Conformal Dynamics

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In this project, I shall consider problems in harmonic analysis arising in geometric analysis and geometric measure theory, and in the theory of elliptic and parabolic partial differential equationsand operators. These questions are interesting because of theirapplicability to real world phenomena (see below), but also because of the deep connections among them. The main directions of the proposed research include the following:1. To develop and apply ``T1/Tb" (i.e., Carleson measure) criteria for the solvability of boundary problems for divergence form elliptic equations and systems.2. To treat various problems in the theory of uniformly rectifiable sets, in the applications of this theory to elliptic and parabolic PDE, and in the theory of quasiconformal mappings.3. To obtain sharp average decay estimates for Fourier transforms, and to apply these to concrete problems including lattice point problems andthe Falconer distance problem. As mentioned above, I propose to work on problems in the area of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic and parabolicpartial differential equations. Roughly speaking, in harmonic analysis one investigates properties offunctions and ``operators" (i.e., mappings which transform one functioninto another) by decomposing them into smaller, constituent pieces,which are easier to understand, and then reassembling the pieces. Thename itself arose by analogy to the decomposition of a musical soundinto its various frequency components, or ``harmonics".geometric measure theory involves the study of the relationship betweengeometric properties of sets, and their ``measures" (the latter aregeneralizations of the notions of length, area, and volume). Partialdifferential equations of elliptic and of parabolic type describe a wide variety of phenoma in the real world, including electrostatics, certain fluid flows and elastic deformations, and various diffusionprocesses such as the conduction of heat, the flow of ground water, certainphenomena arising in the mathematical theory of population biology, and the pricing of options in financial markets. In the last decade the interplay between these different subfields of mathematics has turned out to be afertile ground for investigation, with much exciting work remainingto be done.