Modular Forms, Combinatorial Generating Functions, and Hypergeometric Functions

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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.