Sunday, August 20, 2006

Dennis Overbye answers questions about the Poincaré conjecture

It turned out that Grigory Perelman, a Russian genius who has made the crucial steps to prove the famous problem in mathematics, lives with his mother as an unemployed from about $50 a month. He doesn't care about his fair fraction of the $1 million prize from the Clay Institute. What a cool guy.

Dennis Overbye had a nice article about the Poincaré conjecture. On Friday, he answered many questions of the readers about this topic. The answers are somewhat impressive. They're comprehensible for many readers but full of Thurston's geometrization conjecture, correspondence with Edward Witten, string theory, and many other things. What a remarkable difference from the lousy science journalists such as Susan Kruglinski, Sharon Begley, Robert Matthews, John Cornwell, John Horgan, Paul Boutin, Dietmar Dath, and dozens of others who prefer to write about bitter, non-technical, vague, and deeply flawed attacks against science rather than science itself: medieval bigots who dream about witch-hunts against the scientific heretics.

At the beginning, Dennis Overbye et al. correct the myth that rabbit is topologically a sphere. Of course, its surface is a torus because of the tunnel between the mouth and the "hole below its tail", to put it politely. I would say that genus one is probably an underestimate because there are two more holes in the nose. At least, this would be true for humans but after I saw some pictures of rabbits, I am less sure about the "g=3" lemma for rabbits.

More seriously, the article also discusses whether the proof will have implications in physics because of the obvious connections with perturbative string theory that we will review below.

It is stated that the mathematicians are 100 years ahead of the physicists. Well, it is often the case: they used to work on abstract nonsense such as the Lie groups well before the physicists would find such things relevant. But recently, the comparisons became more subtle and the physicists are often 10 years ahead of the mathematicians - for example in the case of the problems related to mirror symmetry.

More generally, I would say that the validity of a mathematical statement usually has much more far-reaching implications for physics than its detailed proof. In this case, everyone was more or less sure that the conjecture was true anyway. The methods may be important for physics, too. However, the methods used to prove this particular conjecture, while very fascinating from a mathematician's viewpoint, are closely related to tools that have been known in physics for quite some time.

Ricci flows

Strings can propagate on many spacetimes - backgrounds - but you can show that only if the spacetime satisfies the equations of motion (such as the Ricci-flatness of the geometry), the corresponding string theory is consistent (conformally invariant). Still, you can consider strings propagating on various spacetime manifolds that are not Ricci flat - such as various three-dimensional manifolds, for example the three-sphere. If you also considered the string interactions, you would also have to have a critical spacetime dimension but at the level of a trivial worldsheet, it is not necessary.

For Ricci-non-flat spacetimes, the physics on the worldsheet is not scale-invariant. It depends on the scale. If you magnify the worldsheet, it has the same effect as a subtle change of the geometry of the three-manifold. The corresponding dependence of the geometry on the length scale on the worldsheet is essentially called the Ricci flow. The spacetime geometry plays the role of infinitely many coupling constants of the two-dimensional theory living on the worldsheet and the Ricci tensor at a given point is the beta-function for the coupling constants described by the metric tensor at the same point.

The Poincaré conjecture states that everything that looks like a three-sphere at the level of homotopy is homeomorphic to a three-sphere. The Ricci flow is a prescription that can take a manifold that is homotopically a three-sphere - such as the surface of a four-dimensional rabbit without mouth - and deform it so that it eventually becomes an ordinary three-sphere. The Ricci flow - in which you always change the metric by epsilon times the Ricci tensor - makes the manifold increasingly smooth and uniform and one can probably see that the ordinary symmetric three-sphere is the only possible endpoint of such an evolution.

Ask Hamilton, Perelman, Zhu, Cao, and Yau and they will transform this hand-waving into a rigorous mathematical proof. More precisely, they have already done so. Once again, string theory - in the case, its rudimentary methods - is connected with important insights in mathematics. These connections of string theory with deep ideas in mathematics don't prove that string theory is the right theory of our Universe, but they certainly show that string theory is at least a very good way to keep your brain working on sufficiently deep, interesting, and relevant mathematical questions.

And because I believe that newer theories of physics are increasingly deep from a mathematical viewpoint, such connections between string theory and mathematics show that we are going in the right direction, at least as far as one important coordinate in the space of conceivable theories suggests.


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  2. "These connections of string theory with deep ideas in mathematics don't prove that string theory is the right theory of our Universe, but they certainly show that string theory is at least a very good way to keep your brain working on sufficiently deep, interesting, and relevant mathematical questions."

    I see that these attachments and fusions is enough evidence that string/M-theory is the right theory of our multi-flux-warped octonionic black hole throat. I am self-composed ;-)