##
A new improved Riemann Mapping Theorem

*Steven R. Bell **

The aim of my current research is to find a
way to transform complicated regions in the

plane to simpler regions so that problems of
analysis can be solved via finite algebraic

methods.

The Riemann Mapping Theorem states that any
sub-region in the plane without holes can

be conformally mapped to the unit disc. Thus,
it is possible to deform a map of Indiana in

such a way that it becomes a disc and all the
roads cross at the same angles they did on

the original map. This is a useful transformation
for running numerical analysis because the

boundary of the disc is simple, whereas the
boundary of the state is complicated.

The unit disc is an example of a double quadrature
domain: the average of an analytic function

both with respect to area measure and with respect to
boundary arc length measure give the

value at the center.

The new version of the mapping theorem applies
to regions with or without holes and states that

it is possible to conformally map such regions to
double quadrature domains that are *close* to

the original region. Consequently, it is possible
to make subtle changes in a region so that many of

the classical problems of analysis can be solved
via algebra and finite methods, as on a disc,

rather than more difficult limiting methods. Double
quadrature domains also have the desirable feature

that many of the objects of analysis can be "zipped"
down to a small data set consisting of finitely

many numbers.

This research sprouted from an Undergraduate
Research Project of Purdue undergraduate Zack

Sylvan that Professor Bell supervised. The two
went on to extend the work to its present form

with Swedish quadrature domain expert, Björn Gustafsson.

The papers:

S. Bell, B. Gustafsson, and Z. Sylvan, *Szegö
coordinates, quadrature domains, and double*

quadrature domains, Computational Methods and
Function Theory **11** (2011), No. 1, 25-44.
(PDF)

S. Bell, *An improved Riemann Mapping Theorem and
complexity in potential theory,*

Arkiv for matematik **51** (2013), 223-249.
(PDF)

S. Bell, *The Dirichlet and Neumann and Dirichlet-to-Neumann
problems in quadrature,*

double quadrature, and non-quadrature
domains, Analysis and Mathematical Physics,

**5** (2015), 113-135.
(PDF)

* Research supported by the NSF Analysis Program
and Cyber-enabled Discovery and Innovation

Program, grant DMS 1001701