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

通过自守形式微分方程连接 L 函数和弦理论

• 批准号: 2302309
• 负责人: Kimberly Logan
• 金额: $ 16.14万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302309&HistoricalAwards=false
• 关键词: Connections; Between; functions; String; Theory

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.

自动形式表现出数字理论与物理学之间的实质性联系。首先，它们在数字理论中似乎是L功能理论中的基础。 L功能揭示了许多重要数字理论主题，例如素数的分布。在物理学中，自动形式的超对称弦理论的模型对称条件，用于寻找重力散射放大器的能力（假设的重力颗粒）。找到重力散射放大器的高阶能力可以为相对性和实验数据之间的差异提供量子校正。该项目旨在回答围绕L功能和散射放大器理论的许多问题，该问题使用自动形式的研究进行某些弦乐相互作用。为了获得更广泛的影响，PI将领导本科研究项目，继续与Sonya Kovalevsky Day和Navajo Math Circle一起参与，并将为基础教师写一篇开放式访问文本，其中关注活动和课程的数学教师中心，以美国原住民的传统和思想为中心。涉及自动形式的研究是一个共同的问题，这些计划与大多数问题有关。具体而言，PI计划回答有关零（2）L功能的特殊值的许多问题。这些问题中的大多数将L功能的零与某些操作员的光谱联系起来。该项目还解决了有关重力散射放大器的研究引起的许多问题。 PI将对SL（2）溶液的傅立叶模式进行更详细的分析，并通过封闭形式的扩展对解决方案家族进行分类。在研究这些傅立叶溶液的过程中，PI将解决与除数函数的变化卷积总和有关的开放猜想。某些转移的卷积总和还将应用于L功能的子凸度范围。 PI还将计算SL（3）中的光谱溶液，并使用这些技术来证明非分类Eisenstein系列的量子独特性。为了解决这些问题，PI将在功能分析，分析数理论，特殊功能理论和PDES中使用技术。本项目由代数和数字理论计划和既定计划共同资助，以及既定的竞争研究（EPSCOR）的既定计划（EPSCOR）。这奖通过NSF的法定任务，反映了通过评估的诚实构成的构成群体，该奖项已被评估构成群体和众所周知。