+ John R Stallings Partly Solved a Math Puzzle and Cautioned Mathematicians

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John R Stallings Jr, who found a proof for part of the Poincare Conjecture (one of the longest-standing problems in Mathematics) died at the age of 73 last November.

Dr Stallings, born in Arkansas, graduated from the University of Arkansas and finished his doctorate in mathematics at Princeton in 1959.  After a fellowship at Oxford, he taught at Princeton and then became a professor at Berkeley in 1967.

His work largely involved geometry and topology, the study of fundamental properties of shapes.  He later applied that knowledge to the field of geometric group theory, using geometric and topological concepts to prove theorems in algebra.  

The Poincare Conjecture, proposed by Henri Poincare in 1904, says essentially that any shape that does not have any holes, and that fits within a finite space, can be stretched and deformed into a sphere.

Dr Stallings was far from the first mathematician to tackle the Poincare Conjecture; he wasn’t even the first to find a partial solution. 

Stephen Smale of Berkeley was the first.  In 1960 he proved the conjecture for surfaces of five dimensions and higher.

Dr Stallings, then a postdoctoral fellow at Oxford heard this news, but not the details.  He took a swipe.

In a few days, he had come up with his own proof, which worked for dimensions seven and higher.  Less sweeping than Smale’s, Stallings’s proof applied to a slightly different version of the conjecture.  It employed different mathematical techniques.

Barry Mazur, a Harvard mathematician, says “That tells you more about the nature of the problem.  This is a very, very deep geometric problem and every fact of it is not only interesting, but has ramifications.  Different proofs bring out different aspects of a problem.” 

The four-dimensional case was proved in 1982, and in 2003, a Russian mathematician, Grigori Perelman, completed a proof for the thorniest case, of three dimensions.

“How Not to Prove the Poincare Conjecture”

But in 1965, in a paper titled “How Not to Prove the Poincare Conjecture,” Dr Stallings confessed that he had sought, and failed,  to find a final, complete proof.

The paper — about his non-proof — began humorously: “I have committed — the sin of falsely proving Poincare’s Conjecture.  But that was in another country; and besides, until now no one has known about it.”

He explained his errors in that paper.  Then he offered the reason for his confession: it was “in hope of deterring others from making similar mistakes.”

He ended on this musing note:

I was unable to find flaws in my “proof” for quite a while, even though the error is very obvious.  It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong.  Techniques leading to the abandonment of such inhibitions should be cultivated by every honest methematician.

sole source: obituary of John R Stallings, by Kenneth Chang, in the NY Times on 1/20/09.   www.nytimes.com  

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