Proposal number: 1906259
Project Title: Quasiconformal Constructions in Analysis and Dynamics
PI: Christopher Bishop
Program: Geometric Analysis
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Panel Summary
Intellectual Merit
The panel was impressed with the caliber of questions posed
in this proposal and recognized the PI's strong record as
a ``problem solver'' with excellent taste and great skill.
One panelist ranked the proposal at the top of those for
which he served as Primary Reviewer. The panel counted as
a strength that the problems varied in depth with some
suitable for graduate students and also some appropriate
for undergraduate projects. There was some concern that
in some cases the proposal did not adequately set the
problems in context or provide sufficient motivation.
Some of the questions posed seemed to be well known in
the field rather than to reflect a novel vision.
Broader impacts
Strengths: The panel was impressed that the PI mentioned
by name his undergraduate student mentees and had specific
problems listed for undergraduate projects. The panel counted
as positive his plan to write a book.
Weaknesses: The panel felt that the number of graduate
tudents supervised was low, considering the seniority
of the PI.
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Review 1 Summary
The proposal discusses some deep problems in geometric
function theory and complex dynamics which range from
combinatorial geometry, to transcedental dynamics and
to harmonic measure if random motions. Some of the
problems are very interesting (e.g. whether Riemann
surfaces can holomorpically embed in the complex 2-space)
and, if solved, will significantly advance the mathematics
in these fields. The proposed techniques come from
quasiconformal analysis, geometric measure theory
(Question 11) and complex analysis. The PI is a
leading expert in the field of quasiconformal analysis
with very important contributions. In his prior NSF support,
the PI produced research which appeared at the most
prestigious journals and was presented at the ICM 2018.
Finally, most problems are accompanied by a plan of
approach. Thus, it is clear that the proposed research
will be very likely fruitful.
The broader impacts of this proposal are in two directions:
computational infrastructure and educational impact.
For the first part, the PI expects that results related
to meshing problems of the proposal will improve modeling
of surfaces and lead to advances in manufacturing. For the
second part, the PI intends to involve graduate and
undergraduate students in his research, train graduate
students in pure and applied mathematics through a workshop
in computational geometry and produce a book on conformal
fractals. These are very good broader impacts.
Summary Statement
This is the strongest proposal from those that I reviewed.
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Review 2 Summary
The PI has had remarkable success applying his constructions
involving "true trees" to the dynamics of entire functions
and proposes a number of interesting questions and conjectures
continuing this line of research. He also proposes to
continue his investigations regarding computational geometry
and non-obtuse triangulations, as well as his recent work
on certain 4-manifolds. The proposal concludes with problems
about diffusion limited aggregation, and a list of topics that
are perhaps suitable for undergraduates. The proposal is
lacking indications how the PI intends to attack the proposed
questions. However, the PI has an outstanding track record in
solving hard problems so that progress on some of the proposed
problems is very likely.
The PI's algorithms for planar meshing might have applications
in computational questions around surfaces. He has organized a
graduate workshop, is planning to organize another one next year,
and is involved in starting a geometry lab for undergraduate
research.
Summary Statement
This is a strong proposal by an accomplished complex analyst who
has made many fundamental contributions to his areas of research.
Among all proposals I reviewed this year, I rank it in the top third.
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Review 3 Summary
The PI, known for his work in "hard anaylsis" (e.g. work
with holomorphic functions, quasi-conformal mapping, and
harmonic measure), is turning his attention to "quasi-conformal
folding" a new method which lends itself to application in
dynamics. His earlier work on 2-dimensional optimal meshing
will be extended and will give rise to faster algorithms and
more accurate algorithms in computer graphics. The PI has the
talent and the mindset to bring to bear on issues in computer
graphics the new "hard analysis" results he obtains.
The PI's research serves to create a bridge between four
mathematical fields (hyperbolic geometry, classical analysis,
numerical analysis and computational geometry) and thus
encourages and enables more interdisciplinary research among
them. The PI serves as his departments coordinator for high
school research project and is involved in many high quality
synergistic activities involving many levels of the mathematical
community.
Summary Statement
The broader impact involves activities that reach into many
levels of the mathematical community: high school, undergraduate,
graduate, postgraduate and researcher. The PI will address
a variety questions, new questions and also long standing
open questions that the PI has the technical prowess to
answer. An outgrowth of his research are projects that could
be for undergraduate and graduate students.