Proposal Number: 1608577
NSF Program: Geometric Analysis
Principal Investigator: Bishop, Christopher
Proposal Title: Geometric Problems in Conformal Analysis, Dynamics, and Probability
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Panel Summary
The PI seeks to study a wide range of topics including Julia and Fatou sets for transcendental entire functions, removability issues for conformal homeomorphisms and discrete geometry via conformal analysis.
The problems addressed are deep and compelling and likely to set directions for the field. They are also likely to have impact outside of mathematics such as algorithms for computing good triangulations impacting computing and applied numerical analysis.
Prior support has produced quality results; strong papers in excellent journals.
The broader impacts of the proposal are strong with good work around involvement of undergraduate students identifying elements which are accessible to them. The PI has mentored postdoctoral fellows and has supervised PhD research.
The research proposed here is of very high quality backed with a strong track record. The PI will most likely succeed in producing significant advances.
The panel places this proposal in the Highly Recommended for Funding category.
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REVIEW 1:
This proposal seeks to study (a) Julia and Fatou sets for transcendental entire functions, (b) removability issues for conformal homeomorphisms, (c) study of discrete geometry via conformal analysis. For instance, in (a) the PI aims to see whether there are transcendental entire functions whose Julia sets achieve various combinations of packing and Hausdorff measures; whether two entire transcendental functions, with bounded/finite closure of critical and asymptotic values, have the same packing/Hausdorff dimension if they are QC equivalent? Can the non-escaping set of such functions contain wandering domains? In (b) some of the problems posed are: Is the union of two sets that are each removable for conformality also is removable for conformality? If the class of all homeomorphisms of the Riemann sphere that are conformal off of a set contains a non-M\"obius map, then what is the structure of this class (dimensionality, approximation of identity etc.)? Are the Sierpinski carpet and/or the Sierpinski gasket removable? Does a version of Jones-Smirnov theorem hold when chains are replaced by non-connected chains which have pairwise modulus comparable to 1? Is a 2-dimensional Brownian motion removable? In (c) problems such as conforming non-obtuse triangulations for planar graphs (the PI seeks to improve the estimate of the size of such triangulations from O(n^{2.5}) to O(n^2), where n is the number of vertices of the graph), and find algorithms that compute such optimal-sized triangulations. The PI also seeks to find an analog of the measurable Riemann mapping theorem for QC mappings, where the dilatation is prescribed to be piecewise "almost constant".
The broader impacts of the proposal are very good. The PI seeks to construct computer algorithms for computing good triangulations, which in turn has impact in computing and applied numerical analysis.
The PI also identifies elements of the project that are accessible to undergraduate students (the PI has mentored undergraduate students who have then gone on to seek PhD in other institutions).
Summary Statement
The problems proposed here are of very good quality, and the PI's track record indicates that the PI will most likely succeed in addressing them. Some of the PI's recent publications have appeared in Invent. Math. and Acta.
The broader impact of the proposal is also very good. I would rank this as one of the best in the list of proposals assigned to me.
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REVIEW 2:
This is a very well written proposal by an experienced and accomplished mathematician. The research proposed is in three broad areas: the geometry of the stable and unstable sets of dynamical systems generated by transcendental functions; removability questions for conformal maps and in particular their relation to Brownian motion; the interaction of conformal analysis with computational and discrete geometry. There are very many interesting and difficult problems proposed, many of which are extensions of the proposer's earlier work. He uses his tremendous technical ability and insight to work on hard and important problems. For example, he is a master at finding explicit examples and has found counterexamples to a number of long-standing conjectures on the combinatorial structures of dynamically defined sets.
There are many potential applications of the work on computational geometry to computer science and industry. In particular, questions of recognition of patterns, for example facial features, can be approached using some of his results.
The proposer has a strong track record and has published regularly in prestigious journals. He will undoubtedly continue to work at the same high level.
The proposer's work on computational geometry has natural applications to industrial computing. His methods are very suitable for large scale computation and he has had success to date.
The proposer has mentored and worked with a number of students and post-docs.
The proposer has worked with computer scientists and has submitted his work to computer science journals.
The proposer has been very successful with students at various stages of their careers
Summary Statement
This is an excellent proposal by a serious and accomplished mathematician. He will undoubtedly continue to do work at this level and the proposal deserves highest priority for support.
This is among the top four of the proposals in my cohort.
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REVIEW 3:
The part that I am most familiar with is the questions on dynamics of entire transcendental functions acting on the complex plane. The PI has important recent work in this area and has proposed some interesting future topics. This part of the proposal is very strong.
In the context of the five review elements, please
evaluate the strengths and weaknesses of the proposal with respect to broader impacts.
This is the only proposal I have read that has true applications outside of pure mathematics. The PI also suggest several well concrete problems for grad students, undergrads and even high school students.
Summary Statement
The aspects of this proposal that the reviewer feels comfortable assessing a very strong in both intellectual merit and broader impacts.