REVIEW 1:
I rate this proposal as EXCELLENT, very close to the top.
The proposer raises 25 problems/conjectures, each of which
making connections with conformal mappings, probability,
Kleinian groups, quasiconformal analysis, in all ranges
and permutations. It is a splendid list.
The proposer has made powerful impressions in several areas in
geometric function theory, able to do hard estimates with a broad
vision of what seems significant in the broad sense, and is able to
connect the strands together. He has an
active program whose value shows in that students of
top analysts migrate to Stony Brook to learn and
interact with him.
The effect is reciprocal; for example conjecture 15 was settled
in two dimensions with another post-doc, and now proposer offers
a bold way of bringing this to higher dimensions. This is related
to a topic one of his graduate students is pursuing, according
to the proposal.
Brennan's so-called conjecture on the integrability of the
derivative of a conformal mapping has been around for probably 20 years,
but it has provided a vehicle for introducing several exciting
new approaches (all of which have, in the end, failed to resolve the
original problem!). The proposal has a new angle:
his decomposing (which is to appear, and is not accessible from
his web page, so I could not consult it first-hand) of any
conformal map into K-qc homeomorphic and an `expanding'
conformal factors. This leads to
Question 1 which, even if not proved, already shows significant connections
with outstanding work of Astala, Makarov, and the Sullivan convex hull theorem
(which is where his weak bound for Question 1 arises). This leads
to many related and interesting questions, passing though Conjecture 9.
A second set of problems concerns the Ahlfors conjecture on the area of the
limit set of a finitely-generated Kleinian group. It appears that the
proposer has been working on this subject for several years, but the joint
paper (Acta 1997) with Peter Jones on the relation between limit set
and exponent of convergence is an enormous step, and in turn
leads to a path connecting this problem
to a dynamic one involving the decay of heat kernels
The thoroughness and scope of the proposal make it one of the five-ten
best I have review in the last decade. There is a cornucopia of
ideas, even a half-page devoted to problems that would be suitable
for undergraduate research projects. It is a virtuoso display.
REVIEW 2 :
This is a superb proposal of a highly productive mathematician
with a full tool kit allowing him to roam with seeming ease and insight
over complex analysis, geometry and probability. Much of Bishop's past
work is brilliant, in particular his work with Peter Jones and his success
in bringing the study of the hyperbolic convex hull out of the closet.
Among many others, his ideas on bringing the study of the heat kernal
in to study limit sets are exciting---Ahlfors would have appreciated
this very much. Who knows what he will discover next? Bishop has enough
ideas to keep a veritable army at work. His presence at a conference
automatically increases intellectual excitement and interaction between
participants. Anyone who thinks complex analysis is dead should read this
wide ranging and highly stimulating proposal.
REVIEW 3 :
What is the intellectual merit of the proposed activity?
To use the proposer's outstanding technical skill to solve the potentially
very complicated problems in complex analysis related to fractal behavior
at the boundary
What are the broader impacts of the proposed activity?
These problems have been shown to relate to extremal problems in analysis,
to dynamics and to problems in physics such as Brownian motion,
percolation and conformal field theory
Summary Statement
B. is clearly a leading person in this area. His participation in the
special year at the Mittag-Leffler Institute could be of very significant
value. I consider the mix of complex analysis and probability to have
a similar potential as the mix with dynamics has had during the 1990's
(Two Fields medals!)
REVIEW 4:
What is the intellectual merit of the proposed activity?
The project consists of 25 problems; all of them are interesting and
significant, and at least some are extremely difficult. The main theme
is the interplay between three-dimensional hyperbolic geometry and
classical complex analysis.
There are two main directions discussed in the proposal. The first one
is a beautiful new approach to harmonic measure and conformal maps. It
is based on Bishop"s factorization theorem related to a result of Sullivan.
The author has shown how one can reduce various open problems, including
the well-known Brennan"s conjecture, to finding the best factorization
constant.
The second part of the proposal deals with limit sets of Kleinean groups.
The PI has already obtained several results of fundamental importance
in this area. He seems to have clear ideas how to approach some famous
problems as well as a variety of new ones.
Professor Bishop is one of the best complex analysts. He has made remarkable
progress in recent years. His work has been deep, and now he plans
to concentrate on the most challenging problems.
What are the broader impacts of the proposed activity?
Summary Statement
This is an excellent proposal. It should be given the highest priority.
REVIEW 5:
What is the intellectual merit of the proposed activity?
Overall Rating: VERY GOOD/EXCELLENT
The present proposal focuses on important and lively
interactions between analysis and geometry in the
settling of conformal mappings and hyperbolic space.
There are 3 main directions: (1) factorization and
convex hulls, (2) limit sets of Kleinian groups and
(3) harmonic measure.
Topic (1) is the most novel and interesting. In work under
the previous grant, the PI has established an unexpected
connection between Sullivan's work on convex hulls,
the 'factorization' of conformal mappings (compensating
for contraction with a quasiconformal automorphism of the
disk), and a circle of well-known problems in analysis
centering around Brennan's conjecture.
The fresh light provided is promising for progress.
Topic (2) grows from the PI's joint work with Peter Jones
that settled some important and long-standing problems about
the Hausdorff dimension of limit sets Kleinian groups.
This area is becoming well-developed and the previous and
proposed work on infinitely-generated groups is technical and of
less interest than the original breakthrough.
Topic (3) lies mostly square in the PI's area of specialization,
and includes some natural and approachable
ideas concerning the 'multifractal' behavior of
harmonic measure (involving twist and Hausdorff dimension).
The PI's previous work is energetic and substantial.
While some papers from the previous project and technical
and specialized, they support a coherent research program
that is yielding results of lasting interest.
REVIEW 6 :
What is the intellectual merit of the proposed activity?
The proposal includes a number of original ideas. I don't know all the
areas that this proposal touches on, which are quite numerous, reflecting
Bishop's energy and breadth. With respect to Brennan's Conjecture, he
has opened up an entirely new approach, which may well be successful.
In any case, attempts along the lines suggested will open up new avenues
for research.
Bishop's proposal points to interesting connections between hyperbolic
geometry, differential geometry, global analysis and the theory of
functions of one complex variable.
Working at Stonybrook with Yair Minsky around, Bishop has a particularly
good chance to make significant contributions to the study of Kleinian
groups, where he knows considerably more analysis than most people in
the field.
Although I have never met him, I understand that he is a very stimulating
person to talk to, with ideas in many subjects. His work has had a
considerable influence on my own recent research.
What are the broader impacts of the proposed activity?
I don't think this proposal will have more impact than
many proposals, outside the direct impact of the research. However,
Bishop is active enough to be comfortable with several good graduate
students, and I hope he gets them.
Summary Statement
An excellent proposal.