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项目的暑期学校，该项目将于2022年7月在法国高等科学研究所（IHES）举行。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，并在这个要求苛刻和多方面的领域培养未来的研究人员。Langlands计划的核心是通过建立与某些高维“鼓”（称为“算术流形”）频率的联系，来理解丢芬图方程（整数系数的多项式方程）的解。导致费马“最后定理”证明的模块化猜想，就是这种联系的一个实例。许多这方面的世界顶尖专家将在暑期学校发表演讲；45年前，俄勒冈州科瓦利斯（Corvallis）举办了一所具有历史意义的暑期学校，当时的专家们在那里为朗兰兹项目制定了几十年的议程。会议网站：https://indico.math.cnrs.fr/event/6909/Although朗兰兹项目已经扩展得太大了，无法完全覆盖，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，将更新内窥镜及其他领域的现状，迹公式，自同构形式和l函数的周期理论，以及证明泛函性的显式方法。在几何方面，将涵盖局部和全局志村品种和shtukas的发展，以及朗兰兹猜想在函数场上的最新突破，局部朗兰兹猜想，以及与几何朗兰兹程序的关系。在算术方面，将讨论关于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。该奖项反映了美国国家科学基金会的法定使命，并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，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.