Conformal Dynamics

在这个项目中，我将考虑在几何分析和几何测度理论中出现的调和分析问题，以及在 椭圆和抛物型偏微分方程和算子理论。 这些问题之所以有趣，是因为它们对真实的世界现象的可应用性（见下文），但也因为它们之间的深层联系。 本文的主要研究方向包括以下几个方面：1.开发和应用“T1/Tb”（即，Carleson测度）的判别准则.处理一致可求长集理论中的各种问题，以及该理论在椭圆和抛物偏微分方程中的应用，以及拟共形映射理论中的问题。为了得到傅里叶变换的平均衰减估计，并将其应用于具体问题，包括格点问题和法尔科纳距离问题。 如上所述，我建议在调和分析及其应用领域的工作问题，并与几何测量理论和椭圆和parabolicpartial微分方程理论的相互作用。 粗略地说，在调和分析中，人们研究函数和"算子”的性质（即，将一个功能转换为另一个功能的映射），方法是将它们分解为更小的、更容易理解的组成部分，然后重新组装这些部分。 几何测度理论涉及研究集合的几何性质和它们的"测度”（后者是长度、面积和体积概念的概括）之间的关系。 椭圆型和抛物型偏微分方程描述了真实的世界中的各种现象，包括静电、某些流体流动和弹性变形，以及各种扩散过程，如热传导、地下水流动、种群生物学数学理论中出现的某些现象，以及金融市场中的期权定价。 在过去的十年里，这些不同数学子领域之间的相互作用已被证明是一个有待研究的领域，还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.