Selected topics in harmonic analysis

现实生活中遇到的大多数结构都是由简单组件组成的复杂组合。有效的分析需要对这样一个对象进行仔细的分解，以便将整体的特性转化为各个部分，反之亦然。谐波分析使这种分解机制变得精确。它的影响遍及数学内外的各个学科——它是量子物理学和许多现代技术（如信号处理、医学和地震成像）的首选语言。该提案中的问题，虽然植根于谐波分析，但在于其与几何测量理论，可加组合，偏微分方程（PDE）和几个复杂变量等领域的接口，解决了多个数学社区感兴趣的问题。各级都有研究培训的机会。

Most structures encountered in real life are complex assemblies of simpler components. Effective analysis requires careful decomposition of such an object, so that properties of the whole can be translated to the pieces and vice versa. Harmonic analysis makes such decomposition mechanisms precise. Its impact spans disciplines within and beyond mathematics - it is the language of choice in quantum physics and many modern technologies like signal processing, medical and seismic imaging. The problems in this proposal, while rooted in harmonic analysis, lie at its interface with areas such as geometric measure theory, additive combinatorics, partial differential equations (PDE) and several complex variables, addressing issues of interest to multiple mathematical communities. Opportunities for research training exist at all levels. 1. Patterns in sets: Identifying patterns in large but otherwise arbitrary sets is currently a vibrant area of research both in pure mathematics and data science. We aim to build a cohesive theory that explains numerous phenomena concerning configurations in sets, and places them in the context of fractal geometry and Euclidean harmonic analysis on manifolds, while highlighting similarities and differences between the discrete and continuous counterparts. 2. Sets of Directions: Originating from deep open questions in the field such as the Euclidean Kakeya, Bochner-Riesz and restriction conjectures, this genre of problems focuses on behaviour of operators that rely on line segments in many directions. The proposal specifically addresses directional maximal operators, maximal directional Hilbert transforms and their relations with Kakeya-like sets. 3. Smoothing and Decoupling: Decoupling inequalities are currently at the cusp of research in multiple areas. They have proven instrumental in the resolution of long-standing problems in number theory and geometric analysis. Our objective is to explore applications of decoupling in incidence geometry and PDE. 4. Resolution of singularities: Zero sets of functions play a key role in many problems in geometry and analysis. The behaviour of many singular and oscillatory integrals of interest depends on the microlocal structure of these sets. We use an algorithm for resolution of singularities developed earlier to study oscillatory integral operators and jumping numbers of real-analytic functions. 5. Cauchy integral and Menger curvature: The Cauchy integral on a graph is amazingly versatile: it is a reproducing formula for holomorphic functions, a fundamental example of a Calderon-Zygmund singular integral operator and embodies a geometric quantity called the Menger curvature. Do similar analytic and geometric characterizations hold for other kernels of interest? We propose a curvature-based approach for studying certain holomorphic reproducing kernels arising in multidimensional complex analysis.