Mathematical Science: Absolutely Continuous Conjugations

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9400975 Hamilton This award supports mathematical research into problems arising in the modern theory of functions. A central theme of this theory is the deformation of functions by conjugation with bijections. Using tools of classical analysis, potential theory, quasiconformal mapping, boundary values of analytic functions, and conformal mapping two new approaches to old questions in the areas of Riemann surfaces and Complex Dynamics have been developed. The project will continue to exploit the ideas to prove rigidity results on Julia sets of rational functions. For example, if the Julia set is a curve then it has the Fatou property, that is, it is a circle or has dimension greater than one. For Riemann surfaces work will follow on a recent proof of the general version of the Bers Uniformization Theorem to prove a conjecture of Gromov on the uniqueness of the representation. Other work will focus on n-dimensional bilipschitz mappings in the context of quasiconformal mapping. In particular efforts will be made to characterize those which arise as reflections across boundaries of domains. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. The subject exploits the interplay between geometric interpretation with powerful analytic techniques to yield some of the most complete mathematical results in the field of analysis. In this project, variational techniques and geometric reasoning are expected to lead to results of valuable physical significance. ***