points along the Ricci flow.

Then the author shows that some asymptotic properties of complete Riemannian manifolds are preserved by the Ricci flow. Define the aperture $\alpha$ of a complete non-compact Riemannian manifold by $\alpha =\limsup_{s\rightarrow\infty} \Large\frac{\text{diam } S_s}{2s}$, where diam $S_{s}$ is the diameter of the sphere with radius $s$. It is shown that for a complete solution to the Ricci flow with bounded curvature and weakly positive Ricci curvature, the aperture $\alpha$ is a constant. If the Ricci flow on a complete non-compact Riemannian manifold has bounded curvature and the Riemannian curvature is approaching zero at infinity at time zero, then this remains true along the Ricci flow. Suppose we have a complete solution to the Ricci flow on a complete Riemannian $n$-manifold with bounded curvature and weakly positive Ricci curvature, where the Riemannian curvature is approaching zero at infinity; then the asymptotic volume ratio $\nu$ defined by $\nu =\lim_{s\rightarrow\infty} \Large\frac{\text{Vol } B_s}{s^n}$ is constant along the Ricci flow, where Vol $B_{s}$ is the volume of the ball with radius $s$. Another asymptotic property that is preserved by the Ricci flow is the asymptotic scalar curvature ratio $A$ defined by $A =\lim_{s\rightarrow\infty} Rs^2$, where $R$ is the scalar curvature. The author shows that for a complete solution to the Ricci flow with bounded curvature defined for $−∞ < t < T < ∞$, if the initial Riemannian metric either has positive curvature operator or is Kähler with weakly positive holomorphic bisectional curvature, then the asymptotic scalar curvature ratio A is constant.
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The paper is well written and delightful to read. Many examples are worked out in detail to illustrate the idea. It gives a rather complete account and up-to-date references on current research on the Ricci flow.

A large crowd gathered for this year's AGM and justifiably so for, after the society business, the audience were treated to two talks on a recent hot topic in mathematics, namely the Ricci flow. This is a subject which has reached even a lay audience in the past year as it played the central role in Grisha Perelman's recent resolution of the Poincaré conjecture. Perelman proved the Poincaré conjecture by completing a program begun by Richard Hamilton over twenty years ago. Hamilton was the inventor of the Ricci flow and we were lucky enough to have him here as a speaker. His was the second talk but, since it was of an expository nature, it perhaps makes sense to relay it first.

Speaking with his customary élan, Hamilton began by describing the Ricci flow as a 'heat equation' for a Riemannian metric. The idea is that evolving a metric in the direction of minus its Ricci tensor should smooth out the curvature, similar to the effect of diffusion on uneven distributions of heat. This principle is borne out by the fact that many geometric quantities (e.g., curvature, geodesic lengths etc.) evolve by diffusion or diffusion reaction equations under Ricci flow. One might hope, then, that on running Ricci flow, an arbitrary initial metric might converge to a metric of constant curvature. Hamilton went on to explain that for compact surfaces, where constant curvature metrics are always known to exist, that this is precisely what happens, provided one rescales the metric with time. This rescaling is necessary because, for example, positively curved metrics shrink, hence become more positively curved and so shrink even faster. On the other hand, negatively curved metrics grow, albeit at a slower and slower rate.

In three dimensions the picture is much more complicated and singularities form in the metric. One example is that of a 'neck pinch'. This singularity forms in a manifold which has a large cylindrical region, whose 25 cross section is a round sphere, but which is slightly negatively curved along its length. The differing signs in the curvature compete for dominance but, in some cases, the positive curvature wins and the round sphere shrinks to a point in finite time causing a singularity as the manifold 'pinches' along its 'neck'.

Hamilton then gave a beautiful description of his method for understanding singularity formation. He said he thinks of Ricci flow as 'running a movie' of the evolution of a manifold; as a singularity forms you both zoom the movie in to the point where the singularity is forming and simultaneously slow it down. Repeatedly doing this gives, in the limit, another 'movie' which is another Ricci flow and which models the singularity formation. For example, in the neck pinch singularity described above, the singularity model is a regular (product) cylinder collapsing to a line under the Ricci flow. These singularity models have a special property: they are ancient solutions to the Ricci flow; because of the time rescaling the limiting solution has existed for all past times. Such solutions are necessarily very special because, as Ricci flow spreads out curvature and this been going on indefinitely, the resulting metric must be very evenly curved.

Next Hamilton explained why understanding precisely which singularities can form is the key to proving the Poincaré conjecture. The idea is to take a simply connected three-manifold with an arbitrary metric and run Ricci flow, hoping it will converge to a round sphere. After a time you will encounter a singularity which you hope to recognise as some sort of neck pinch; for example, a singularity whose model is a collapsing cylinder. If that is the case, then you can cut the manifold across the neck, glue two caps onto the holes and try to carry on the flow. Hopefully, after a finite time and a finite number of surgeries, you will be left with a collection of round spheres. This means you can recognise your original manifold as made up of the connected sum of several spheres and so homeomorphic itself to a sphere.

The main obstacle to this plan was that, pre Perelman, one of the potential singularity models did not have a cylindrical region for one to cut across. This model is known as the 'cigar', for it resembles the product of a cigar shape and a line. One of Perelman's major advances in the study of the Ricci flow was to give an in-depth description of what happens at regions of very high curvature, where the singularities are forming. Using this analysis he is able to rule out this bad sort of singularity; as Hamilton put it, 'Perelman showed that it was close, but no cigar!'