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) *

S. Bell, B. Gustafsson, Ruminations on Hejhal's theorem about the Bergman and Szegö
kernel, Analysis and Mathematical Physics, in press. (PDF)

S. Bell, Real algebraic geometry of real algebraic Jordan curves in the plane and the
Bergman kernel, Analysis Mathematica, in press. (PDF)

S. Bell, Just analysis: The Poisson-Szegö-Bergman kernel, Journal of Geometric Analysis,
in press. (PDF)

* Research supported by the NSF Analysis Program and Cyber-enabled Discovery and Innovation
Program, grant DMS 1001701