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。 我在谐和分析方面的研究是受测不准原理的谐和分析版本的启发。这一原理在哲学上与海森堡测不准原理有关，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严格和定量的解释。我研究在某种意义上“小”的函数/运算符/度量，并试图理解其后果。 1.李群在物理学中经常被用来描述真实世界的几何图形。轨道测量是这种环境中的基本组成部分，支撑力很小。我的项目的第一部分是了解它们的变换的大小与相关的几何和代数结构之间的关系。这对于李群的分析是很重要的。 2.Sidon集可以根据其上支持傅里叶变换的函数的解析性质来定义。尽管它们在直觉上很小，但它们有一个复杂的结构。在项目的第二部分中，我们的目标是用更容易理解的集合来刻画Sidon集。 3.项目的第三部分是使用分形分析中的工具来研究与Cantor集相关的自然度量的几何。这些动物之所以令人感兴趣，是因为它们的病态行为。在简单的例子中发现了令人惊讶的几何图形。我们的目标是确定这种奇怪的行为是否广泛传播，以及为什么会发生这种行为。

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 by the harmonic analysis version of the Uncertainty Principle. This principle is philosophically related to the Heisenberg Uncertainty Principle in that, 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 rigorous and quantitative interpretations of this principle. I study functions/ operators/measures that are "small", in some sense, and seek to understand the consequences of this. 1. Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and have small support. The first part of my project is to understand the relationship between the size of their transforms and the associated geometric and algebraic structures. This is important for analysis on Lie groups. 2. Sidon sets can be defined in terms of the analytic properties of the functions whose Fourier transform is supported on the set. Although they are small in an intuitive sense, they have a complicated structure. In the second part of the project our objective is to characterize Sidon sets in terms of sets that are simpler to understand. 3. The third part of the project is to use tools from fractal analysis to study the geometry of natural measures that are associated with Cantor sets. These are of interest because of their pathological behaviour. Surprising geometry has been discovered in simple examples. Our goal is to determine whether this odd behaviour is wide spread and why it is occurring.