Panel Summary
Intellectual merit. The PI is a clear leader in his field with an excellent
track of solving difficult problems and of a recognized high technical
ability. In this proposal he lists a large number of interesting and
important problems on conformal and quasiconformal mappings, many of
them of a computational nature.
The panel felt that this is important work that should
be supported and that the PI has the ability to make
significant progress on hard problems.
Broader impact. Some of the results of a more computational
nature may be of interest in applications. The PI has
been involved in organizing conferences. The PI has had
three graduate students over his career.
The PI is requesting funding for two graduate students.
His record in this regard is not as strong as one
could fairly expect of someone with such a strong
standing in the field, and does not support funding
in this respect. The panel also felt that the PI
should consider getting involved in the exposition
of mathematics to more general audiences, such as
in summer schools where he could reach students from
minorities and other under-represented groups.
Recommendation. The panel recommends this proposal
for funding with high priority. Some panel members
expressed doubt as to whether it deserves funding
for five years, in view of weaknesses concerning broader impact.
REVIEW 1:
What is the intellectual merit of the proposed activity?
The proposal contains no less than 22 problems and
conjectures, mostly concerning conformal and quasiconformal
mappings in the context of computational geometry.
The main topics are the computational complexity
of the fast conformal mapping, distortion of quasiconformal
maps, and questions around Koebe's conjecture that every
finitely connected planar domain is conformally equivalent
to a circle domain. Other proposed work addresses meshes
and triangulations and the straightening of chord-arc
curves.
The PI sets out the various problems with appropriate
details and some ideas for solutions. This is a rich
area of study and well worth undertaking from a mathematical point of view.
What are the broader impacts of the proposed activity?
Conformal mappings are being used in computer vision, imaging (for example
of the brain), pattern recognition, and geography. Meshes and triangulations
are of course also important in computer vision as well as
numerical analysis. The PI is organizing a multi-disciplinary
workshop at which Mumford will be a speaker.
The PI also includes a thoughtful summary of his past
involvement of undergraduates and graduates in his
research, as well as plans for the future. His area
of research certainly seems eminently suitable for this purpose.
Summary Statement
This is a very carefully thought out and well-written proposal.
The PI has an extremely good research record, with a very high
proportion of papers appearing in top journals. He is clearly
an expert in all aspects of the proposed research. I rate this
proposal E/V and strongly recommend that it be funded.
REVIEW 2:
What is the intellectual merit of the proposed activity?
The PI propose to study $2$- and $3$-dimensional geometry which
come from classical complex analysis, the theory of quasiconformal
mappings, hyperbolic geometry and computational geometry. He has
shown that ideas from hyperbolic and computational geometry could
help a problem of classical analysis e.g., efficiently compute
conformal maps onto planar domains. The PI also found good meshes
on a domain and improved numerical methods of computing conformal
maps.
What are the broader impacts of the proposed activity?
The PI has developed new interactions of classical analysis with
applied mathematics and computer science. The PI is planning to
hold workshops. The PI taught his recent work into graduate
courses and undergraduate seminars.
REVIEW 3:
What is the intellectual merit of the proposed activity?
a) Intellectual merit of the proposed activity (Excellent)
The PI continues to work in two-dimensional quasiconformal
geometry with emphasis on computational problems based
on numerical experiments. He is successful in both areas.
Perhaps the best recent result of him is the construction
of an A1 weight not comparable to any quasiconformal
Jacobian (a 13 years old problem of Stephen Semmes).
I cannot comment much about his computational geometry
(linear time Riemann mapping theorem, or an iterative
method for conformal mappings). Another fine result
of the PI from prior NSF support is about distortion
of disks under normalized quasiconformal mappings which
are conformal outside the disks. The conjecture by
Astala, Clop Mateu, Orobitg and Uriarte-Tuero was
interesting and true for d=2, by trivial means. Basically,
the area of f(E) equals the area of E. The PI disproved
it for 0<2, which was not obvious at all. However,
analogous question concerning d-dimensional content
is even more interesting and important for understanding
distortion of Hausdorff measures under quasiconformal mappings.
The answer is known (in the positive) for d =1 and d = 2,
and remains open for other cases. The most impressive
is his factorization of a conformal map f : D->. as
f = g.h, with h being 8-quasiconformal and |g'| > ..
The problem has originated from the efforts to prove
Brennan's conjecture. Factoring by 2-quasiconformal
map would suffice, but it is not always possible.
Never mind, as a consequence of the factorization
he instead obtained beautiful result that any simply
connected domain can be mapped to the disk by a
locally Lipschitz homeomorphism and any quasi-disk can
be mapped to a disk by a Lipschitz homeomorphism of the plane.
There are many more recent results by the PI, but one
needs some preliminaries to state them. It is a
well-thought out proposal by a first rate researcher.
It attacks, like his previous work, specific technical
issues. I am certain one can expect the same high
level in his future research. He will continue study
of the distortion by quasiconformal mappings, Problems
15 and 16. Problem 18 is especially appealing: Can
a quasiconformal map from a surface to the plane
send positive area to zero area? The PI proposes new
questions about conformal welding, BMO topology of
chord-arc curves, and algorithms for fast conformal
mappings. These are the areas in which I am not familiar
to make a fair judgment of depth and importance. This
is an excellent proposal, and it should be given top
priority for funding. I rate the intellectual merit
as Excellent
What are the broader impacts of the proposed activity?
b) Broader impact of the proposed activity (Very Good-Excellent)
The PI supervised 3 PhD students (I am not sure who
of those actually completed their PhD), and 2 postdoctoral
scholars. Among them Karen Lundberg (female mathematician)
is actively involved in the proposed research.
No doubt the broader impact of the proposal is focused on the
computational aspects of the conformal geometry, which might create
broader interaction between computer science and classical analysis.
This is also a "key to the dissemination of new ideas to other research
communities, motivating students to enter mathematics, providing accessible
problems for undergraduate and graduate research and communicating with
non-technical audiences". The PI is seriously involved in this type of
broader impact activity (various lectures at Mathematics and Computer
Departments). In his proposal the PI clearly specifies problems related
to his research that might be suitable for graduates or even undergraduates.
I believe that the proposed integration of research into education and
the PI's activity involves some training of graduate students and
pos-docs. My mild criticism, however, is that the proposed activity does
not really target broad audiences via expository articles or mini-courses
or series of lectures. Such lectures (when addressed to really general
audiences) usually bring together prospective and young researchers,
possibly from underrepresented groups, introducing them and helping them
to establish professional connections with experts in the field. The
guiding principle here is that the background and broader connections
provided in these lectures allow the students, women and underrepresented
minority groups, to enjoy more specialized conferences in the future.
The requested funds for graduate students in the proposal will only help
those directly involved with the research of the PI. I rate the broader
impact of the proposal as Very Good-Excellent.
REVIEW 4:
What is the intellectual merit of the proposed activity?
In this proposal Chris Bishop poses a large number of
problems on conformal and quasiconformal mappings of plane
domains. In the first half of the proposal, the problems arise out
of how to compute the map from a Euclidean polygon onto a disk. He
already has an algorithm to compute a $1+\epsilon$ quasiconformal
map from an n-gon whose complexity is $O(n p \log p)$ where $p=|\log
\epsilon|.$ He now asks is current algorithm optimal - can the $O(p \log p )$
term be improved. Can he use a "faster" FFT (fast fourier transform)
algorithm since he only needs an approximation of the FFT. What
about a faster method for Schwarz-Christoffel mapping? He has a
conjecture on bit complexity. Can he extend the algorithm to
circular arc polygons. What about Koebe domains? And this is only
the beginning.
These problems seem to be his current interest and he has three
preprints in the last year on them.
In the second part of the proposal, Bishop asks a number of
questions about chord-arc curves: among others, is the space of them
connected in the BMO topology? can a chord-arc curve be
straightened by an expansive motion? He also has a number of
problems about conformal welding.
These problems are more in the line of what Bishop has done for most
of his 20 year career. He has a strong track record of finding good
counterexamples to conformal (and quasiconformal mapping problems).
These, in turn, often show what the theorems should be. He has
terrific technical ability and a broad outlook.
What are the broader impacts of the proposed activity?
As broader impact, Bishop plans to run a conference, from his
current grant, bringing computational and applied mathematicians
together with pure mathematicians to discuss problems like the ones
in this proposal. He outlines a long list of problems suitable for
graduate and undergraduate students. He seems to be attracting more
graduate students with his new interests than he had in the past.
Summary Statement
In sum, Bishop is a very productive, high powered mathematician. He
has set himself an impressive program of research and undoubtedly
will be successful with it. He definitely deserves support. I rate
this proposal excellent.
REVIEW 5:
What is the intellectual merit of the proposed activity?
There are potential applications in theoretical computer science.
What are the broader impacts of the proposed activity?
The principal investigator plans to organize workshops and interact with
the applied community.
Summary Statement
This proposal concerns some technical problems in computational conformal
mapping.The principal investigator is a well established research
mathematician.
REVIEW 6:
What is the intellectual merit of the proposed activity?
The general area of this proposal is classical complex analysis very
broadly construed. Specific problems in the proposal deal with efficient
algorithms in computational conformal geometry, the geometry of chord-arc
curves in the plane, distortion of quasiconformal maps, and conformal welding.
The PI is a leader in the field and has an outstanding
record of contributions to the area with many publications
in top journals. The PI's choice of problems shows a broad vision of the field.
Many of his ideas are very innovative and may lead to a solution of
Koebe's longstanding problem on uniformization by circle domains or to
a solution of the important problem of characterizing the plane up to
bi-Lipschitz homeomorphism, for example.
Given the PI's past record, it is guaranteed that the proposed activity
will lead to top-quality research with high impact.
What are the broader impacts of the proposed activity?
There are connections of the proposed research with applied areas such
as pattern recognition and computer vision. The PI intends to co-host
a workshop that will foster the interaction between computer scientists
and mathematicians. The PI also lists specific problems for
graduate and undergraduate research.
Summary Statement
This is an excellent proposal
and should be funded with highest priority.
From a purely scientific point of view I rank it highest
among the proposals I have reviewed. On the other
hand, in relation with his scientific standing, the PI's record of advising
PhD students, mentoring post-graduates and general service to the
mathematical community could be better. Therefore, I rank the proposal
2nd out of 9.