Aspects of thinness in harmonic analysis

谐波分析师寻求发展数学理论，帮助找到可能出现的问题的解决方案，从数学物理，电气工程和其他数学分支。 我在谐波分析方面的研究是由不确定性原理的谐波分析版本所激发的。这个原理与海森堡测不准原理有着密切的联系，粗略地说，它说关于一个函数的信息必须通过对它的傅里叶变换的相应控制的损失来“支付”，反之亦然。这在应用中具有重要意义。例如，它意味着一个人不能拥有一个在时间和频率上都有限制的无线电信号。 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/操作符/度量，并试图理解其后果。 1.李群经常用于物理学中，因为它们描述了真实世界的几何。轨道测量是这一设置的基本组成部分，并且具有较小的支持。我的项目的第一部分是理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.西顿集可以根据其傅里叶变换在集合上得到支持的函数的解析性质来定义。虽然它们在直观意义上很小，但它们具有复杂的结构。在项目的第二部分，我们的目标是描述西顿集的集合，更容易理解。 3.该项目的第三部分是使用分形分析的工具来研究与康托集相关的自然测度的几何。这些是有趣的，因为他们的病理行为。在简单的例子中发现了令人惊讶的几何学。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。

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.