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Scrunched-up dimensions untangled

British physicist Stephen Hawking may claim that extra dimensions provide the key to understanding the "grand design" of the universe, but it's Chinese-American mathematician Shing-Tung Yau who actually figured out how those extra dimensions work.

In his new book, "The Shape of Inner Space," Yau and his co-author, Steve Nadis, touch upon the work that led to the discovery of theoretical "Calabi-Yau spaces" — and the cosmic implications of multidimensional geometry.

The typical representation of a Calabi-Yau space looks like twisted web of a crumpled-up piece of paper. There's something elegant about its look — in fact, Calabi-Yau paperweights were voted the most popular gewgaw for holiday giving in last year's Cosmic Log Geek Gift Guide contest. But these shapes aren't just abstract art: String theorists believe that every single point in our universe is actually a compactified Calabi-Yau space in six dimensions.

Why would they think that? It's because the best theory they've been able to come up with for the universe's grand design requires 10 dimensions to make all the mathematics come out right. Because we can only perceive three dimensions of space and one dimension of time, they suggest that the other six dimensions curled up into near-nothingness when our universe took shape.

Yau was the one who worked out the mathematics for the curled-up spaces. At first, he was trying to disprove a conjecture about complex geometry that was proposed by another mathematician named Eugenio Calabi. But then Yau came around to the view that Calabi was actually right, and in 1976 he published the proof that laid the groundwork for the concept of Calabi-Yau spaces. String theorists eventually seized upon the concept in their explanations for the universe's 10-dimensional structure. Later, the theorists threw in an extra dimension to make it 11, because that helped make sense out of five different subtheories. It's that 11-dimensional view of the universe, known as M-theory, that Hawking is touting as the groundwork for the grand design.

"The Shape of Inner Space" delves deeply into the math behind M-theory. It also traces Yau's life story, which started with his birth in China in 1949 and and Hong Kong and eventually brought him to Harvard. There are plenty of career highlights along the way: In 1982 Yau won the math world's most prestigious prize, the Fields Medal, for proving the Calabi conjecture. In 2006 he played a role in the tale of Russian mathematician Grigory Perelman's refusal to accept the Fields Medal for proving the famous Poincare conjecture. And Yau is also known for his high-profile criticism of the Chinese educational system and scientific establishment.

You can read all about that and more (including Yau's early yen for kung-fu novels) in an extended interview on Discover magazine's website. During my telephone chat with Yau this week, we focused more on the cosmic perspective. Here's an edited transcript of the Q&A:

Cosmic Log: Recently there's been a lot of talk about multiple dimensions. Stephen Hawking also recently wrote a book, I hear ... and he talked about how M-theory was really the secret to the "grand design." But in your book, you point out that the grand design really has a lot to do with geometry. So a lot of people wonder what's going on with multiple dimensions that we can't directly perceive. What difference does it make for our understanding of the universe?

Shing-Tung Yau: We physicists always try to bridge two important discoveries in physics together. One is general relativity, and the other is quantum mechanics. There were many efforts to discover such a unification field, [but none was successful] until string theory came along. So far that's the only consistent theory. In order to make quantum mechanics consistent with general relativity, there is no other choice but to make space-time be 10-dimensional. On the other hand, we do have to try to understand the space-time that we perceive. So we make six dimensions very, very tiny.

We still see the four-dimensional space-time that we experience in general physics, special relativity and all that. This six-dimensional space is what we call "internal space." There are a lot of models for this, but the most effective and useful model is Calabi-Yau space, where We can do all the calculations. And in fact, if we can choose the right Calabi-Yau space, we should be able to calculate the properties of the particles in the universe. But the trouble is, right now, we have many candidates for these calculations. One day, if we have the fundamental physics that can lead us to know how to calculate such geometry, and if we pick the right geometry, we will be able to calculate the masses of all the particles in the universe.

In any case, a great number of important discoveries have been made, in terms of philosophical principles as well as mathematics. So as a mathematician I'm very excited about this.

Q: Some people say that our universe could be just one among 10 to the 500th power possibilities. I guess that has led some people to say that the study of the structure of our universe can do nothing more than catalog one of those possibilities, and that there's no particular reason why the laws of physics work one way rather than another way. The only answer would be, "Well, this is just the way it is." Just as one cave is more habitable than another because of the way it's laid out, it just so happens that the geometry of our universe makes it more suitable for life, and there's no use trying to figure out why that is.

A: That's one point of view. I do not necessarily take that point of view. The fact that we can already see that there is a finite number of possibilities is very exciting. There could have been an infinite number of possibilities for space-time. But right now we know of only a finite number, and that should be considered a good starting point.

Other people have called this the "cosmic landscape." Maybe that's what they think, but that's not what I believe. There must be some more fundamental principle in order to choose the right geometry to tell what the universe is supposed to be.

Q: What do you think are the most promising experimental avenues for determining the geometry of the universe?

A: If supersymmetry can be found, that would be very exciting for string theory. The whole theory is based on this supersymmetry. It may be possible to find supersymmetric partners in the Large Hadron Collider. If supersymmetry exists, most people would start to think that string theory has a strong foundation.

First, the experimental data have to come out. Then it will be time to look for more concrete statements about the geometry, and hopefully based on that, we can try to make more concrete statements about the geometry of the universe. Right now we don't have the experimental data, so we cannot say much. On the other hand, we can do a lot of theoretical calculations, using computers, and the calculations come out to be very beautiful. They have led to many important discoveries in math itself.

Some of the discoveries in mathematics are extraordinary. The calculations have solved problems that we didn't know how to solve for 100 years.

Q: Do you have particular solutions in mind?

A: Mathematicians have been looking at polynomial equations. You want to know how many solutions are there. We didn't know how to do this for a long, long time. In some important cases, string theory has inspired us to find the right formula. The formula is extremely complicated. If we didn't know much about string theory, I believe that even now, we still would not be able to find such a formula. But based on the inspiration and the intuition from string theory, we found the formula, and we also proved it. We proved it independently of whether string theory itself is right or wrong.

Q: So string theory already has yielded important scientific advances, even if it hasn't unraveled the deepest mysteries yet.

A: That's right. Many, many important problems in mathematics and geometry were solved, and that's why many mathematicians are actually looking to string theory, because it provides intuitions that we did not have before.

Q: The mathematics of string theory is beautiful, but there's also a beauty to these depictions of Calabi-Yau spaces. How do you feel about those representations? Are they simplications of what you've done, or do you see them as tributes to what you've done?

A: Those pictures are grossly simplified versions of what we know. We cannot really draw a Calabi-Yau space, because it is a six-dimensional space. We use computers to make some slice of these Calabi-Yau spaces. What we see is far away from being the original picture of what we think in our heads, but it's useful, I think, even to look at the slice. In our original philosophical description for the space, we can do a huge amount of things — in algebra and number theory, many problems can go back to this space.

I constructed this space eight years before string theory became an important subject. I just thought that it was exotic and beautiful. I couldn't imagine that any other space could be more beautiful. We went on to many discoveries in math. And then, to my great surprise, the string theorists came along, and all these great people in string theory came to ask me how this space was constructed. I was really excited by the possibility that such spaces could be used to understand nature.

Q: In the past, you've talked a lot about China's status in science, math and technology, so I can't let you go without asking at least one question about global competitiveness and the worries that people have about American innovation.

A: There is absolutely no question that America will be the leader for a very long time. I think it would take a long time for any Chinese universities to be as good as our top universities. It may take 50 years, it may take a century. So America's science and technology is really far ahead of China's. I don't think Americans should worry about it. But on the other hand, of course, there are a lot of good Chinese scientists coming up, and they are very competitive. I wish that they would be as original as our American colleagues.

So it's very interesting to have a good group of Chinese scientists competing with us. They will make us better. I always feel it is important to have challenges — to have people competing with us. Otherwise we'll think that we are just great and will not evolve toward a better scientific world. So I don't think there'll be any negative effect from this competition. It will be positive — for both sides.

Update for 7 p.m. ET Oct. 3: Toward the latter part of the book, Yau and Nadis take the concept of a 10-dimensional (or 11-dimensional) universe into some speculative frontiers. For example, what would happen if those rolled-up dimensions were to "un-scrunch" themselves? That phenomenon, known as decompactification, would be a very bad thing for the universe as we know it. It's the kind of stuff that science-fiction nightmares are made of.

Where would you look for evidence of extra dimensions? On the widest scale, you might look for anomalous areas in the cosmic microwave background (the so-called "big bang afterglow") that could be seen as signs that our bubble universe has bumped into a neighboring bubble. You might also look for evidence of cosmic strings at work. Yau and Nadis say cosmic strings could be detected by instruments to come such as the Gaia satellite or the Large Synoptic Survey Telescope.

On the smallest scale, you might look for supersymmetric particles in the Large Hadron Collider, as Yau said in the Q&A, or you could look for special types of particles leaking back and forth through extra dimensions. Kaluza-Klein particles would have unusual masses or spins due to their sojourn through the multidimensional realm.

More about extra dimensions:


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