Thursday, August 17, 2006

"Elusive proof, elusive prover"

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.

From this NYTimes story about a century-old conjecture, and about Grigory Perelman, the Russian mathematician who made the key breaktrhough needed for proving it. Just how hard has it been to prove this conjecture?

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.


  1. Subrahmanya said...

    You may like this

  2. gaddeswarup said...

    Jhansi saw the news and reminded me that I once wrote a paper on the Poincare conjecture. "But that was in another country; besides nobody knows about it". It was some simplification of Papakyriokopoulos's approach and it was clear soon that it would not work.
    The best persons in India to consult about this are, I think, in Bangalore. There is one Harish Seshadri in I.I.Sc who knows some of the technology and Siddhrda Gadgil in I.S.I, Bangalore centre who is an expert on 3-maniflods. I am not even attempting to understand the solution; too many new techniques( I want to understand other things now). Even after Perelman's three papers, it took experts three and half years and a thouseand pages of hard mathematics. But Harish can probably give you an outline of the ideas. Regards,