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.