Proposal Number: 1006309
Performing Organization:SUNY Stony Brook
NSF Program: Geometric Analysis
Principal Investigator: Bishop, Christopher
Proposal Title: Analysis of conformal and quasiconformal maps
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Panel Summary #1
Proposal Number: 1006309
Panel Summary:
This is one of the best proposals considered by the panel. The PI has
made important contributions to harmonic measure and Kleinian groups;
this work was deep and difficult. Recently he has moved to the numerical
computation of conformal maps and the computational complexity of the
algorithms. The proposal is thorough and ambitious; it shows great
breadth over the area of complex analysis. The PI tackles interesting
problems and is expected to make significant progress on a number of
problems proposed. The work done under the grant will have a real
impact on complex analysis.
The broader impact is an important feature of the proposal. The PI has
made contributions to numerical techniques used in conformal mapping.
He works with both undergraduate and graduate students. He has had
several PhD students with more in the pipeline. In addition, the PI
has organized a conference bringing together experts from various areas
of numerical conformal mapping.
The panel places this proposal in the "Highly Recommended for Funding"
category. This proposal was read to the panel and the panel concurred
that the summary accurately reflects the panel discussion.
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Review #1
Rating: Excellent
REVIEW:
What is the intellectual merit of the proposed activity?
The PI proposes to continue his investigations into
numerical conformal mapping (various techniques), into Brennan's
conjecture and related matters, into DLA and related matters concerning
analytically defined fractal curves, and into the Koebe conjecture
and related issues. The PI has a strong track record in these areas, and
he has a large tool chest to work with. Thus investigations tend not
to narrowly focused since he has a lot of tools to work with.
What are the broader impacts of the proposed activity?
Successful research in the area can have applications to physics
and applied math, e.g, Mumford has asked relevant questions whose
solutions may bear on research in visit. The PI has a bag full of
easier to hard problems which he uses to attract undergrad and
grad students.
Summary Statement
The PI is a wonderful researcher who works on a broad range of very
interesting problems in subjects related to conformal mapping. It is
not easy to find significant new results in these old areas, but
he finds them. The PI is constantly
evangelizing both undergrad and graduate students with a rich
variety of easier to very hard problems to attract them to work
in the subject.
Rating: E
Rating: E
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Review #2
Rating: Multiple Rating: (Excellent/Very Good)
REVIEW:
What is the intellectual merit of the proposed activity?
The intellectual merits include a better understanding of the theory
of conformal and quasiconformal maps and ways that ideas from
classical analysis and computational geometry can improve existing
algorithms for numerically computing them.
As stated in the proposal, conformal mapping remains the most effective
way to solve various problems in potential theory and is an important
technique in meshing, graphics and pattern recognition (as illustrated
in the work of David Mumford).
The PI's work on these problems uses ideas and techniques from several
different areas (classical analysis, hyperbolic geometry, computational
geometry and numerical analysis) and serves as a bridge between these ?elds.
1) This increases the number of people who become aware of interesting
problems, and
2) increases the number of ideas that might be applied to solve them.
The PI has used material from previous proposals in courses for both
graduates and undergraduates that emphasize the interconnectedness of the
areas listed above.
What are the broader impacts of the proposed activity?
Summary Statement
The PI outlines 25 different problems to study. The proposal is quite
thorough, and contains many ideas about how to tackle these problems.
It is an ambitious proposal, but the PI has seems to have a stellar
track record in this area of mathematics.
Grade: Between Very Good and Excellent
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Review #3
Rating: Excellent
REVIEW:
What is the intellectual merit of the proposed activity?
The PI plans to study geometric properties of
conformal and quasiconformal maps. The PI has an extremely strong
record from prior NSF support with an impressive number of papers
appearing in Annals and Acta. He is well known for his outstanding
work on harmonic measure and Kleinian groups. In more recent years,
the PI has studied algorithms for the numerical computation of
conformal maps including deep and difficult works on a
linear time algorithm, conformal welding, and the closeness of the
initial guess for the Schwarz-Christoffel algorithm via Delauney
triangulation. The proposal includes a number of interesting problems
related to the convergence of the Schwarz Christoffel algorithm using
initial guesses due to Davis and to Driscoll and Vavasis. The latter
two researchers won a prize from SIAM for their method, which is quite good in
practice but lacks a convergence proof. The PI's own ''fast mapping
method'' works in linear time. Numerical implementation of the latter
could be quite useful. In the process of analyzing some of these
methods, he uses the iota map of a region to the disk which is
approximately conformal. He proposes a modification to the iota map
with possible application to the well-known Brennan conjecture. Other
interesting problems are concerned with the conformal spectrum of
harmonic measure, DLA and a generalization of the Koebe conjecture.
What are the broader impacts of the proposed activity?
Improved conformal mapping would have impact on other sciences that
use conformal mapping. For example, D. Mumford and coauthors have
applied conformal mapping to the study of pattern recognition and
computer vision (how to find the best representative in a database
from the shadow of an object). ThePI demonstrated the wide
interest in conformal mapping by hosting a workshop attended by
applied mathematicians, computer scientists and mathematicians devoted
to these problems. He has supervised 3 PhD students in the
last ten years, and has more in the pipeline. He proposes a number of
computational problems that are appropriate for undergraduate research
projects.
Summary Statement
This proposal is head-and-shoulders above all other proposals
I've seen this year. The past work is truly top-notch, the proposed
problems are difficult, but reasonable given the inventiveness and
creativity the PI has shown in past work.