Selected topics in harmonic analysis

在真实的生活中遇到的大多数结构都是由较简单的部件组成的复杂组件。有效的分析需要仔细地分解这样一个对象，这样整体的属性就可以转化为碎片，反之亦然。谐波分析使这种分解机制精确。它的影响跨越了数学内外的学科-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。在这个建议中的问题，虽然植根于调和分析，在于它的接口领域，如几何测度理论，加法组合，偏微分方程（PDE）和几个复杂的变量，解决问题的兴趣，以多个数学社区。各级都有研究培训的机会。*1.集合中的模式：** 在大型但任意的集合中识别模式目前是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凝聚力的理论，解释了许多现象的配置集，并将它们放在分形几何和欧氏调和分析的背景下流形，同时突出离散和连续对应之间的相似性和差异。 2.方向集：** 起源于该领域的深度开放问题，如欧几里得Kakeya，Bochner-Riesz和限制条件，这种类型的问题侧重于在许多方向上依赖线段的运营商的行为。该提案具体解决了方向极大算子，最大方向希尔伯特变换及其与Kakeya类集的关系。 3.平滑和解耦：** 解耦不等式目前处于多个领域研究的风口浪尖。他们已被证明有助于解决长期存在的问题，在数论和几何分析。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.奇点的解析：** 零函数集在几何和分析中的许多问题中起着关键作用。许多奇异和振荡积分的行为取决于这些集的微局部 * 结构。我们使用一个算法的解决方案的奇异性研究振荡积分算子和跳跃数的实解析函数。5.柯西积分和门格尔曲率：* 图上的柯西积分是惊人的通用：它是全纯函数的再生公式，是Calderon-Zygmund奇异积分算子的基本例子，并体现了一个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内核？我们提出了一种基于曲率的方法来研究某些全纯再生核在多维复分析中产生的。

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.