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CAREER: Maass Forms, Modular Forms, and Applicati

职业：Maass 表格、模块化表格和应用

基本信息

批准号：
1449679
负责人：
Amanda Folsom
金额：
$ 37.32万
依托单位：
Amherst College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1449679&HistoricalAwards=false
关键词：
CAREER Maass Forms Modular Applicati

项目摘要

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.

该奖项支持数论、组合学和测谎论交叉领域的研究。特别是，P.I.试图确定弱Maass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。该项目的主要目标包括研究量子模形式、顶点算子代数迹函数和分次维，以及组合Q-级数的自同构性质。此外，P.I.还将把所有级别的一些教育和外展项目纳入奖项目标。也就是说，P.I.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一个数学丰富项目。P.I.还将担任MATHCOUNTS外联组织耶鲁大学-纽黑文分会的教师顾问，该组织是在纽黑文公立学校推广数学的全国性组织的一个本科生分支机构。P.I.还寻求通过研究合作、指导和推广计划，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。数论是数学中最古老的分支之一，至今仍是一个广泛而活跃的研究领域。模形式发挥了许多基本作用；例如，它们是费马最后定理、朗兰兹程序、黎曼假设和Birch和Swinnerton-Dyer猜想证明的核心，并在组合学、密码学、数学物理和许多其他领域产生应用。P.I.将研究模形式的天然亲属，即弱Maass形式及其推广。虽然最近取得了一些进展，但缺乏全面的理论。这项研究旨在帮助理解这些函数的作用，不仅在数论和模形式中，而且在组合学和测谎理论中也是如此。