Tuesday, December 26, 2006

Poincare, Perelman and the Universe


SCIENCE magazine's 2006 Breakthrough of the Year has been justifiably awarded to Grigori Perelman's proof of the Poincare conjecture. Kea is very good at writing about maths. Perelman's achievement culminated years of work in isolation, the barriers of skepticism, and others claiming credit when they realised he was right. His work has great implications for theories of the Universe.

Topology is the study of surfaces that can undergo stretching. It applies to our physical world via surface integrals and boundaries. We learned at a very young age that a donut is Genus One, topologically the same as a teacup. Henri Poincare, the father of modern topology, conjectured in 1904 that a 3-dimensional space with a "trivial" fundamental group must be a hypersphere, the boundary of a sphere in 4-dimensional Space. Poincare's conjecture defied proof for nearly a century.

Richard Hamilton proposed a solution based upon a "Ricci Flow." In this theory, regions of high Ricci curvature tensor R_{ab} would diffuse out, in a manner analogous to heat flow, until a surface of equal curvature (a sphere) was achieved. Extending the Ricci flow to 3 dimensions seemed insurmountable. Perelman met Hamilton while in the US, returned to his native Russia in 1995, then spent 7 years trying to solve the problem.

In November 2002 Perelman began posting outlines of a proposed proof on the internet. Though some mathematicians realised he was on to something, there was some skepticism. For one thing, bits of the proof were incomplete. Did people ignore Perelman, say he didn't have a theory, and write unfair things about him? Some may have; but Bruce Kleiner, John Lott, John Morgan and Gang Tian aided in completing and publishing Perelman's work. By Spring 2006 the proof was complete, and the world realised what had been accomplished.

The Poincare Conjecture applies directly to cosmology. It says that the only possible shape for our 3-dimensional Universe is the surface of a 4-dimensional sphere. Einstein calculated that if the Universe contained enough density, gravity would curve it into such a sphere. This was the Einstein Static Universe, but he quickly realised that the same gravity would cause a sphere to collapse. He did not consider that expansion would prevent a collapse, for there was little evidence for an expanding Universe. To support this spherical space, Einstein added a "cosmological constant," a purely hypothetical repulsive force.

Alexander Friedmann and Georges Lemaitre independently found solutions to the Einstein equations for an expanding Universe. Working atop Mount Wilson in California, Edwin Hubble and Milton Humason found that redshift of galaxies was related to distance, indicating that our Universe was expanding. In a 1931 visit to Mount Wilson, Einstein conferred with Hubble and peered through the telescope. The world's most famous scientist happily conceded that Lemaitre and Hubble were right. He dropped the cosmological constant, calling it his "greatest blunder." In a 1932 paper with Wilhelm de Sitter, Einstein expressed preference for an asymptotically expanding Universe of "critical" density (Omega) = 1.

Poincare and Perelman indicate that a sphere is the only possible shape. This shape can only be maintained without collapsing if there is a certain density. That density is not "critical" but the stable density. If the Universe were not of the stable density , then matter would be created via pair production until that density was reached. That is why baryons are exactly 4.507034% of the Universe.

As you have doubtless heard, Perelman has prematurely retired from mathematics and public life. He has left his job at the Steklov, a mathematical institute in St. Petersburg. Though he is qualified for the Fields Medal and the Clay Prize, he has refused them. He is said to be tired and disappointed by the lack of ethical standards in mathematics. (This year some unnamed mathematicians published a paper implying that they completed the proof first.) Perelman sounds quite modest and reasonable when quoted in the New Yorker:

"I can't say I'm outraged. Other people do worse. Of course, there are many mathemticians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest."

"It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated."

Grigori Perelman's achievement is immensely important. His behaviour shows that he was motivated by the challenge of solving a problem; not by fame or prizes. It is shameful that petty behaviour of others has driven a real talent like Perelman out of the field. With the harsh treatment given to new ideas and lone researchers, his actions are completely understandable.

5 Comments:

Blogger Kea said...

Happy New Year, Louise. Good to see a post on cool maths! Thanks.

1:53 PM  
Blogger L. Riofrio said...

Thanks. Happy Year 007 to you too! I miss your posts on maths, which are much more comprehensive than mine.

7:14 PM  
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