RUI: Harmonic Maass Forms and Quantum Modular Forms

RUI：谐波马斯形式和量子模形式

• 批准号: 2200728
• 负责人: Amanda Folsom
• 金额: $ 27.33万
• 依托单位: Amherst College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200728&HistoricalAwards=false
• 关键词: RUI; Harmonic; Maass; Forms; Quantum

This project studies questions about modular forms, broadly speaking, which are certain symmetric complex functions that have played key roles in mathematics and number theory. For example, the theory of modular forms was important in the 1995 proof of Fermat's Last Theorem, a conjecture which remained unsolved for centuries. Modular forms are also connected to the Riemann Hypothesis, a major unsolved conjecture, and yield applications to combinatorics, mathematical physics, cryptography, and more. Research goals include studying the theory and applications of mock modular forms, harmonic Maass forms, quantum modular forms, and related functions, which are more modern relatives to modular forms. While these subjects have seen substantial developments within the last 20 years, a comprehensive theory is still lacking. Some research projects within this RUI award are specifically designed for undergraduate research. Undergraduate research mentoring and training and writing for broad mathematical audiences are also major components of this project. An overarching goal of this RUI project is to understand roles played by the holomorphic parts of harmonic Maass forms (mock modular forms) and related functions, for example, holomorphic quantum modular forms, torus knots and quantum modularity, partial theta functions, quantum Jacobi forms, mock Jacobi forms, and combinatorial applications. Methods and goals build on the P.I.’s earlier work in these areas, and include tools and results from the growing theory of harmonic Maass forms, the developing subject of quantum modular forms, and connections between these two areas, among others.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.

这个项目研究关于模形式的问题，广义地说，这是某些在数学和数论中发挥关键作用的对称复函数。 例如，模形式的理论在1995年证明费马大定理中很重要，这是一个几个世纪以来一直没有解决的猜想。 模形式也与黎曼猜想（Riemann Hypothesis）有关，这是一个主要的未解猜想，并产生了组合数学、数学物理、密码学等方面的应用。研究目标包括研究模拟模形式，谐波马斯形式，量子模形式和相关函数的理论和应用，这些都是模形式的现代亲戚。 虽然这些主题在过去20年中有了很大的发展，但仍然缺乏一个全面的理论。 RUI奖项中的一些研究项目是专门为本科生研究设计的。 本科生研究指导和培训和写作广泛的数学观众也是这个项目的主要组成部分。 这个RUI项目的首要目标是了解调和马斯形式（模拟模形式）和相关函数的全纯部分所扮演的角色，例如，全纯量子模形式，环面结和量子模块化，部分θ函数，量子雅可比形式，模拟雅可比形式和组合应用。 方法和目标建立在PI之上该奖项反映了NSF的法定使命，并被认为是值得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来支持的。

项目成果

Hook length bias in odd versus distinct partitions

奇数分区与不同分区中的钩长度偏差

• 发表时间: 2023
• 期刊: Séminaire lotharingien de combinatoire
• 作者: Ballantine, Cristina;Burson, Hannah;Craig, William;Folsom, Amanda;Wen, Boya
• 通讯作者: Wen, Boya

Hook length biases and general linear partition inequalities

• DOI: 10.1007/s40687-023-00402-1
• 发表时间: 2023-03
• 期刊: Research in the Mathematical Sciences
• 影响因子: 1.2
• 作者: C. Ballantine;Hannah E. Burson;William Craig;A. Folsom;Boya Wen

Equidistribution and partition polynomials

• DOI: 10.1007/s11139-023-00762-w
• 发表时间: 2022-09
• 期刊: The Ramanujan Journal
• 作者: A. Folsom;Joshua Males;Larry Rolen

On Best Practices for the Recruitment, Retention, and Flourishing of LGBTQ+ Mathematicians

关于 LGBTQ 数学家的招募、保留和发展的最佳实践

• DOI: 10.1090/noti2709
• 发表时间: 2023
• 期刊: Notices of the American Mathematical Society
• 作者: Buckmire, Ron;Folsom, Amanda;Goff, Christopher;Hoover, Alexander;Nakao, Joseph;Sather-Wagstaff, Keri
• 通讯作者: Sather-Wagstaff, Keri

Periodic partial theta functions and q-hypergeometric knot multisums as quantum Jacobi forms

量子雅可比形式的周期性偏 theta 函数和 q 超几何结多重和

• DOI: 10.1016/j.jmaa.2023.127727
• 发表时间: 2023
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Amanda Folsom