RUI: Harmonic Maass Forms and Quantum Modular Forms

这个项目研究关于模形式的问题，广义地讲，模形式是在数学和数论中扮演关键角色的某些对称复函数。例如，模形式理论在1995年费马最后定理的证明中是重要的，这是一个数百年来一直没有解决的猜想。模形式还与黎曼假设相联系，这是一个尚未解决的主要猜想，并产生了组合学、数学物理、密码学等方面的应用。研究目标包括模拟模形式、调和Maass形式、量子模形式和相关函数的理论和应用，这些都是模形式更现代的关系。虽然这些学科在过去的20年里有了长足的发展，但仍然缺乏全面的理论。该奖项中的一些研究项目是专门为本科生研究设计的。面向广大数学受众的本科生研究、辅导、培训和写作也是该项目的主要组成部分。这个RUI项目的主要目标是了解调和Maass形式(模拟模形式)的全纯部分和相关函数所起的作用，例如全纯量子模形式、环面纽结和量子模性、部分theta函数、量子Jacobi形式、模拟Jacobi形式和组合应用。方法和目标建立在P.I.S早期在这些领域的工作基础上，包括不断发展的调和Maass形式理论、发展中的量子模形式主题以及这两个领域之间的联系等的工具和结果。该奖项反映了美国国家科学基金会的法定使命，并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 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.