Conformal Dynamics

在这个项目中，我将考虑几何分析和几何测度理论中的调和分析问题，以及椭圆型和抛物型偏微分方程和算子理论中的调和分析问题。这些问题很有趣，因为它们适用于现实世界的现象（见下文），但也因为它们之间的深刻联系。本文拟开展的主要研究方向包括：1。发展并应用“T1/Tb”（即Carleson测度）判别椭圆型方程和系统散度边界问题的可解性。处理一致可整集理论中的各种问题，讨论该理论在椭圆型和抛物型偏微分方程中的应用，以及拟共形映射理论中的各种问题。获得傅里叶变换的尖锐平均衰减估计，并将其应用于具体问题，包括格点问题和Falconer距离问题。如上所述，我打算研究调和分析及其在几何测度理论、椭圆型偏微分方程和抛物型偏微分方程理论中的应用和相互作用。粗略地说，在调和分析中，人们研究性质、函数和“算子”（即将一个函数转换为另一个函数的映射），方法是将它们分解成更容易理解的更小的组成部分，然后将这些部分重新组合起来。这个名字本身是通过类比将音乐声音分解成各种频率成分或“谐波”而产生的。几何测度理论涉及对集合的几何性质及其测度之间关系的研究（后者是长度、面积和体积概念的概括）。椭圆型和抛物型偏微分方程描述了现实世界中各种各样的现象，包括静电、某些流体流动和弹性变形、各种扩散过程，如热传导、地下水流动、人口生物学数学理论中的某些现象，以及金融市场中期权的定价。在过去的十年中，这些不同数学分支之间的相互作用已被证明是研究的沃土，有许多令人兴奋的工作有待完成。

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.