| In the 7 September issue of Science is an article, "Surprise Proof of an Old
Conjecture," written for a general audience but a good deal better
than what has hit the newspapers.
The conjecture proposed by Ludwig Bieberbach in 1916 is that if an analytic
function of the form
2 3
z + a z + a z + ...
2 3
never assumes any value more than once on the unit disk, then the absolute
value of the kth coefficient a is never more than k for all k.
k
The Science article outlines the history of the work on this conjecture. The
proof itself, by a hitherto undistinguished Purdue mathematician, Louis de
Branges, has not yet been published, but the Soviets who verified the proof
(no American mathematician would read it) have circulated it within the
mathematical community.
To quote the article, "The importance of the conjecture is mainly that
it has proved so difficult and that so much useful mathematics was developed
as researchers tried to resolve it. Mathematicians agree that it is too soon
to say whether de Branges's methods or the very fact that he resolved the
Bieberbach conjecture will have significance for mathematics in general. But
the lack of any immediate practical applications does not diminish the
importance of the result in the mathematics community."
Val
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| For further reading on the conjecture and related problems, see "Conformal
Invariants - Topics in Geometric Function Theory", by Lars Ahlfors.
(McGraw-Hill 1973). Some very pretty mathematics here that I haven't
thought about in years...
Just another ex-mathematician -- Jerry
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