Fourier analytic techniques in geometry and analysis

约瑟夫·傅立叶在19世纪初的一个重大发现是，某些函数可以写成简单的“波函数”的无限和。这种分解现在被称为傅里叶级数，在数学和更广泛的科学中有广泛的应用，例如在信号处理和求解复杂的微分方程式中。傅里叶变换描述了傅里叶级数收敛的速度，即分解中的波幅随频率增加的衰减率。一种看法是，傅里叶变换衰减得越快，函数一开始就越像波。因此，关于原始对象的一些几何信息被傅里叶衰减捕捉到。这项研究项目考虑测量(质量分布)的傅里叶变换，它类似于函数。众所周知，一个度量的傅里叶变换编码了大量关于它的几何结构的信息，例如关于它的维度、曲率属性和算术共振。我们在几个具有挑战性的环境中研究了傅里叶变换及其编码的几何信息。例如，我们考虑当原始测量在标准几何操作下被扭曲时，例如将二维空间中的测量投影到直线上时，它是如何受到影响的。关于集合和测度的Hausdorff维度的类似问题是几何测度论的核心，我们将在这个方向上建立经典结果的傅立叶分析类比。我们还考虑了概率环境下的傅里叶变换。布朗运动是一个基本的随机过程--最初观察到的是悬浮在水中的一粒花粉遵循的看似随机的路径--也是我们的原型。我们将考虑与布朗运动和相关过程有关的自然(随机)测度的傅里叶变换。最后，我们将考虑动态不变测量，我们将使用热力学形式中的转移算符和其他工具来分析傅里叶衰变。

A powerful discovery of Joseph Fourier in the early 1800s was that certain functions could be written as an infinite sum of simple 'wave-like functions'. Such a decomposition is now known as a Fourier series, and has had wide-ranging applications across mathematics and wider science, for example in signal processing and in solving complicated differential equations. The Fourier transform describes how quickly the Fourier series converges, i.e. the decay rate of the amplitudes of the waves in the decomposition as frequency increases. One way of viewing this is that the faster the Fourier transform decays, the more wave-like the function was to begin with. Thus, some geometric information about the original object is captured by Fourier decay. This research project considers the Fourier transform of measures (mass distributions), which are analogous to functions. It is well known that the Fourier transform of a measure encodes a lot of information about its geometric structure, for example concerning its dimension, curvature properties, and arithmetic resonances. We investigate the Fourier transform, and the geometric information it encodes, in several challenging contexts. For example, we consider how it is affected when the original measure is distorted under standard geometric operations, such as projecting a measure in 2 dimensional space onto lines. Similar questions about the Hausdorff dimension of sets and measures are at the heart of geometric measure theory and we will establish Fourier analytic analogues of classical results in this direction. We also consider the Fourier transform in probabilistic settings. Brownian motion is a fundamental random process - first observed as the seemingly random path a grain of pollen follows when suspended in water - and is our archetypal example. We will consider the Fourier transform of natural (random) measures associated with Brownian motion and related processes. Finally, we will consider dynamically invariant measures, where we will use transfer operators and other tools from the thermodynamic formalism to analyse the Fourier decay.