Aspects of Thinness in Harmonic Analysis

调和分析员寻求发展数学理论，以帮助找到解决数学物理、电气工程和其他数学分支可能出现的问题的方法。*我对调和分析的研究在一定程度上是受到测不准原理的调和分析版本的推动，测不准原理是一种与海森堡测不准原理松散相关的指导哲学。粗略地说，它说，关于一个函数获得的信息必须通过相应的傅立叶变换失去控制来支付，反之亦然。这在应用程序中具有重要意义。例如，它意味着一个人不能有一个在时间和频率上都受限的无线电信号。*在我的研究中，我对这一原则进行了定性和定量的解释。我研究在某种意义上或薄或小的函数、度量和集合，我的目标是理解这一点的后果。长期以来，数学家一直对这一主题感兴趣，因为众所周知，“薄”物体在许多问题中都很重要，并且可以表现出有趣的现象。例如，魏尔斯特拉斯著名的连续的、无处可微的函数的例子是一个三角级数，它的变换有很薄的支撑，而康托集，一个薄但不可数的集合，在数学的许多分支中都很重要。最近，调和分析中的薄度概念与其他数学领域，如数论和组合学，发现了重要的联系。*我的研究计划将有两个主要部分。*1.稀疏集：Sidon集可以根据其上支持傅里叶变换的函数的分析性质来定义。尽管它们在直觉上很薄，但它们的结构却很复杂。我们这部分程序的目标是用更容易理解的更稀疏的集合来刻画Sidon集的特征，这是调和分析中的一个基本和长期存在的问题。*2.弱支撑度：**(A)经典的Cantor测度是一个著名的同时具有小支撑度和(某种)小傅立叶变换的例子。它在整个数学中都很有趣，因为它有病态的行为，但又容易处理。在程序的第二部分，我们将开发技术来量化Cantor类度量和其他度量的局部行为，这些度量来自迭代构建，但有重叠。这些都是当前分形几何学中的重要话题。*(B)李群描述真实世界的几何学，因此在物理学中经常被使用。轨道度量是这一背景下的基本组成部分，与康托度量一样，轨道度量的支撑度和变换都很小。我的研究计划的最后一部分是了解这类薄度量的光滑性及其相关的几何结构。我预计这将在李群的调和分析中有重要的应用。

Harmonic analysts seek to develop mathematical theories that help to find solutions to problems that might have arisen from mathematical physics, electrical engineering and other branches of mathematics. ****My research in harmonic analysis is motivated, in part, by the harmonic analysis version of the Uncertainty Principle, a guiding philosophy loosely related to the Heisenberg Uncertainty Principle. Roughly speaking, it says information gained about a function has to be `paid for' by a corresponding loss of control on its Fourier transform, or vice versa. This is significant in applications. For instance, it implies that one cannot have a radio signal that is bounded in both time and frequencies. ****In my research, I develop qualitative and quantitative interpretations of this principle. I study functions, measures and sets that are thin or small, in some sense, and my goal is to understand the consequences of this. This topic has long been of interest to mathematicians for it is well known that `thin' objects are important in many problems and can exhibit interesting phenomena. For example, Weierstrass' famous example of a continuous, nowhere differentiable function was a trigonometric series with thin support of its transform, and the Cantor set, a thin but uncountable set, is important in many branches of mathematics. Recently, there have been important connections found relating thinness ideas from harmonic analysis with other areas of mathematics such as number theory and combinatorics.****My research program will have two major components. ****1. Thin sets: Sidon sets can be defined in terms of analytic properties of the functions whose Fourier transform is supported on the set. Although they are thin in an intuitive sense, their structure is complicated. The objective of this part of our program is to characterize Sidon sets in terms of thinner sets that are simpler to understand, a fundamental and long-standing problem in harmonic analysis. ****2. Thinly supported measures: **(a) The classical Cantor measure is a well-known example of a measure which has both small support and (somewhat) small Fourier transform. It is of interest throughout mathematics because of its pathological behavior and yet tractability. In the second part of the program we will develop techniques to quantify the local behavior of Cantor-like measures and other measures that arise from an iterative construction, but have overlap. These are important, current topics in fractal geometry.****(b) Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and like Cantor measures have small support and small transform. The final part of my research program is to understand the smoothness properties of this class of thin measures and their associated geometric structures. I anticipate this will have important applications for harmonic analysis on Lie groups.**