Summer School on the Langlands Program

朗兰兹项目暑期学校

• 批准号: 2201510
• 负责人: Tasho Kaletha
• 金额: $ 5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201510&HistoricalAwards=false
• 关键词: Summer; Langlands; Program

This award will fund US-based graduate students and postdocs to participate at the Summer School on the Langlands program, to be held in July 2022 at the Institut des hautes études scientifiques (IHES) in France. The Langlands program is one of the most expansive and consequential research programs in today's mathematics, spanning a breathtaking range of topics, from number theory to mathematical physics, and the summer school aims to highlight connections between various subfields, and train future researchers in this demanding and multifaceted area. At its heart, the Langlands program aims to understand solutions of diophantine equations (polynomial equations with integer coefficients) by establishing connections with the frequencies of certain higher-dimensional "drums" called "arithmetic manifolds." The modularity conjecture, which led to the proof of Fermat's "last theorem," is one of the instances of such connections. Many of the world's foremost experts on the subject will be speaking in the summer school; the event is taking place 45 years after a historic summer school in Corvallis, Oregon, where the experts of the time set the agenda of the Langlands program for several decades. Meeting website: https://indico.math.cnrs.fr/event/6909/Although the Langlands program has expanded too much to be covered in its entirety, the summer school will highlight connections between the representation-theoretic, arithmetic, and geometric aspects of it. On the analytic side, there will be updates on the status of endoscopy and beyond, trace formulas, the theory of periods of automorphic forms and L-functions, and explicit methods for proving functoriality. On the geometric side, developments around local and global Shimura varieties and shtukas will be covered, together with recent breakthroughs on the Langlands conjectures over function fields, the local Langlands conjectures, and relations to the geometric Langlands program. On the arithmetic side, new geometric insights on moduli spaces of Langlands parameters and Galois representations will be discussed, together with their connections to modularity lifting theorems and derived structuresThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项将资助美国的研究生和博士后参加将于2022年7月在法国高等科学研究所（IHES）举行的朗兰兹项目暑期学校。朗兰兹计划是当今数学中最广泛和最重要的研究计划之一，涵盖了从数论到数学物理的一系列令人惊叹的主题，暑期学校旨在突出各个子领域之间的联系，并培养未来的研究人员在这个要求苛刻和多方面的领域。朗兰兹计划的核心是通过与某些被称为“算术流形”的高维“鼓”的频率建立联系，来理解丢番图方程（整数系数的多项式方程）的解。导致费马“最后定理”证明的模块性猜想就是这种联系的一个例子。许多世界上最重要的专家将在暑期学校发言;这次活动是在俄勒冈州科瓦利斯历史性的暑期学校45年后举行的，当时的专家在那里制定了朗兰兹计划的议程几十年。会议网站：https://indico.math.cnrs.fr/event/6909/Although朗兰兹计划已经扩大了太多，无法全面覆盖，暑期学校将突出它的代表性理论，算术和几何方面之间的联系。在分析方面，将更新内窥镜和超越的状态，迹公式，自守形式和L函数的周期理论，以及证明泛函的显式方法。在几何方面，将涵盖围绕本地和全球志村品种和shtukas的发展，以及最近在函数场上的朗兰兹代数上的突破，本地朗兰兹代数以及与几何朗兰兹纲领的关系。在算术方面，将讨论朗兰兹参数和伽罗瓦表示的模空间的新几何见解，以及它们与模块化提升定理和衍生结构的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。