Spectral Asymptotics of Laplace Eigenfunctions

该研究项目属于谱渐近领域，研究流形（曲面和曲率空间）上高频拉普拉斯特征函数的行为。特征函数的物理类似物是驻波，特征值可以认为是它们对应的频率。高频特征函数和它们所在的流形几何之间的相互依赖关系是从量子物理学到数论的广泛领域的核心。的确，特征函数是Schrödinger方程的稳态解，它们的特征值是相应的能量。为了说明与数论的联系，精确计算平面环面上给定频率的特征函数的数量的任务相当于计算整数可以表示为（比如说）两个平方的和的方法的数量。该项目旨在开发新的工具来促进对谱渐近的理解，谱渐近与看似不同的数学和科学领域的相互联系使其研究特别有价值。作为研究项目的一部分，PI打算开发和使用来自微局部分析和傅立叶积分算子理论的工具来改进描述高频特征函数行为的各种公式，特别是描述底层几何对这些公式的影响。PI打算在流形积的Weyl定律的剩余项的猜想方面取得进展，发展用于谱渐近和几何测量理论的傅里叶积分算子的一般多线性理论，并进一步探索谱量与流形中相应几何构型之间的联系。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。

The research project falls within the field of spectral asymptotics, which studies the behavior of high-frequency Laplace eigenfunctions on manifolds (surfaces and spaces with curvature). The physical analogues of eigenfunctions are standing waves, and the eigenvalues may be thought of as their corresponding frequencies. The interdependence between high-frequency eigenfunctions and the geometry of the manifold on which they live is central to a broad range of fields from quantum physics to number theory. Indeed, eigenfunctions are steady-state solutions to the Schrödinger equation, and their eigenvalues are the corresponding energies. To illustrate the connection to number theory, the task of accurately counting the number of eigenfunctions of a given frequency on the flat torus is equivalent to counting the number of ways an integer can be expressed as the sum of, say, two squares. This project aims to develop new tools to advance understanding in spectral asymptotics, whose interconnectedness to seemingly disparate areas of mathematics and science make its study particularly valuable. As part of the research project, the PI intends to develop and use tools from microlocal analysis and the theory of Fourier integral operators to refine a variety of formulas describing the behavior of high-frequency eigenfunctions, and in particular describing what effect the underlying geometry has on these formulas. The PI intends to make advancements towards a conjecture on the remainder term of the Weyl law for products of manifolds, to develop a general multilinear theory of Fourier integral operators for use in both spectral asymptotics and geometric measure theory, and to further explore the connection between spectral quantities and the presence of corresponding geometric configurations in the 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.