Fourier analytic techniques in geometry and analysis

约瑟夫·傅立叶在19世纪早期的一个重大发现是，某些函数可以写成简单的“波状函数”的无穷和。这种分解现在被称为傅立叶级数，并在数学和更广泛的科学领域有广泛的应用，例如在信号处理和求解复杂的微分方程中。傅里叶变换描述了傅里叶级数收敛的速度，即随着频率的增加，波的振幅在分解过程中的衰减率。一种理解方法是傅里叶变换衰减得越快，函数一开始就越像波。因此，关于原始物体的一些几何信息被傅立叶衰减捕获。本研究项目考虑测度的傅里叶变换（质量分布），它类似于函数。众所周知，一个测度的傅里叶变换编码了很多关于它的几何结构的信息，例如关于它的维度、曲率特性和算术共振。我们研究傅里叶变换，和几何信息编码，在几个具有挑战性的背景下。例如，我们考虑当原始度量在标准几何操作下被扭曲时它是如何受到影响的，例如将二维空间中的度量投影到直线上。关于集合和测度的豪斯多夫维数的类似问题是几何测度理论的核心，我们将在这个方向上建立经典结果的傅里叶解析类似物。我们还考虑了概率情况下的傅里叶变换。布朗运动是一个基本的随机过程——首先观察到的是花粉颗粒悬浮在水中时看似随机的运动轨迹——这是我们的典型例子。我们将考虑与布朗运动和相关过程相关的自然（随机）测度的傅里叶变换。最后，我们将考虑动态不变的措施，我们将使用传递算子和热力学形式的其他工具来分析傅里叶衰减。

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.