Proposal Number: 1305233
NSF Program: Geometric Analysis
Principal Investigator: Bishop, Christopher
Proposal Title: Quasiconformal methods in analysis, geometry and dynamics
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Panel Summary
Intellectual Merit:
The intellectual merit of this proposal is outstanding.
Not only does the proposal contain an impressive breadth
of project topics and techniques (including geometric analysis,
combinatorics, and speeds of algorithms), but the PI provides
concrete plans of attack for each of them. The PI has a superb
on several of his proposed problems.
Broader impacts:
The broader impacts of the proposal are good, but could be improved.
The PI does work with a number of graduate students, however the PI
is encouraged to reach out more to the mathematical community,
beyond mentoring graduate students. The panel encourages the PI
to take more of a leadership role in his field.
Results from prior support:
The PI had support during 6/10 through 5/13 on an NSF grant, during
which time he obtained many significant results. In particular, he
disproved conjectures of Epstein, Eremenko and Lyubich, related to
entire functions with finite (resp. bounded) critical set. As a crucial
tool he showed that every compact planar set can be approximated by
trees associated with Shabat polynomials. Another line of research
is his interesting work in computational geometry, on conforming
meshes of planar straight-line graphs. He very successfully applied
ideas from conformal mappings, notably his fast conformal mapping
algorithm, to problems in computational geometry.
Summary rationale for recommendation:
The PI is a leader in his field, and the proposal contains a rich
variety of problems and proposed methods of attack that the PI's
past record suggests are within his power to achieve. The panel
recommends this proposal for funding.
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REVIEW 1:
The PI produced an impressive amount of very interesting results,
answering numerous questions from various areas. In particular,
he disproved conjectures of Epstein, Eremenko
and Lyubich, related to entire functions with finite (resp. bounded)
critical set. As a crucial tool he showed that every compact planar
set can be approximated by trees associated with Shabat polynomials.
Another line of research is his interesting work in
computational geometry, on conforming meshes of planar straight-line
graphs. He very successfully applied ideas from conformal mappings,
notably his fast conformal mapping algorithm, to problems in computational geometry.
He proposes to work on a large number of natural questions, such as
generalizations of Belyi functions to non-compact surfaces, determination
of the order of certain entire functions from combinatorial data,
constructions of non-obtuse triangulations. He also proposes innovative
approaches to long-standing conjectures (such as Brennan's, Astala's,
and straightening of chord-arc curves in BMO).
With his powerful new tool of "qc-folding", his expertise in geometric
function theory, and his impressive track record in delicate geometric
constructions, he is likely to make good progress on several of his proposed problems.
The broader impact lies in the practical applications of optimal meshing
and in the interdisciplinary character of the research. Several of the
proposed problems are suitable for graduate students.
This is an excellent proposal, one the two best I have read this year,
and very likely to lead to interesting new results.
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REVIEW 2:
The proposal contains many interesting problems. The PI has a great
track record. One strength of the proposal is that the PI applies
a broad range of techniques to solve interesting problems in different areas.
In the context of the five review elements, please
evaluate the strengths and weaknesses of the proposal with respect to broader impacts.
The broader impact part is good.
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REVIEW 3:
This is an excellent proposal, one of the best I have reviewed.
The PI has a superb record in publishing first rate mathematics and
this proposal suggests that he will continue to do so. The PI works
in a broad range of mathematics including geometric analysis,
combinatorics, speeds of algorithms. He applies a broad range of
techniques to solve interesting problems. The problems in the current
proposal are carefully formulated and interesting.
The broader impact is persuasive. The PI has had a number of Ph.D. students.
He also makes a persuasive argument that his work is of interest to
scientists outside the usual world of theoretical mathematicians,
which certainly qualifies as broader impact.
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