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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

146.0. "Bieberbach Conjecture" by HARE::STAN () Thu Sep 13 1984 02:23

What is the Bieverbach Conjecture?
It has something to do with analytic functions of complex variables.

A recent newspaper article reported that the conjecture has been
proven; but the article failed to say what the conjecture is.
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146.1SUPER::MATTHEWSSun Sep 16 1984 06:2730
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
146.2RANI::LEICHTERJWed Sep 19 1984 02:536
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