Proposal Number: 1305233 NSF Program: Geometric Analysis Principal Investigator: Bishop, Christopher Proposal Title: Quasiconformal methods in analysis, geometry and dynamics -------------------------------------------------------------- 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. ------------------------------------------------------------------ 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. ------------------------------------------------------------------ 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. ------------------------------------------------------------------ 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. ------------------------------------------------------------------