Three problems in Harmonic Analysis

在这个数学研究项目中，重点研究谐波分析中的三个不同问题。第一个是关于一大类非阿贝尔齐次李群的散度旋度型估计，其中包括海森堡群作为最简单的非平凡模型。第二个问题是关于一类伪微分算子的构造，它们适用于两个不同的标志；然后，这些算子将用于研究某些在研究几个复杂变量时自然出现的偏微分方程的解算子。第三个问题是关于一类椭圆部分具有保形不变性的非线性波动方程，我们引入保形几何元素来研究这类波动方程。从分析的角度来看，这些问题提出了有趣的挑战。广义上讲，谐波分析通过将函数或信号分解成不同的频率来研究它们。它的应用非常广泛，从信号处理和断层扫描到相对论和量子力学等物理学问题。Yung和他的合作者将研究谐波分析中一些深刻而基本的问题，以及它与其他数学领域（如几个复杂变量和共形几何）的联系。特别是，在其中一个拟议的项目中，将对某些非线性波如何在空间中传播进行研究。这项研究将与该领域的专家以及一些刚刚开始研究生涯的年轻研究人员合作。

In this mathematics research project, the focus is on three different problems in harmonic analysis. The first one is on a divergence-curl type estimate on a large class of non-abelian homogeneous Lie groups that includes the Heisenberg group as the simplest, nontrivial model. The second one is about the construction of a certain class of pseudodifferential operators that are adapted to two different flags; these operators will then be used to study the solution operators of certain partial differential equations that arise naturally in the study of several complex variables. The third problem is about a class of nonlinear wave equations whose elliptic parts exhibit conformal invariance, and we bring in elements from conformal geometry in studying these wave equations. These problems present interesting challenges from the analytical point of view.Broadly speaking, harmonic analysis studies functions or signals by decomposing them into different frequencies. Its applications are abundant, ranging from signal processing and tomography to problems arising from physics such as relativity and quantum mechanics. Yung and his collaborators will investigate some deep and fundamental questions in harmonic analysis, along with its connections to other areas of mathematics such as several complex variables and conformal geometry. In particular, in one of the proposed projects, a study of how certain nonlinear waves propagate through space will be carried out. The research will be in collaboration both with faculties that are experts in the field, as well as some young researchers who are just starting their research careers.