The Poincare conjecture

Has the Poincare Conjecture been solved? Possibly. Read this recent news report about a new proof by an obscure Russian loner, Grisha Perelman. The Conjecture is one of the famous Millennium Problems in mathematics.

“This is arguably the most famous unsolved problem in math and has been for some time,” said Bruce Kleiner, a University of Michigan math professor reviewing Perelman’s work.

Here is the clearest statement I can find of what the whole thing means:

To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes — the sphere — there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe.

Or try this:

[the] work has huge implications for our understanding of partial differential equations. PDEs (as they are known in the trade) are the mainstay of physics and engineering. Mazur notes that physicists and engineers use PDEs to model everything from the flow of water to the buildup of heat in aircraft engines. “I would expect this work to have enormous applications in many fields of science,” he says.

There may also be applications for scientists studying DNA…Some kinds of DNA wrap themselves into knot formations that can be insanely difficult to decipher. But Mazur says Thurston’s classification [referring to related work] may provide a way to calculate the exact nature of any knot – so in theory it could be used to work out the structure of knotty DNA molecules.

The upper reaches of mathematics can often seem absurdly detached from life down here on planet earth, but Mazur points out that you can never know where things might lead. He cites the case of James Clerk Maxwell. In the late 19th century Maxwell worked out the equations of electromagnetism. “At the time it would have been easy to write off Maxwell’s ideas about invisible forces as a mystical abstraction,” Mazur says. But Maxwell’s work laid the foundations for the development of radio, and hence the communications revolution. Every time we turn on the TV or pick up a cellphone or log onto a WiFi system we are reaping the rewards of Maxwell’s equations.

Another bottom line: Perelman will receive a million dollars if his result stands up. Alex says this is another win for bounty hunters!