Summer School on the Langlands Program

该奖项将资助美国的研究生和博士后参加朗兰兹计划暑期学校，该计划将于2022年7月在法国高级科学学院(IHES)举行。朗兰兹项目是当今数学中最广泛和最有意义的研究项目之一，涵盖了从数论到数学物理的一系列令人惊叹的主题，暑期班旨在突出各个子领域之间的联系，并在这个要求苛刻和多方面的领域培训未来的研究人员。朗兰兹计划的核心是通过与某些高维“鼓”(称为“算术流形”)的频率建立联系来理解丢番图方程(具有整数系数的多项式方程)的解。导致费马“最后定理”证明的模数性猜想就是这种联系的一个例子。这一主题的许多世界顶尖专家将在暑期学校发表演讲；这次活动是在俄勒冈州科瓦利斯的一次历史性的暑期学校45年后举行的，当时的专家在那里制定了数十年来朗兰兹项目的议程。会议网站：https://indico.math.cnrs.fr/event/6909/Although朗兰兹计划已经扩展得太多，不能完全涵盖，暑期学校将强调它的表示理论、算术和几何方面之间的联系。在分析方面，将有关于内窥镜和其他方面的最新进展，迹公式，自同构形和L函数的周期理论，以及证明函数性的明确方法。在几何方面，将涵盖围绕本地和全球Shimura品种和Shtuka的发展，以及在函数域上的朗兰兹猜想、本地朗兰兹猜想以及与几何朗兰兹程序的关系方面的最新突破。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的几何见解，以及它们与模提升定理和派生结构的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.