Connections Between L-functions and String Theory via Differential Equations in Automorphic Forms

自守形式证明了数论和物理学之间的实质性联系。首先，它们出现在数论中，作为L-函数理论的基石。L-函数揭示了许多重要的数论主题，如素数的分布。在物理学中，自守形式是超对称弦理论的对称条件模型，并被用来寻找引力子（假设的引力粒子）的散射振幅系数。发现引力子散射振幅的高阶系数可以为相对论和实验数据之间的差异提供量子校正。这个项目旨在回答围绕L-函数理论的一些问题，并使用自守形式的研究来解决某些弦相互作用的散射振幅。为了更广泛的影响，PI将领导本科生研究项目，继续参与Sonya Kovalevsky Day和Navajo Math Circle，并将为小学教师编写一份开放获取的数学教科书，重点关注以美洲原住民传统和思想为中心的活动和课程。涉及自守形式的微分方程的研究是连接该项目中大多数问题的共同线索。具体来说，PI计划回答一些与GL（2）L-函数的零点和特殊值有关的问题。这些问题中的大多数都与L-函数的零点和某些算子的谱有关。 该项目还解决了引力子散射振幅研究中出现的一些问题。PI将对SL（2）解的傅立叶模式进行更详细的分析，并通过闭合形式展开对一族解进行分类。在研究这些傅立叶解的过程中，PI将解决一个与除数函数的移位卷积和有关的公开猜想。某些移位卷积和也可以应用于L-函数的次凸性界。PI还将计算SL（3）中的谱解，并使用这些技术证明非简并Eisenstein级数的量子唯一遍历性。为了解决这些问题，PI将使用泛函分析，解析数论，特殊函数理论，该项目由代数和数论计划以及刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF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进行评估，被认为值得支持的搜索.

Automorphic forms demonstrate a substantial link between number theory and physics. First, they appear in number theory as building blocks in the theory of L-functions. L-functions shed light on many important number theoretic topics such as the distribution of prime numbers. In physics, automorphic forms model symmetry conditions of supersymmetric string theory and are used to find coefficients of the scattering amplitude for gravitons (hypothetical particles of gravity). Finding higher order coefficients of the graviton scattering amplitude may provide a quantum correction to the discrepancy between relativity and experimental data. This project seeks to answer a number of questions centered around the theory of L-functions and scattering amplitudes for certain string interactions using the study of automorphic forms. For broader impacts, the PI will lead undergraduate research projects, continue her involvement with the Sonya Kovalevsky Day and the Navajo Math Circle, and will write an open access text on math for elementary teachers with a focus on activities and curriculum that centers Native American traditions and ideas​.The study of differential equations involving automorphic forms is a common thread connecting most of the questions addressed in this project. Specifically, the PI plans to answer a number of questions relating to the zeros and special values of GL(2) L-functions. Most of these questions relate the zeros of L-functions to the spectrum of certain operators. The project also addresses a number of questions arising from the study of scattering amplitudes for gravitons. The PI will conduct a more detailed analysis of the Fourier modes of the SL(2) solutions and classify a family of solutions through a closed form expansion. In the course of the study of these Fourier solutions, the PI will address an open conjecture relating to a shifted convolution sum of divisor functions. Certain shifted convolution sums also have applications to subconvexity bounds for L-functions. The PI will also compute a spectral solution in SL(3) and uses these techniques to prove quantum unique ergodicity for non-degenerate Eisenstein series. To address these problems, the PI will use techniques in functional analysis, analytic number theory, the theory of special functions, and PDEs.This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).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.