Applications of Automorphic Forms in Number Theory and Combinatorics

自守形式在数论和组合学中的应用

• 批准号: 1363265
• 负责人: Ling Long
• 金额: $ 4.3万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-03-01 至 2015-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363265&HistoricalAwards=false
• 关键词: Applications; Automorphic; Forms; Number; Theory

The proposal is to support an international research conference for 60 or more participants entitled "Applications of Automorphic Forms in Number Theory and Combinatorics", to be held April 12-15, 2014, at Louisiana State University in Baton Rouge, Louisiana. This conference aims to bring together top experts along with junior researchers and graduate students in the active fields of number theory and automorphic forms. Distinguished speakers include Abel prize winners Jean-Pierre Serre and John Tate. From the synergy of a gathering of researchers with related interests but complementary knowledge and skills, advances are expected in research programs for the participants, dissemination of important current work by experts to a broad audience, and professional development for junior researchers and graduate students. Number theory has long fascinated and challenged curious minds; in recent years, it is becoming an indispensable tool for many practical applications including communication, coding theory, and cryptography. In the 1960's and 70's Langlands revolutionized number theory with broad conjectures linking algebraic number theory to automorphic forms, which, if proved, would unify large areas of mathematics. The proof of the Shimura-Taniyama-Weil conjecture and Serre's conjecture for degree-2 Galois representations over finite fields provided dramatic examples. Applications of these results have been seen in many areas of number theory including the theory of noncongruence modular forms. The coefficients of automorphic forms, central to this theory, often appear when counting interesting mathematical objects, including integer partitions. Among the applications of modular forms, one of the most unexpected is in the construction of expanders, which are also known as Ramanujan graphs. In recent years, there are applications of Ramanujan graphs to coding theory and cryptography. This conference will enable direct communication between both seasoned experts and junior researchers about current progress on these topics. This will not only encourage new collaborations and research projects, but we also expect distinguished experts to use their insights to guide the research trajectories of many junior participants. This conference will impact society in a few distinct ways. First, we expect significant professional development, especially for graduate students and junior researchers, through contact with distinguished plenary speakers, and participation in special sessions. In addition, we expect broad dissemination of the results of this conference both through our website, where we will post slides of talks and posters, and through a potential proceedings volume for the meeting. Finally, we expect this conference to draw publicity to the number theory currently being done in the south, and work being done by female number theorists. Conference website: https://www.math.lsu.edu/nt2014/

该提案是为了支持将于2014年4月12日至15日在路易斯安那州巴吞鲁日的路易斯安那州立大学举行的题为“自守形式在数论和组合学中的应用”的国际研究会议，该会议将有60名或更多的与会者参加。 本次会议的目的是汇集顶级专家沿着与初级研究人员和研究生在数论和自守形式的活跃领域。 杰出的演讲者包括阿贝尔奖得主让-皮埃尔·塞尔和约翰·泰特。 从具有相关兴趣但互补知识和技能的研究人员的聚集的协同作用，预计参与者的研究方案将取得进展，专家向广大受众传播重要的当前工作，以及初级研究人员和研究生的专业发展。 数论长期以来一直吸引着好奇的头脑，近年来，它正在成为许多实际应用中不可或缺的工具，包括通信，编码理论和密码学。在20世纪60年代和70年代，朗兰兹革命性地将数论与代数数论和自守形式联系起来，如果被证明，将统一数学的大领域。 志村-谷山-韦伊猜想（Shimura-Taniyama-Weil conjecture）和塞尔猜想（Serre's conjecture）在有限域上的二次伽罗瓦表示的证明提供了戏剧性的例子。 这些结果在数论的许多领域都有应用，包括非同余模形式理论。自守形式的系数是这个理论的核心，经常出现在计算有趣的数学对象时，包括整数分割。 在模形式的应用中，最令人意想不到的是扩张器的构造，也被称为Ramanujan图。 近年来，Ramanujan图在编码理论和密码学中有应用。本次会议将使经验丰富的专家和初级研究人员之间就这些主题的当前进展进行直接交流。 这不仅将鼓励新的合作和研究项目，我们还希望杰出的专家能够利用他们的见解来指导许多初级参与者的研究轨迹。这次会议将以几种不同的方式影响社会。 首先，我们期望通过与杰出的全体演讲者接触和参加特别会议，获得显著的专业发展，特别是对研究生和初级研究人员。 此外，我们希望通过我们的网站广泛传播这次会议的成果，我们将在网站上张贴会谈幻灯片和海报，并通过可能的会议记录卷广泛传播这次会议的成果。 最后，我们希望这次会议能够宣传目前在南方正在做的数论，以及女性数论家正在做的工作。会议网址：https://www.math.lsu.edu/nt2014/