Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity

这个项目将研究算术量子混沌中的一些悬而未决的问题，算术量子混沌是数论和理论物理之间的一个多学科领域。量子混沌的数学研究涉及到量子力学和经典力学之间的相互作用。我们的目标是理解后者的混沌行为，如遍历台球流，是如何通过前者的混沌行为来反映的。这个项目的主要任务之一是在与数论有额外联系的环境中研究这些关系：研究素数及其分布。作为UVA数论REU的一部分，PI还将指导本科生和高中生，并继续培训博士生。PI将研究涉及L函数次凸界应用的自同构型解析理论中的问题。一个需要解决的主要问题是限制算术量子唯一遍历性，它涉及到模曲面上曲线上自同构形的质量均匀分布的研究。此外，PI还将证明Rankin-Selberg L-函数的新的次凸界。证明这些结果的方法将依赖于谱互易的新实例，它们是L函数的矩的恒等式，并将使用与自同构形及其L函数的积分相关的周期公式来证明。该奖项反映了美国国家科学基金会的法定使命，并已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。

This project will investigate some open questions in arithmetic quantum chaos, a multidisciplinary field at the interface of number theory and theoretical physics. The mathematical study of quantum chaos involves the interplay between quantum mechanics and classical mechanics. The goal is to understand how chaotic behavior of the latter, such as ergodic billiard flow, is reflected by chaotic behavior of the former. One of the main tasks of this project is to investigate these relations in settings where there are additional ties to number theory: the study of prime numbers and their distribution. The PI will also supervise undergraduate and high school students as part of the UVA number theory REU, as well as continue to train PhD students.The PI will investigate problems in the analytic theory of automorphic forms that involve applications of subconvex bounds for L-functions. One major problem to be resolved is restricted arithmetic quantum unique ergodicity, which involves the study of the mass equidistribution of automorphic forms on curves on the modular surface. The PI will additionally prove new subconvex bounds for Rankin-Selberg L-functions. The methods to prove these results will rely on new instances of spectral reciprocity, which are identities for moments of L-functions, and will be proven using period formulae relating integrals of automorphic forms and their L-functions.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.