Modular Forms, Combinatorial Generating Functions, and Hypergeometric Functions

数学的一个分支称为数论，它起源于对整数和整数值函数的研究。数论的美好之处之一是，当深入研究时，看似简单的问题可以发展成丰富而复杂的发现。PI将调查一些与模和自同构形式有关的问题，这些问题在上个世纪的数论中的许多主要问题中发挥了核心作用。通过这项研究，国际数学社将指导本科生研究生，吸引博士生参与研究，为研究生举办地区性数论会议，利用和发展实践在研究指导和课堂教学中建立包容性，并继续与众多专业女性研究人员合作。国际数学社将研究模形式、调和Maass形式、模拟模形式和量子模形式之间的关系，这是模形式理论的一个主要研究领域。从历史上看，组合生成函数的例子是各种类型的模块化行为的丰富来源，确定组合生成函数的模块化的一般理论将是有价值的。为了更好地理解组合生成函数在模型理论中的作用，PI将参与涉及各种组合秩母函数的模块化的项目，以及与分拆理论相关的组合函数的不等式及其矩。PI还将从事各种类型的超几何函数的研究，这些函数是许多数学领域的重要组成部分，应用于代数、分析、算术几何、组合数学、数学物理和数论。PI在这一领域的项目包括Ramanujan-Sato型级数和与算术三角群有关的超同余、推广的van Hamme超同余猜想和超几何函数的p-进理论，它可以给出超几何L-函数的p-进方面的显式描述。超几何级数理论中的项目旨在加强我们对超几何函数在更大的模块和自同构形式景观中的作用的理解。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。

The branch of mathematics called number theory evolved out of the study of integers and integer-valued functions. One of the beautiful things about number theory is that seemingly simple questions, when deeply investigated, can blossom into rich and intricate discoveries. The PI will investigate a number of problems that relate to modular and automorphic forms, which have played a central role in many major problems in number theory over the last century. Through this research, the PI will mentor undergraduate research students, engage PhD students in research, hold a regional number theory conference for graduate students, utilize and develop practices to build inclusivity in research mentoring and classroom teaching, and continue collaborations with numerous professional women researchers.The PI will investigate relationships between modular forms, harmonic Maass forms, mock modular forms, and quantum modular forms, which is a major area of study in modular forms theory. Historically, examples arising from combinatorial generating functions have been a rich source of varied types of modularity behavior, and it would be of value to determine a general theory for the modularity of combinatorial generating functions. The PI will engage in projects involving modularity of various combinatorial rank generating functions, as well as inequalities for combinatorial functions related to partition theory and their moments in order to better understand the role of combinatorial generating functions in modular forms theory. The PI will also engage in research on hypergeometric functions of various types, which are an important component of many areas of mathematics and have applications to algebra, analysis, arithmetic geometry, combinatorics, mathematical physics, and number theory. Projects of the PI in this area include Ramanujan-Sato type series and supercongruences related to arithmetic triangle groups, generalized van Hamme supercongruence conjectures, and a p-adic theory of hypergeometric functions which may yield explicit descriptions of p-adic aspects of hypergeometric L-functions. The projects in the theory of hypergeometric series are designed to strengthen our understanding of the role of hypergeometric functions in the larger modular and automorphic forms landscape.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.