CAREER: Eigenfunctions, Weyl Laws, and Random Waves

大量的物理现象，从波的传播到量子粒子的位置，都是由一些特殊函数的行为决定的，这些函数可以解决所谓的亥姆霍兹方程。这项研究计划的首要目标是了解这些函数的性质如何取决于所研究空间的潜在几何形状。该项目包括两项主要的教育工作。首先是为即将入学的研究生建立工作小组；它的目标是让学生培养一套强大的技能来申请助学金，并为教学做好准备。第二是开发半经典分析课程的在线材料，当没有提供面对面的课程版本时，学生可以使用。理解拉普拉斯特征函数的行为在数学物理中是至关重要的，自18世纪以来一直在研究。特别是，高能本征函数的集中性质的具有挑战性的研究已经成为广泛工作的主题，并且为研究拉普拉斯本征函数而开发的工具几乎对光谱理论和几何分析的每个领域都产生了深远的影响。该项目涉及研究者开发的一个框架，通过使用半经典分析，从特征函数在相空间中的集中和传播行为中提取有关特征函数结构的信息。当前的研究旨在适应和发展这些方法，通过研究它们的逐点增长、L^p-范数、相关的两点Weyl定律以及在随机波中的应用，来更深入地理解特征函数的行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of some special functions that solve what is known as the Helmholtz equation. The overarching goal of this research program is to understand how properties of these functions depend upon the underlying geometry of the space being studied. The project includes two main educational efforts. The first is the creation of working groups for incoming graduate students; its goal is to prepare the students to develop a strong set of skills to apply for grants and to prepare them for teaching. The second is the development of online materials for a course on semiclassical analysis that will be available for students to use whenever the in-person version of the course is not offered. Understanding the behavior of Laplace eigenfunctions is of fundamental importance in mathematical physics and has been studied since the 1700s. In particular, the challenging study of concentration properties of high energy eigenfunctions has been the subject of extensive work, and the tools developed for studying Laplace eigenfunctions have had a profound impact on nearly every area of spectral theory and geometric analysis. This project concerns a framework developed by the investigator to extract information on the structure of eigenfunctions from their concentration and propagation behavior in phase space via the use of semiclassical analysis. The current research aims to adapt and develop these methods to reach a deeper understanding of eigenfunction behavior by studying their pointwise growth, L^p-norms, associated two-point Weyl Laws, and applications to random waves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.