From Polyakov to Perelman: History of the Ricci Flow (and a Millennium Prize)

March 19, 2010

It has just been announced by the Clay Mathematics Institute, that G. Perelman has won the Millennium prize for “resolution of the Poincare conjecture”.

Previously I’ve written about the Poincare conjecture and Perelman’s celebrated proof of it, which was based on the so-called Ricci flow equation. Now let me talk more about how this story is related to physics, and where this equation first appeared in history.

The Ricci flow is an analog of the heat equation for geometry – it is a diffusive process acting on the metric of a Riemannian manifold. It is a special case of so-called geometric evolution equations, that describe the deformation of the metric of Riemannian manifolds driven by their curvature.
Actually, what is usually called the Ricci flow equation, constitutes a very simple equation. Let M be a n-dimensional compact manifold with positive definite metric g_{ij}. The Ricci flow equation is

dg_{ij}/dt = -2 R_{ij}(g)

where t is an external parameter defined on some interval and i,j =1, 2, …, n; the Ricci tensor with components R_{ij} is calculated from the Riemann tensor as the trace R_{ij}=R^k_{ikj}.

In physics it was introduced by Friedan in the 1980’s in the renormalization-group (RG) flow of two-dimensional sigma models. Such sigma models describe the propagation of strings in curved backgrounds. A variant of the Ricci flow appeared already in the 1970’s in the renormalization group studies of non-linear sigma models; the original computation was performed by Polyakov who considered the class of O(n) sigma models with target space metric that of the round sphere on S^{n-1} for n>2. Polyakov (1975) originally considered the O(n) nonlinear sigma models. Here the radius of the sphere is the only parameter of the theory, and as such, it serves as the inverse of its coupling constant, i.e. R ~1/g. These two-dimensional models are not conformally invariant in the quantum regime, as it was found that the coupling constant runs with respect to the world-sheet renormalization scale parameter \Lambda to lowest order in perturbation theory.

The computation of the perturbative beta function was later extended by Friedan (1980) to encompass two-dimensional non-linear sigma models with generalized coupling given by the target space metric g_{ij} of arbitrary Riemannian manifolds. Here

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}-\frac{\alpha'^2}{2}R_i^{\, klm}R_{jklm}+...

where the dots stands for higher order curvature corrections, and the RHS is a perturbative expansion in \alpha'. 
The first-order term in \alpha' is

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}.

Now, to see that the Ricci flow is essentially the same as RG flow, we identify the “time” t with the energy scale \Lambda through t=\log(\Lambda^{-1}), then \Lambda \sim e^{-t} and so large positive t corresponds to small energy \Lambda: flow in t is therefore a flow from the ultra-violet (UV) regime to the infra-red (IR) regime, or from small to large distance scales. The calculation above of Friedan led to the development of the sigma model approach to string theory, since the requirement of conformal invariance for the world-sheet quantum field theory gives rise to the vacuum Einstein equations in target space, R_{ij}=0 plus higher order corrections. Further generalizations can be considered by including other massless modes of the string, such as the dilaton \phi and the anti-symmetric torsion field B_{ij} to the action of the world-sheet sigma model.

Perelman’s proof (2003) of the Poincare  conjecture is related in an interesting way to this physical model. Perelman’s proof uses an “entropy” functional {\cal F}. Let {\cal M} be the space of smooth Riemann metrics on M. Define {\cal F}:{\cal M}\times C^\infty (M)\rightarrow R by

{\cal F}(g,f) =\int_M\left(R+|\nabla f|^2\right)e^{-f}dV

this is basically the Einstein-Hilbert action in general relativity, where f is a “dilaton” field; roughly speaking we go from Perelman to Polyakov by setting f=\phi. To my knowledge, the full extent of the connection between string theory and Perelman’s proof has not really been understood yet.

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The Abel Prize won by John Griggs Thompson and Jacques Tits

March 27, 2008

Today, the name of the winner of the 2008 Abel Prize was announced by the president of the Norwegian Academy of Science and Letters. The winner is John Griggs Thompson (U. of Florida) and Jacques Tits (College de France). Their work is in algebra and group theory, and in particular in the classification of finite simple groups.

The Abel Prize is the mathematicians analog of the Nobel Prize. From the homepage of the Abel Fund:

“The Niels Henrik Abel Memorial Fund was established on 1 January 2002, to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.”

The prize was first proposed to be part of the 1902 celebration of 100th anniversary of norwegian mathematician Henrik Abel’s birth; however, for various historical reasons, the prize was first to be awarded beginning in 2002, on the 200th anniversary of Abel’s birth.

The former laureates are: 2007:  S. R. Srinivasa Varadhan, (Courant Institute of Mathematical Sciences, New York University). 2006:  Lennart Carleson (Royal Institute of Technology, Sweden).
2005:  Peter D. Lax (Courant Institute of Mathematical Sciences, New York University). 2004:  Michael F. Atiyah (University of Edinburgh) and Isadore M. Singer (MIT). 2003:  Jean-Pierre Serre (Collège de France).


Dr. Thompson and Dr. Tits will hopefully have a nice trip to Oslo in late May, when they will receive their prizes from King Harald of Norway. Rumor has it, that Tits will buy a gift for his wife, and Thompson will pay for his familys trip to Oslo.

A good review (by Ron Solomon) of the problem of classifying simple finite groups can be found here [pdf].


Perelman, the Poincare Conjecture and a Fields Medal

August 22, 2006

Poincare’s conjecture (1904) is related to the nature of three-dimensional manifolds, or spaces. It is in many ways deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.

(The conjecture is fundamental to topology, and incidentally one of the $1 million Millenium Prize Problems awarded by The Clay Mathematics Institute of Cambridge, Massachusetts).

In three dimensions, it is hard to discern the overall shape of something; so for example, when we envision the surface of a sphere, we are really seeing a two-dimensional object embedded in three dimensions. To understand the problem it is therefore easier to start with two dimensions.

The topology of two-dimensional manifolds or surfaces has been well understood since the 19th century. In fact, there is a simple list of all possible smooth compact orientable surfaces. Any such surface has a well-defined genus g ≥ 0, which can be described intuitively as the number of holes; and two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have
the same genus.

This means that all such two-dimensional surfaces are classified by a single integer, g. A surface of genus zero is topologically a sphere (an apple), a surface of genus one is a torus (a doughnut) and a surface of genus two is, well a dougnut with two handles (or whatever you want to call it), and so on.

In the case of two dimensions, it is easy to see how the conjecture should be understood: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

The corresponding question in higher dimensions is much more difficult. The most basic example of such a manifold is the three-dimensional unit sphere, that is, the set of all points (x,y,z,w) in four-dimensional Euclidean space which have distance exactly 1 from the origin: x^2 + y^2 + z^2 + w^2 = 1.

A distinguishing feature of the two-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere, or that “you can’t lasso a two-sphere”.

In mathematical terms, the Poincare Conjecture can be phrased as follows: If a compact three-dimensional manifold M has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M is homeomorphic to the sphere S^3 ?

From the first, the apparently extremely simple nature of this statement has led mathematicians to overdo it. For about 100 years, several mathematicians have tried to prove the conjecture. But they all failed. Well, except from one, it seems.

Grigori Perelman’s proof is based on a study of the geometry of the infinite dimensional space consisting of all Riemannian metrics on a given smooth three-dimensional manifold.

The metric g_ij on a Riemannian manifold determines the length between points according to ds^2 = gij dx^i dx^j. (From the first and second derivatives of this metric tensor, one can compute the Ricci curvature tensor R_ij , and the scalar curvature R; for example, Euclidean flat space has R_ij = R = 0, while a usual three-dimensional sphere of radius r, has Ricci curvature R_ij = 2g_ij /r^2 and scalar curvature R = 6/r^2).

Perelman’s proof is based on the study of the Ricci flow, i.e. the solutions to the differential equation:

dg_ij/dt = −2R_ij ,

where t is like an external time-parameter. In other words, the metric is required to change with time so that distances decrease in directions of positive curvature.

Perelman’s succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process; however, the fates of more complicated objects were previously thought to be problematic. As the Ricci flow progressed, kinks and neck pinches – or singularities – could appear, pinch off and even shrink away. Perelman has showed that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started (his papers at arXiv.org).

For his work, Dr. Perelman has just been awarded the Fields Medal (an analog of the Nobel Prize) and as said before, also the Millenium Prize; but Perelman seems to have refused to accept the award (news.google.com). What a coolheaded guy 😉

The other winners of the Fields Medal are, Terence Tao, Wendelin Werner, and Andrei Okounkov.

In the context of string theory, the Ricci flow represent the world-sheet renormalization group (RG) flow; Perelman’s proof might have important implications for understanding space-time singularities.

To be continued…


Why is Mathematics Supposed to be so Difficult?

December 19, 2005

(This article was originally posted on blogger.com on Jul 7, 2005)

This post is a response to a recent article by Tine Byrckel in the Danish newspaper Information about “The Mystery of Mathematics”.

Tine’s main concern is two-fold. First of all why so few Danish women (at high-school level and up) are interested in mathematics and secondly, why math in high-school is presented as a mysterious subject, which definitions, theorems etc. you should just accept – as they are – since they are not motivated in any way, and that the core ideas in math and the important intuition about math are not given much time, if any at all.

Why women tend to show little interest in mathematics and physics, at least in Denmark and many other countries, I cannot really explain, but only guess about. One could imagine that this is an effect of both biological and cultural factors (there could be other reasons, but I will not discuss them here). Globally it is correct, that women are underrepresented in both math and physics (and in a number of other “hard sciences”). However, locally there can be big differences. I would estimate that at most 10% of those who are studying physics at the Niels Bohr Institute in Copenhagen are female, while the corresponding number in, for example, one of the biggest universities in Lisbon of Portugal is much closer to 50%. As far as I remember,  the percentage of women studying physics at the universities of Portugal and Italy is quite close to 50%. But the number of those female students, which end up graduation with a Ph.D., is probably much less than 50%. This could suggest that it is in many ways a cultural phenomenon. With respect to the biological side, one could imagine that the males and females different ways of thinking could manifest itself in the way that men seem to be more attracted to math and physics than women are. However, actually I don’t believe that this question is something that can be settled on the basis of science. What we are seeing is – in broad terms – only an instant view, and we don’t know how the situation is going to develop in the future, even with or without any governmental “regulations”. As an example, has the number of women attending medical school at the universities of Denmark increased quite tremendously in recent years, and there are now more females than males at these schools. 

I would like to add some comments about how math and physics (at least in Denmark) are taught in both high school and at the universities, but some of these comments might also apply to the situation in the US, for example. Generally I believe that the quality of the teaching of math and physics in most high schools is simply too low, and that could explain why many women tend to choose other fields of study before entering college. That there are, from my point of view, also many problems with the way math is taught at university, only makes the situations worse. But why should males react differently than females to the way math and physics is taught in high school?

I am criticizing the education in math and physics at high schools for several reasons. Generally, the teachers are simply not good enough. I suspect, among many other things, that the reason behind this is, that it is the lesser successful students at university, which later choose to teach in high school. I’m just saying this based on my own experience and based on my experience with private teaching of high school students while I was studying at the Niels Bohr Institute.

Here I would like to add a personal story. I’ve just recently taught a female first-year high school student in math and physics. The physics book was (not to my surprise) written by their own teacher, and contained a surprising high number of errors. Here I’m not talking about misspellings, misprints or the like, but about actual misconceptions of fundamental physics or wrong and imprecise definitions. In this book vectors are, as always, used in connection with Newton’s laws of motion combined with a plenty of fine illustrations of cars, bricks and much else, which are being affected by forces, which again are represented by vectors. But it is not explained any place in the book what a vector really is, only that you can add two or three of them, but not that you can multiply them with real numbers, subtract them and define an (inner) product of vectors. And there is not much help in the companion book from the math course, since here there are simply no chapters about vector calculus!

So the teaching is also poor since the books are too poor, and obviously not written or edited by experts. The books are not bad just because of the things I mentioned above, but also because they try to learn you lots of useless stuff. Classical geometry and trigonometry (about triangles etc.) might have been useful to be taught about around 50 years ago, but to teach about a huge number of theorems about the angles of triangles, the ratio between different angles, the ratio between the edges of similar triangles, cosine- and sine-relations and so on, simply has no or very little usability today, at least that is how the situation looks like for a student. Of course there are a few definitions and theorems, which could prove good to know (for example, you cannot escape knowing what cosine and sine means, if you are studying physics or engineering), but those fundamental things can be written down on a couple of pages compared with the actual 20-30 pages filled with silly theorems and definitions which no high school student possibly could remember.

I am of course aware, that there are an enormous number of applications of trigonometry. Fields which make use of trigonometry include astronomy (and hence navigation), optics, analysis of financial markets, medical imaging, architecture and so on, but the central point is not whether trigonometry is relevant, but rather what part of trigonometry you should choose to teach high school students about and how you should make them interested in the subject. I don’t know who is responsible for choosing the literature, which is to be taught from in high school, but I don’t think that those who are, have had any contact with the university community (or fundamental research) for many years, if ever.

All this said, I can understand why many students find mathematics mysterious. However, I don’t think that mathematics is so mysterious, no more mysterious than you make it look and feel like. This also depends a lot on how you actually teach math. It happens quite often at university that the teacher in a certain branch of mathematics (algebra, classical analysis, group theory and so on) present the subject as one huge pile of definitions (or axioms) and theorems which you are then supposed to spend a lot of time proving. This is simply not the way a mathematician thinks about math, or the way that the field develops. In my view, the good teachers are recognized by their ability to explain and visualize the central ideas behind the math.