The cause of death was prostate cancer, reported the University of California, Berkeley, where Dr. Stallings had been a member of the faculty since 1967. His death was not announced by the university until last Monday.

The conjecture, proposed by Henri PoincarŽ in 1904, essentially says 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 PoincarŽ Conjecture. He was not even the first to find a partial solution. That was Stephen Smale of Berkeley, who in 1960 proved the conjecture for surfaces of five dimensions and higher.

When Dr. Stallings, then a postdoctoral fellow at Oxford, heard news of Dr. Smale's accomplishment but not the details, he took a swipe at it. Within a few days, Dr. Stallings had come up with his own proof, which worked for dimensions seven and higher.

Even though Dr. Stallings's proof was in a sense less sweeping than Dr. Smale's, it applied to a slightly different version of the conjecture and employed different mathematical techniques.

"That tells you more about the nature of the problem," said Barry Mazur, a Harvard mathematician. "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."

In a paper titled, "How Not to Prove the Poincaré Conjecture" in 1965, Dr. Stallings confessed that he had sought to find a final, complete proof. The paper began humorously: "I have committed -- the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now no one has known about it."

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.

Born in Morrilton, Ark., John Robert Stallings Jr. graduated from the University of Arkansas in 1956 and finished his doctorate in mathematics at Princeton in 1959. After the fellowship at Oxford and a stint teaching at Princeton, Dr. Stallings became a professor at Berkeley in 1967.

Dr. Stallings retired in 1994 but continued to supervise graduate students through 2005.

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

The latter work won him the Frank Nelson Cole Prize in 1970, which was awarded by the American Mathematical Society once every five years for the top work in algebra.

Dr. Stallings is survived by his longtime companion, Marjorie Mulcahy.

In his 1965 paper about his nonproof of the Poincaré Conjecture, after he had explained his errors "in hope of deterring others from making similar mistakes," Dr. Stallings ended on a 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 mathematician."

By KENNETH CHANG

Published: January 18, 2009 © New York Times