# Bieberbach conjecture

Ludwig Bieberbach.

The Bieberbach conjecture is a celebrated conjecture made by the German mathematician
Ludwig Bieberbach (1886–1982) in 1916, which was finally proved, after
many partial results by others, by Louis de Branges of Purdue University
in 1984.^{1} Bierberbach is infamous in the history of mathematics
because of his outspoken anti-Semitism during the Nazi era. Following the
dismissal of Edmund Landau (1877–1938) from the University of Göttingen,
Bierberbach wrote: "This should be seen as a prime example of the fact that
representatives of overly different races do not mix as students and teachers...
The instincts of the Göttingen students felt that Landau was a type who
handled things in an un-German manner."

Bieberbach's conjecture (BC) stemmed from the Riemann conjecture (RC), which
makes a claim about any region of a plane that is simply-connected (in other
words, any region, however complicated, that doesn't have any holes). RC
says there must be some function, or mapping, such that every point in the
arbitrary region is associated with one and only one point inside a circle
with unit radius. Complex functions are best suited to plane-to-plane mappings
and are often easier to work with if they can be represented as a power
series. For example, given the complex number *z*, the function
ez can be expressed as the infinite series 1 + *z* + *z*^{2}/2!
+ *z*^{ 3}/3! + ... Bieberbach guessed that there is a link
between the conditions imposed on a function by RC and the numerical coefficients
of the terms in a power series that represents the function. The BC says
that if a function gives a one-to-one association between points in the
unit circle and points in a simply-connected region of the plane, the coefficients
of the power series that represents the function are never larger than the
corresponding power. In other words, given that *f*(*z*) = *a*_{0} + *a*_{1}*z* + *a*_{2}*z*^{ 2} + *a*_{3}*z*^{ 3} + ... then |*a*_{n}| ≤ *n* |*a*_{1}| for each *n*.

### Reference

1. Branges, L., de "A Proof of the Bieberbach Conjecture." *Acta Math*.,
154: 137-152 (1985).