Applications of Automorphic Forms in Number Theory and Combinatorics

该提案是为了支持一个名为“自同态形式在数论和组合中的应用”的国际研究会议，该会议将于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。本次会议旨在汇集数论和自同构形式活跃领域的顶尖专家、初级研究人员和研究生。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。通过聚集兴趣相关但知识和技能互补的研究人员的协同作用，与会者的研究计划有望取得进展，专家们的重要当前工作将传播给广泛的受众，初级研究人员和研究生的专业发展。长期以来，数论一直吸引并挑战着好奇的头脑；近年来，它已成为通信、编码理论和密码学等许多实际应用中不可或缺的工具。在20世纪60年代和70年代，朗兰兹通过将代数数论与自同态形式联系起来的广泛猜想彻底改变了数论，如果这些猜想得到证明，将统一数学的大部分领域。有限域上二阶伽罗瓦表示的Shimura-Taniyama-Weil猜想和Serre猜想的证明提供了生动的例子。这些结果的应用已经在数论的许多领域被看到，包括非同余模形式理论。自同构形式的系数是该理论的核心，经常出现在计算有趣的数学对象时，包括整数分区。在模形式的应用中，最令人意想不到的是展开图的构造，它也被称为拉马努金图。近年来，拉马努金图在编码理论和密码学中得到了广泛的应用。这次会议将使经验丰富的专家和初级研究人员能够就这些主题的最新进展进行直接交流。这不仅将鼓励新的合作和研究项目，而且我们也期待着杰出的专家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，我们期待显著的专业发展，特别是研究生和初级研究人员，通过与杰出的全体会议发言人接触，并参加特别会议。此外，我们希望通过我们的网站（我们将在网站上发布演讲幻灯片和海报）以及可能出版的会议论文集广泛传播本次会议的成果。最后，我们希望这次会议能够宣传目前在南方所做的数论，以及女性数论学家所做的工作。会议网站：https://www.math.lsu.edu/nt2014/

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/