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。* * * * * * 1。集合中的模式：******识别大型但任意集合中的模式是目前纯数学和数据科学中一个充满活力的研究领域。我们的目标是建立一个内聚理论来解释关于集合中构型的许多现象，并将它们置于分形几何和流形欧几里得调和分析的背景下，同时强调离散和连续对应物之间的异同。* * * * * * 2。方向集：******起源于欧几里得Kakeya， Bochner-Riesz和限制猜想等领域的深度开放问题，这类问题侧重于在许多方向上依赖线段的算子的行为。本文特别讨论了方向极大算子、极大方向希尔伯特变换及其与类kakeya集合的关系。* * * * * * 3。平滑与解耦：******解耦不等式目前在多个领域处于研究的前沿。它们在解决数论和几何分析中长期存在的问题方面已被证明是有用的。我们的目标是探索解耦在入射几何和偏微分方程中的应用。* * * * * * 4。奇异性的解决：******零集函数在几何和分析中的许多问题中起着关键作用。许多奇异积分和振荡积分的行为取决于这些集合的微局部结构。我们使用先前提出的奇异解算法来研究实解析函数的振荡积分算子和跳数。* * * * * * 5。柯西积分和门格尔曲率：******图上的柯西积分具有惊人的通用性：它是全纯函数的再现公式，是卡尔德隆-齐格蒙德奇异积分算子的一个基本例子，体现了称为门格尔曲率的几何量。类似的解析和几何特征是否适用于其他感兴趣的核？我们提出了一种基于曲率的方法来研究多维复数分析中出现的某些全纯再现核。

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.