Mathematical Sciences: Oscillatory Integrals, Singular Integrals, and Their Applications

摘要：潘潘的研究将集中在谐波分析和相关领域的许多问题中自然出现的各种奇异积分算子和振荡积分算子，包括复分析和偏微分方程理论。随着这些积分在大量数学问题中的应用，它们的重要性得到了越来越多的认识。虽然这一领域的研究已经深入而富有成果，但仍有更多的东西有待了解。该项目将涵盖许多当前感兴趣的主题，完成后将在全面理解这些算子和数学分析中的相关问题方面迈出一大步。所研究的对象构成了热传导和波动等物理现象数学建模的重要组成部分。它们也出现在涉及电路和信号处理的工程问题中。因此，获得对这些物体的理解不仅具有理论价值，而且对其他科学和工程领域也具有指导意义。

Abstract: Pan Pan's investigation will be focused on various singular integral and oscillatory integral operators which arise naturally in many problems in harmonic analysis and related fields including complex analysis and the theory of partial differential equations. With applications to a large number of problems in mathematics, these integrals have received an increasing amount of recognition for their importance. While research in this area has been intensive and fruitful, much more remains to be understood. This project will cover many topics of current interest, which when finished will make a big stride toward the full understanding of these operators and related problems in mathematical analysis. The objects under investigation form an important part of the mathematical modeling of physical phenomena such as heat conducting and wave motion. They also appear in engineering problems involving electric circuits and signal processing. Thus, gaining understanding of such objects will not only have theoretical values but also provide guidance in other areas of science and engineering.