CAREER: Maass Forms, Modular Forms, and Applicati

该奖项支持在数论，组合学和李理论的交叉研究。特别是，P.I.试图确定更精确的关系和弱马斯形式和它们的推广，（非全纯）雅可比形式和组合q-超几何级数之间的相互作用。主要的计画目标包括量子模形式、顶点算子代数迹函数和分次维数的研究，以及组合q-级数的自守性质。 私家侦探此外，还将把各级教育和推广计划纳入奖励目标。 也就是私家侦探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计划。 私家侦探他还将担任耶鲁大学纽黑文分校的数学外展组织的教师顾问，该组织是促进纽黑文公立学校数学的国家组织的本科生分支。私家侦探还寻求通过研究合作、指导和外联方案，为研究生、本科生和博士后研究员，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;它们是费马大定理、朗兰兹纲领、黎曼假设、伯奇和斯温纳顿-戴尔猜想等证明的核心，并在组合学、密码学、数学物理和许多其他领域产生应用。私家侦探将研究模形式的自然亲属，即弱马斯形式及其推广。 虽然最近已经取得了进展，但缺乏一个全面的理论。 拟议的研究旨在帮助理解这些功能的作用，不仅在数论和模块化的形式，而且组合学和李群理论。

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.