Proposal Number: 2303987
NSF Program: Cofunded by Geometric Analysis and Analysis
Principal Investigator: Bishop, Christopher
Proposal Title: Quasiconformal analysis, optimal triangulations and fractal geometry
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Panel Summary
Intellectual Merit:
The panel affirmed that PI is a leader of the field, and the proposed set of problems
is ambitious and exciting. The proposal contains a very large number of different themes
from a very wide range of topics, studying properties of Weil-Petersson curves, optimal
triangulations of polygons, and properties of Julia sets of transcendental entire functions.
The panel agreed that the proposal is very well written, giving many illustrative examples
and explaining very well the PI's previous results on these topics. The track record of the
PI and the results from the prior NSF support were deemed excellent.
Broader Impacts:
In regard the Broader Impacts, the panel noted that the PI has recently been very active
in inspiring young mathematicians. He maintains an in-depth webpage (that includes materials
for online courses on topics stemming from PI's research under prior NSF support), and that he
co-authored an important monograph on fractals in probability and analysis. The proposal contains
a list of problems suitable for undergraduate students and plans for a future graduate workshop
and an online graduate class. The panel deemed the Broader Impact component as excellent.
Summary and Recommendation:
The panel placed this proposal in the Highly Competitive category.
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Proposal Review 1
In the context of the five review elements, please
evaluate the strengths and weaknesses of the proposal
with respect to intellectual merit.
The proposal offers 3 main directions of inquiry within the
general area of conformal, quasiconformal, and hyperbolic
geometry. These are the same 3 areas described in the summary of
results from prior NSF support:
1. The study of a class of closed curves called Weil-Peterson that
generalize smooth curves in an important ways (with many
connections to other fields of mathematics).
2. The study of optimal triangulations, a topic with important
consequences in applied mathematics generally.
3. The study of fractals arising in transcendental dynamics.
The PI displays an impressive array of results, advances, new
problems, conjectures and applications. The questions that are
described range from very specific to more open-ended ones. Still
this reviewer would have appreciated a bit more discussion around
for instance Problems 8 and 9.
In the context of the five review elements, please
evaluate the strengths and weaknesses of the proposal with respect to broader impacts.
The broader impacts of this proposal are potentially very
large. The PI has already started building connections to other
areas of mathematics, especially more applied ones. Such
endeavors need to be supported to prevent the natural tendencies
for siloization prevalent in the sciences.
Summary Statement
I rank this proposal in the top third of all the proposals I reviewed.
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Proposal Review 2
In the context of the five review elements, please
evaluate the strengths and weaknesses of the proposal with respect to intellectual merit.
The PI proposes applications of (quasi)conformal and hyperbolic geometry to solve
deep problems about (1) Weil-Petersson (WP) curves and related objects, (2) triangulations
of planar straight line graphs, and (3) fractal sets such as transcendental Julia sets and
some random fractals.
The PI is a leading scientist with an outstanding track record incl. several publications
in Annals and Acta. Closely related to the present proposal are the articles on transcendental
1-dimensional Julia sets (Invent. Math., 2018) and a traveling salesman theorem for
Jordan curves (Adv. Math., 2022).
A major product of the PI's recent NSF support are numerous characterizations of WP curves
(via beta numbers, hyperbolic convex hulls, knot energies, inscribed polygons…). These
characterizations open directions for future research, which the PI proposes to explore.
If successful, this will advance understanding within the respective fields, but the true
innovative potential arises through the connection of different areas of mathematics,
computational and numerical methods, and perhaps even physics.
The proposal includes innovative and highly topical questions originating from the PI’s recent
research, but also, e.g., a conjecture by Wendelin Werner. There is a good mix of explicit
problems with concrete proof strategies and open-ended questions that leave room to explore
new connections (e.g., WP curves and SLE, non-obtuse triangulations and dynamics).
In the context of the five review elements, please evaluate the strengths
and weaknesses of the proposal with respect to broader impacts.
This proposal has very good potential for broader impacts in multiple directions: training
and inspiration for students, possible applications in image modeling (by extending the PI's
work on triangulation to three dimensions) and in the experimental study of growth of natural
phenomena modeled by fractals.
The interdisciplinary spirit of the proposal feels genuine and is backed up by the PI's
involvement in an interdisciplinary program (Center for Finance) at Stony Brook, his talks
at computer science conferences, and citations from outside pure mathematics for results
from his prior NSF support.
The PI has recently been very active in inspiring early-career mathematicians. He maintains
an in-depth webpage (incl. materials for online courses on topics from prior NSF support)
and he co-authored a beautiful book on fractals in probability and analysis. The proposal
lcontains a list of problems suitable for undergraduate students and plans for a future graduate
workshop and an online graduate class.
The funds requested to support one graduate student for the duration of the project are
adequate. Based on the PI's past mentoring experience, he is well qualified to guide the student.
However, the student's role regarding the different components of the research plan is not
clearly described.
Summary Statement
This is an outstanding proposal with great potential to advance knowledge in and across fields.
The expected broader impacts in several directions are convincingly presented and backed up by
the proposer's track record. I ranked this proposal in the top third of all the proposals I reviewed.
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Proposal Review 3
In the context of the five review elements, please evaluate the strengths
and weaknesses of the proposal with respect to intellectual merit.
The PI is a leader of the field, and the proposed problems are ambitious and exciting.
The proposal contains a very large number of problems, from a very wide range of topics,
studying properties of Weil-Petersson curves, optimal triangulations of polygons, and
properties of Julia sets of transcendental entire functions. The proposal is very well
written, giving many illustrative examples and explaining very well the PI's previous
results in these topics. There is not much discussion about how he intends to solve the
proposed new problems, but given the PI's excellent track record I have no doubt that he
will be able to solve several of them, and will make very substantial progress on many others.
In the context of the five review elements, please evaluate the strengths
and weaknesses of the proposal with respect to broader impacts.
I found the broader impact to be also very impressive. He is passionate about supervising
a very large number of projects with junior people (graduate, undergraduate, and also for
high school students). His work on triangulations has real life applications in physics,
computer vision, biology, etc, providing faster algorithms. He is collaborating with
researchers in some of these areas, and his work was cited/used by them.
Summary Statement
This is an excellent proposal.