From: "Charles Matthews" <charles@sabaki.demon.co.uk>
Subject: Re: Riemann mapping theorem
Date: Mon, 29 Jan 2001 09:23:04 -0000
Newsgroups: sci.math

brosha wrote

>What level of mathematical ability would be required before one would be
>able to understand this theorem and its application?

Leaving aside possible application, the statement of the theorem is quite
simple.  Proofs are difficult (and are for the professionals - the history
of Riemann's variational proof is instructive).

One needs the idea of "conformal mapping".  This can be made conceptual in
terms of the cartographer's dilemma: a map projection from the Earth's
surface to a flat chart cannot be completely faithful.  For example the
familiar Mercator projection distorts areas, making Greenland over-sized and
understating the area of the African continent.

A conformal mapping is one that doesn't distort angles.  We need the case
here where you map one planar area by another one.  If you draw some grid of
curves on the first, mapped to another grid of curves on the second, the
mapped curves will cross at the same angles as before.  Nothing else is
guaranteed about the mapping, which is supposed though to be 1-1 and
continuous at least (probably I should say smooth for simplicity).

The question solved by the RMT is this: which plane areas can be conformally
mapped this way by the unit disc of points of distance less than 1 from the
origin?  The answer is: any region that is topologically a candidate (namely
a simply-connected open set - where simply-connected means every loop inside
the region can contract to a point without moving outside, and open means
"no edge points included").

An application as mooted to the brain surface would be exploiting the
theorem in cases where the region to be mapped could be extremely irregular
in shape.  The distortions of scale would be large.

Charles