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.