CAREER: Maass Forms, Modular Forms, and Applicati

该奖项支持数论、组合学和李论交叉领域的研究。特别是，P.I.试图确定弱质量形式及其推广，（非全纯）Jacobi形式和组合q-超几何级数之间更精确的关系和相互作用。主要项目目标包括研究量子模形式，顶点算子代数迹函数和渐变维数，以及组合q系列的自同构性质。私家侦探还将把各级教育和推广计划整合到奖项目标中。也就是说，pi已经开始与纽黑文公立学校合作，并将继续为中小学生开发数学丰富课程。私家侦探还将担任耶鲁大学-纽黑文MATHCOUNTS Outreach分会的指导老师。MATHCOUNTS Outreach是一个在纽黑文公立学校推广数学的全国性组织的本科生分支。P.I.还寻求通过研究合作、指导和外展计划，为研究生、本科生和博士后，包括各级数学领域的妇女和女孩，增加研究和教育机会。数论是数学最古老的分支之一，今天仍然是一个广泛而活跃的研究领域。模形式扮演了许多基本的角色；它们是证明费马大定理、朗兰兹程序、黎曼假设、伯奇猜想和斯温纳顿-戴尔猜想的核心，并在组合学、密码学、数学物理和许多其他领域得到了应用。pi将研究模形式的自然亲戚，即弱质量形式及其推广。虽然最近取得了一些进展，但缺乏一个全面的理论。所提出的研究旨在有助于理解这些函数的作用，不仅在数论和模形式，而且在组合学和李论。

This award supports research at the intersection of number theory, combinatorics, and Lie theory. In particular, the P.I. seeks to determine more precise relationships and interplay between weak Maass forms and their generalizations, (non-holomorphic) Jacobi forms, and combinatorial q-hypergeometric series. The major project objectives include a study of quantum modular forms, vertex operator algebra trace functions and graded dimensions, and the automorphic properties of combinatorial q-series. The P.I. will additionally integrate a number of educational and outreach programs at all levels into the award objectives. Namely, the P.I. has begun a collaboration with the New Haven Public Schools, and will continue to develop a mathematics enrichment program for elementary and middle school students. The P.I. will also act as faculty advisor to the Yale University-New Haven chapter of MATHCOUNTS Outreach, an undergraduate arm of the national organization which promotes mathematics in New Haven Public Schools. The P.I. also seeks to enhance research and educational opportunities for graduate students, undergraduate students, and postdoctoral fellows, including women and girls in mathematics at all levels, through research collaboration, mentoring, and outreach programs. Number theory is one of the oldest branches of mathematics, and continues to be a field of extensive and active research today. Modular forms have played many fundamental roles; they are central to the proof of Fermat's Last Theorem, the Langlands program, the Riemann hypothesis, and the Birch and Swinnerton-Dyer conjecture, for example, and yield applications in combinatorics, cryptography, mathematical physics, and many other areas. The P.I. will study natural relatives of modular forms, namely weak Maass forms and their generalizations. While recent developments have been made, a comprehensive theory is lacking. The proposed research seeks to contribute to the understanding of the roles of these functions not only within number theory and modular forms, but also combinatorics and Lie theory.