Shimura Varieties and Automorphic Forms with Arithmetic Applications

数论研究整数、素数和用整数或有理数求解方程。在数字时代，数论在算法、密码学和数据安全中至关重要。现代数学已经看到越来越多的数论和其他领域之间的相互作用，从统一的角度来看。一个主要的例子是朗兰兹纲领，它包括一个庞大的网络和开放式的问题。即使是部分的进展也导致了惊人的结果，如费马大定理的验证和佐藤-泰特猜想的解决，塞尔猜想，等等。本项目旨在拓宽对Langlands纲领及其相关问题的理解，主要研究以下问题：（1）群GSp（2n），GSO（2n）和SO（2n）的整体Langlands对应的新实例;（2）Igusa簇的全稳定迹公式及其应用;（3）超尖点表示的局部调和分析问题，如不变维数;（4）完成了非准分裂酉群的内窥镜分类。这项研究的结果将促进进一步的进展，并激发新的调查。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory studies integers, prime numbers, and solutions of an equation in integers or rational numbers. In the digital age, number theory has been essential in algorithms, cryptography, and data security. Modern mathematics has seen increasingly more interactions between number theory and other areas from a unifying perspective. A primary example is the Langlands program, comprising a vast web of conjectures and open-ended questions. Even partial progress has led to striking consequences such as verification of Fermat's Last Theorem and resolution of the Sato-Tate conjecture, the Serre conjecture, and more. This project aims to broaden understanding of the Langlands program and related questions.The research focuses on the following topics: (1) new instances of the global Langlands correspondence for the groups GSp(2n), GSO(2n), and SO(2n); (2) a full stable trace formula for Igusa varieties and applications; (3) local harmonic analysis questions on supercuspidal representations such as invariant dimensions; and (4) a completion of the endoscopic classification for non-quasi-split unitary groups. The results of this research will stimulate further progress and inspire new investigations. Graduate students will be supported to take part in these projects.This 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.