Collaborative Research: Microlocal Concentration and Propagation in Spectral Theory

在自然界中，波型方程的解出现在量子粒子的波函数、音乐厅的声学和引力波的散射等广泛的现象中。 在这些情况下，存在与空间的基础几何形状相关的某些基本频率和振动模式。 这些频率和模式对于理解波浪现象的行为起着至关重要的作用。例如，它们出现在音乐厅的声学设计和乐器的振动中。 该项目开发基于传播现象的工具来研究空间几何形状与相应模式行为之间的微妙关系。这项研究将为一般空间的振动模式结构提供新的见解。 对于具有非正曲率的空间，许多定量的规律性结果（通过 L^p 范数测量）是已知的。 然而，在限制较少的几何假设下，这些类型的估计是不可用的。 该项目的主要目标有两个：(1) 描述导致单个模态 L^p 范数增长的集中现象，以及 (2) 利用这种理解仅基于基础流形的动态特性来生成模态和频率的估计。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

In nature, solutions of wave-type equations appear in phenomena as far reaching as wave functions of quantum particles, the acoustics of concert halls, and scattering of gravitational waves. In these situations, there are certain fundamental frequencies and vibrational modes associated to the underlying geometry of the space. These frequencies and modes play an essential role in the understanding of the behavior of wave phenomena. They appear, for example, in the acoustical design of concert halls and the vibration of instruments. This project develops tools, based on propagation phenomena, to study the delicate relationship between the geometry of the space and behavior of the corresponding modes. This research will provide new insight into the structure of vibrational modes for general spaces. Many quantitative regularity results (as measured by L^p norms) are already known for spaces with non-positive curvature. However, these types of estimates are unavailable under less restrictive geometric assumptions. The main goals of this project are two fold: (1) to describe the concentration phenomena that result in L^p norm growth for individual modes, and (2) to use this understanding to produce estimates for modes and frequencies based solely on dynamical properties of the underlying manifold.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.