Harmonic analysis, additive combinatorics and geometric measure theory

分形图是表现出多尺度结构的对象，并且可以被分配一个不一定是整数的“维度数”。它们在自然界中无处不在(植物生长、景观形成)，也可以由人类活动(通信系统、基础设施网络)产生。在数学中，它们出现在数学物理或偏微分方程等领域的许多问题中，并在几何测度论和动力系统中得到了广泛的研究。在某种意义上，通过迭代相对简单的复制规则构建的分形对象可以表现出非常高的复杂性，并且其行为方式很难预先预测。 这项研究的第一个目标是发展基于调和分析和加性组合学的研究这类结构的新方法。传统上，调和分析在分形学研究中的应用相对有限。我计划发展一个关于分形集的奇异和振荡积分理论，仿照经典调和分析中光滑流形的类似理论(包括限制估计、最大和平均算子以及微分定理)，但也结合了只在分形域中出现的新特征。这项工作将借鉴调和分析和最近发展起来的加性组合数学领域的见解和方法。虽然已经获得了一些基础性的结果，证明了这是一个可行的和有趣的理论，但许多进一步的问题仍然悬而未决。特别是，分形集可以表现出调和解析行为，对于流形或在加法组合学中研究的离散对象都没有确切的相似之处，我想研究这些新现象。 第二个目标是将这些发展应用于维度理论、几何测度论和动力系统中的问题。几何测度论中的一个核心问题涉及投影、切片、交集、算术和以及指定维度集合的差。特别重要的是在动力系统和数学物理中出现的分形集，例如吸引子或不变集。例如，说出一个“典型”的分形图投影或切片应该是什么样子可能很容易，但要证明一个关于特定投影或切片的类似陈述却非常困难。同样，在许多情况下，定性结果很容易证明，但要获得定量估计要难得多。调和分析过去在这样的背景下很有用，我预计我计划开发的新方法将导致进一步的进步。

Fractals are objects that exhibit a multiscale structure and can be assigned a "number of dimensions" that is not necessarily integer. They are ubiquitous in nature (plant growth, landscape formation), and can also be produced by human activity (communication systems, infrastructure networks). In mathematics, they arise in many problems in fields such as mathematical physics or partial differential equations, and are studied extensively in geometric measure theory and dynamical systems. Fractals often display chaotic behaviour, in the sense that a fractal object constructed by iterating a relatively simple replication rule can exhibit very high levels of complexity and behave in ways that are very difficult to predict in advance. The first goal of the proposed research is to develop new methods of studying such structures, based on harmonic analysis and additive combinatorics. Traditionally, the use of harmonic analysis in the study of fractals has been relatively limited. I plan to develop a theory of singular and oscillatory integrals for fractal sets, modelled after the analogous theory for smooth manifolds in classical harmonic analysis (including restriction estimates, maximal and averaging operators, and differentiation theorems), but also incorporating new features that only occur in the fractal setting. This work will draw on the insights and methods from both harmonic analysis and the more recently developed field of additive combinatorics. While some foundational results have already been obtained, proving that this is a viable and interesting theory, many further questions remain open. In particular, fractal sets can exhibit harmonic-analytic behaviour that does not have exact analogues either for manifolds or for the discrete objects studied in additive combinatorics, and I would like to investigate these new phenomena. The second goal is to apply these developments to questions in dimension theory, geometric measure theory and dynamical systems. A central family of questions in geometric measure theory concerns projections, slices, intersections, and arithmetic sums and differences of sets of specified dimensionality. Of particular significance are fractal sets that arise in dynamical systems and mathematical physics, such as attractors or invariant sets. For example, it can be very easy to say what a "typical" projection or slice of a fractal should look like, but very difficult to prove a similar statement about a specific projection or slice. Similarly, there are many situations where qualitative results are easy to prove, but it is much harder to obtain a quantitative estimate. Harmonic analysis has been useful in such contexts in the past, and I expect that the new methods I plan to develop will lead to further advances.