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

自同构形式证明了数论和物理学之间的本质联系。首先，它们在数论中作为L函数理论的基石出现。L函数揭示了许多重要的数论问题，如素数的分布。在物理学中，自同构形式模拟了超对称弦理论的对称条件，并被用来求出引力子(假设的引力粒子)的散射幅度系数。找到引力子散射振幅的高阶系数可能会对相对论和实验数据之间的差异提供量子修正。这个项目试图通过对自同构形式的研究来回答一些围绕L理论的问题--某些弦相互作用的函数和散射幅度。对于更广泛的影响，PI将领导本科生研究项目，继续参与索尼娅·科瓦列夫斯基日和纳瓦霍数学圈，并将为小学教师编写一份开放获取的数学文本，重点是以美国土著传统和思想​为中心的活动和课程。涉及自同构形式的微分方程式的研究是连接该项目中大多数问题的共同主线。具体地说，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。这些问题大多与L函数的零点与某些算子的谱有关。该项目还解决了研究引力子散射幅度时产生的一些问题。PI将对SL(2)解的傅里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。某些移位卷积和也适用于L函数的次凸界。PI还将计算SL(3)中的谱解，并使用这些技术证明非退化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、特殊函数理论和偏微分方程组的技术。该项目由代数和数论计划和既定的激励竞争研究计划(EPSCoR)联合资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.