Panel Summary:
This is an excellent proposal. The PI is one of the most original and
imaginative researchers in his area. The problems proposed, on conformal
collapsing maps, Kleinian groups and connections of 3-dimensional geometry
to numerical analysis of conformal mappings, are very interesting and
innovative, some of them extremely difficult. He has an excellent record.
His work covers a wide area, using very different and new ideas.
This proposal is placed in the top group and recommended for funding.
Review 1:
What is the intellectual merit of the proposed activity?
The PI proposes to study geometric properties of conformal and quasiconformal
mappings. The project is concerned with constructing conformal collapsing
maps, the geometry of Kleinian limits sets, and connections between
3-dimensional hyperbolic geometry, computational geometry and numerical
conformal mappings. These are important aspects of the area. In the
proposal, the PI proposed a number of questions (actually twenty five
questions or conjectures ), which include some well known conjectures
such as Koebe"s circle domain conjecture and the Ahlfors conjecture on
Kleinian group. Most of the conjectures are explicitly stated. The PI
is an active researcher with a very good track record. The results from
the prior NSF support in the past three years contain more than ten
papers with very good results such as the interpolation theorem for
extension of homeomorphism and the disproof of Rudin"s orthogonality
conjecture. The PI has made substantial contributions to the field.
What are the broader impacts of the proposed activity?
The PI included in the proposal a couple of questions suitable for
undergraduate research projects. The proposed work will promote the
integration of research and education at both graduate and undergraduate
education. It might also have applications to other disciplines such as
modeling fluid flow and mesh generalization.
Summary Statement
The proposal seems to have too many questions. However, I still give
it an overall ranking of excellent. The PI is a very active researcher
and well-qualified for the work. Substantial progresses from this
proposal are expected. I strongly recommend funding.
Review #2
What is the intellectual merit of the proposed activity?
The first part of the proposal involves characterizing the circle
homeomorphisms that are "conformal weldings"; i.e., maps that interpolate
between the interior and exterior Riemann mapping for a closed Jordan curve.
These conformal weldings have applications, for example, to complex dynamics.
Bishop plans to apply an old method of Moore of collapsing sets in the
plane to this problem. He also proposes to apply this method to the Koebe
conjecture on infinitely connected domains. His planned approach to these
two problems seems promising, and Bishop has obtained preliminary results
on the first problem.
(I am not able to comment on the second part of the proposal on limit sets
of Kleinian groups.)
In an applied direction, Bishop used the "medial axis" of a domain to
construct faster algorithms to compute the Schwartz-Christoffel transformation
for polygonal domains (as mentioned in Prior Results). This work and
the proposed Problem 23 are based on experimental observations by applied
mathematicians. I find the methods here to be more interesting than the
results, and I expect these methods will result in new applications as
well as theoretical results. (The proposed problems in this area are
less precisely stated than in the other parts of the proposal.)
What are the broader impacts of the proposed activity?
The proposed research on the medial axis, which was stimulated by
experimental observations, has relevance to computer vision and computational
geometry
It seems that there are many potential topics here for Ph.D. research.
(Bishop also mentions possible undergraduate research projects, but there
is no indication that he has previously involved undergraduates in research.)
Summary Statement
Bishop has an excellent track record and the proposal states a number
of interesting problems, for which the methods of his past research look
promising. This proposal should be given priority for support.
Review #3
What is the intellectual merit of the proposed activity?
Bishop is one of the most innovative researchers in geometric complex
analysis and related topics. Has an impressive record of research, and
the proposal gives strong indications of this continuing in the future.
His most recent work covers a exceptionally wide spectrum of topics:
conformal welding, interpolation of conformal mappings, introduction of
methods of three dimensional hyperbolic geometry to conformal mappings,
applications to numerical analysis, several interesting works on Kleinian
groups, Rudin's orthogonal conjecture, conformal dimension and properties
of quasiconformal mappings.
The proposal has three different research programs, each of them important
and promising. The first on conformal collapsing suggest a unification
and an approach to several questions in geometric analysis in one complex
dimension. The unifying aspect comes through understanding the conformality
properties of Moore's theorem, and I found this part of the proposal
especially innovative. Many of the problems suggested here are notoriously
difficult, but even success with partial results would have an impact.
The second theme considers Kleinian groups and is continuation of Bishop's
earlier work. Several central conjectures such as Ahlfors' conjecture are
listed here, but it is not clear if there are new methods to approach
this set of problems. On the other hand, the last theme, on applications
of ideas from the three dimensional topology to the numerical analysis
of conformal mappings and Schwarz-Christoffel formula I find again very
innovative and promising for a major breakthrough in the near future.
What are the broader impacts of the proposed activity?
Bishop has very wide range of interests, from topology to analysis and
numerical applications. He interacts and does research with specialists
from many different areas of mathematics, and this certainly will have
a strong impact in the mathematical community. Particularly promising
in this respect is his new approach to numerical analysis of conformal
mappings using methods arising in three dimensional hyperbolic geometry.
Summary Statement
Bishop is one of the most innovative researchers in geometric complex
analysis and related topics.
He has a very wide range of interests, from topology to analysis and
numerical applications. He interacts and does research with specialists
from many different areas of mathematics. The proposal indicates a very
innovative and original approach to several problems in geometric complex
analysis.