Use Freefind to search this site (or the web)
Site search Web search

powered by FreeFind

Tuesday, August 15, 2006

Two cool articles

I commend Mark Johnson for alerting me to two excellent articles that contain stuff about mathematics.

The first one deals with my research area: topology.

The second one is an interview with Field's Medalist Edward Witten on String Theory.

Topology: the Poincare Conjecture and Pearlman's proof

August 15, 2006

Elusive Proof, Elusive Prover: A New Mathematical Mystery

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.

“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”

In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.

Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”

But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.

“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.

Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.

Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”

William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.

“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”

Depending on who is talking, Poincaré’s conjecture can sound either daunting or 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, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.

In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.

In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: 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.

With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.

Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.

“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.

In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.

Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.

“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1986.

Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.

In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.

Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.

Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.

“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.

It was Dr. Perelman who broke the logjam. He was able to show 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. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.

Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.

Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.

In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.

“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.

Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.

Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.

Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.

But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.

“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”

Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.

In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures..

Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”

Both works were supported by the Clay institute, which has posted them on its Web site, Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (

Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.

James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.

“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.

In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.

Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.

Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”

PBS inverview with Edward Witten on String Theory

Viewpoints on String Theory
Edward Witten

The Elegant Universe homepage

Many physicists consider Ed Witten to be Einstein's true successor. A mathematical physicist at the Institute for Advanced Study in Princeton, New Jersey, Witten has been awarded everything from a MacArthur "genius grant" to the Fields Medal, the highest honor in the world of mathematics. His contributions to string theory have been myriad, including the time in 1995 when he gave the then somewhat moribund field a much-needed boost by showing how the five different variations of the theory then competing with one another actually all belonged under one umbrella. In this interview, Witten talks about how the big bang could have coughed up a string so large that it might still survive in the universe and be visible with telescopes—and other ideas to make your head spin.

Note: For a definition of unfamiliar terms, see our glossary.

Spreading out particles

NOVA: What is string theory?

Witten: String theory is an attempt at a deeper description of nature by thinking of an elementary particle not as a little point but as a little loop of vibrating string. One of the basic things about a string is that it can vibrate in many different shapes or forms, which gives music its beauty. If we listen to a tuning fork, it sounds harsh to the human ear. And that's because you hear a pure tone rather than the higher overtones that you get from a piano or violin that give music its richness and beauty.

So in the case of one of these strings it can oscillate in many different forms—analogously to the overtones of a piano string. And those different forms of vibration are interpreted as different elementary particles: quarks, electrons, photons. All are different forms of vibration of the same basic string. Unity of the different forces and particles is achieved because they all come from different kinds of vibrations of the same basic string. In the case of string theory, with our present understanding, there would be nothing more basic than the string.

NOVA: Why does something as simple as replacing points with strings make such a huge difference?

Witten: It's indeed surprising that replacing the elementary particle with a string leads to such a big change in things. I'm tempted to say that it has to do with the fuzziness it introduces. So the particle is spread out. But it turns out that everything about spacetime is a little bit spread out; it's blurred. You have to start doing some calculations to really see it. It's hard to explain it just in words or by drawing pictures.

“You enter a completely new world where things aren’t at all what you’re used to.”

Spreading out the particle into a string is a step in the direction of making everything we're familiar with fuzzy. You enter a completely new world where things aren't at all what you're used to. It's as surprising in its own way as the fuzziness that much of physics acquired in light of quantum mechanics and the Heisenberg uncertainty principle.

NOVA: In reading about string theory and in talking to people about physics in general, we hear a lot about string theory being beautiful, but what does that mean? What's beautiful about it?

Witten: Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle. It's the kind of beauty that might be hard to explain to a person from a different walk of life who doesn't deal with science or math professionally. But the beauty of Einstein's equations, for example, is just as real to anyone who's experienced it as the beauty of music. We've learned in the 20th century that the equations that work have inner harmony.

Now there must be skeptics out there who will tell you that these beautiful equations might have nothing to do with nature. That's possible, but it's uncanny that they are so graceful and that they capture so much of what we already know about physics while shedding so much light on theories that we already have.

Enter fuzziness

NOVA: Can you give us an example of something concrete that string theory does for physics that goes beyond previous theories?

Witten: In Einstein's general relativity the structure of space can change but not its topology. Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything. You can imagine a bowling ball and you can imagine a coffee cup that has a handle—the coffee cup is different topologically because there's a handle. Even if you could bend it or stretch it, as long as you don't break it, it's still got that handle, which makes it topologically different.

There was a long history of speculation that in quantum gravity, unlike Einstein's classical theory, it might be possible for the topology of spacetime to change. And it turned out in string theory in the late '80s and early '90s we actually were able to calculate examples where you could really see changes in the topology of spacetime. That was fun because it was very concrete—you could understand it pretty well. And it illustrated how the theory went beyond Einstein's general relativity as understood before in a very nice and down-to-earth way.

NOVA: How does string theory allow you to change the topology of spacetime?

Witten: Quantum mechanics brought an unexpected fuzziness into physics because of quantum uncertainty, the Heisenberg uncertainty principle. String theory does so again because a point particle is replaced by a string, which is more spread out. And even though it's a naïve statement, it leads in the right direction: when we study it more deeply, we find that in string theory, spacetime becomes fuzzy.

So imagine now we have this coffee cup. If the handle is big enough, you can see it's there. But if you had a very small handle, because of the fuzziness of spacetime, you couldn't tell if it was there or not. Then it could disappear. That fuzziness of spacetime leads to the possibility that the topology can change.

Extra dimensions required

NOVA: String theory requires 10 dimensions of space. Does that complicate the theory or does it solve problems?

Witten: Technically you need the extra dimensions. At first people didn't like them too much, but they've got a big benefit, which is that the ability of string theory to describe all the elementary particles and their forces along with gravity depends on using the extra dimensions. You have that one basic string, but it can vibrate in many ways. But we're trying to get a lot of particles because experimental physicists have discovered a lot of particles. The electron and its heavy cousins the neutrinos, the quarks, photons, gravitons, and so on. There is really a big zoo of elementary particles that you're trying to explain. Having those extra dimensions and therefore many ways the string can vibrate in many different directions turns out to be the key to being able to describe all the particles that we see.

NOVA: But why do we think that they might actually exist? We certainly can't see them.

Witten: We see light waves with our eyes, but most of the other particles take 20th-century equipment to discover them. As for the forces, electromagnetism and gravity we experience in everyday life. But the weak and strong forces are beyond our ordinary experience. So in physics, lots of the basic building blocks take 20th- or perhaps 21st-century equipment to explore.

“I would conclude that extra dimensions really exist. They’re part of nature.”

As far as extra dimensions are concerned, very tiny extra dimensions wouldn't be perceived in everyday life, just as atoms aren't: we see many atoms together but we don't see atoms individually. In a somewhat similar fashion, our experiences and our observations would average over the extra dimensions if those were small enough.

NOVA: So just because we don't see them doesn't mean the theory is wrong?

Witten: The theory has to be interpreted that extra dimensions beyond the ordinary four dimensions the three spatial dimensions plus time are sufficiently small that they haven't been observed yet. So we would hope to test the theory, conceivably directly at accelerators. I suspect that's a long shot. More likely we'll do it indirectly by making more precise calculations about elementary particles based on the existence of extra dimensions.

NOVA: Do you think extra dimensions actually exist, or are they a mathematical device?

Witten: If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways. They're beyond our ordinary experience just like atomic nuclei are. On the other hand, we don't understand the theory too completely, and because of this fuzziness of spacetime, the very concept of spacetime and spacetime dimensions isn't precisely defined. I suspect that the fuzziness of spacetime will play more of a role in the eventual answer than we understand now. [To try to picture a fourth spatial dimension, see Imagining Other Dimensions.]

NOVA: If these extra dimensions exist, does string theory offer any explanation of why there are apparently three space dimensions larger than the rest?

Witten: That's a big problem that has to be explained. As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map.

Verifying string theory

NOVA: It seems like the standard criticism of string theory is that it isn't testable. How do you respond to that criticism?

Witten: One very important aspect of string theory is definitely testable. That was the prediction of supersymmetry, which emerged from string theory in the early '70s. Experimentalists are still trying to test it. It hasn't been proved that supersymmetry is right. But there is a very precise relationship among the interaction rates of different kinds of particles which follows from supersymmetry and which has been tested successfully. Because of that and a variety of other clues, many physicists do suspect that our present decade is the decade when supersymmetry will be discovered. Supersymmetry is a very big prediction; it would be interesting to delve into history and try to see any theory that ever made as big a prediction as that.

NOVA: What are some of the other ways that string theory could be confirmed experimentally?

Witten: There are a lot of conceivable ways we could get experimental information that would help with string theory. Explorations of cosmology, studying the cosmic background microwave radiation and hopefully finding gravitational waves left over from the big bang and studying their properties are very plausible avenues for eventually testing string theory, although there isn't yet to my thinking a satisfactory theoretical understanding of what to expect.

But it's conceivable that the big bang could have produced a string so large that it would be present in today's universe and visible in telescopes, perhaps discoverable by the satellites that are now mapping out the microwave sky. If that were discovered, it would be a dramatic confirmation of the existence of strings. Still, that's a story that will develop over the next decade or two as the experiments progress and conceivably as the theory progresses.

NOVA: How likely do you think it is that string theory will be proven correct?

Witten: Well, I don't have a crystal ball. You know, the theory of neutron stars was tested and the same is true of the theory of black holes and the theory of gravitational waves. A lot of the theories that were there in the '20s and '30s that looked like they were way beyond reach were eventually tested. They were tested because there were new technologies, there were new instruments, there were newer things found in the sky. Things happened that you couldn't foresee. That's what happens in science.

“I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past.”

So when you ask me how string theory might be tested, I can tell you what's likely to happen at accelerators or some parts of the theory that are likely to be tested. But I also have to point out that part of the answer is the unknown. Just as the theory of neutron stars, black holes, gravity waves, and so many other things were tested because of things that nobody foresaw, there are just so many ways that nice surprises could happen that would lead to new advances in string theory. There are all kinds of possibilities, like literally seeing a string in a telescope if nature has chosen to be kind to us in that particular way. I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past in science. But if I could tell you what they were, they wouldn't be surprises.

NOVA: Do you think it's possible that string theory will turn out to be wrong, or at least some branch of knowledge that just isn't connected to nature?

Witten: I guess it's possible that string theory could be wrong. But if it is in fact wrong, it's amazing that it's been so rich and has survived so many brushes with catastrophe and has linked up with the established physical theories in so many ways, providing so many new insights about them. I wouldn't have thought that a wrong theory should lead us to understand better the ordinary quantum field theories or to have new insights about the quantum states of black holes.

The question reminds me a little bit of the question about interpreting fossils. When fossils were first explored 100 or 200 years ago, some people thought they were traces of past life that had survived in the rocks and others thought that they had been placed there at the creation of the universe by the creator in order to test our faith. So I guess string theory might be wrong, but it would seem like a kind of cosmic conspiracy.

NOVA: It's been said that string theory really belongs to the 21st century. Do you agree?

Witten: Back in the early '70s, the Italian physicist, Daniele Amati reportedly said that string theory was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark. How wise it was is so clear from the fact that 30 years later we're still trying to understand what string theory really is. What Amati meant was that usually the physical theory isn't developed until there are more or less the concepts and ideas in hand for making sense out of it. By the time Einstein developed general relativity, he actually knew what he was doing.

But string theory wasn't like that. The first traces appeared in 1968 with the Venetziano model. Nobody at the time had the conception that could have led to string theory in a clear way or understood what it was. It was something incredibly beautiful, a trail that people followed without understanding what it was. We've come through 30 years of remarkable discoveries, and we can see a lot of puzzles still ahead.

NOVA: Where does inspiration in this field come from for you?

Witten: You have to be open-minded because ideas come from different places. You can think about something in one way for a long time and it seems like the only way to think about it, but it really isn't. Somebody could make a suggestion that really sounds naïve. It might even be naïve, but it could have an important element of the truth in it. And it could be truth that one's overlooking. So it's really hard to state a general rule. If one could say the general rule about where to find inspiration, we would just teach it to our students and then science would be much more straightforward.


Post a Comment

<< Home

View Counter Statsfree hit counter
frontline plus