Applications of the Trace Formula to Shimura Varieties and the Langlands Program

该研究项目涉及数论方面的工作，数论是数学的一个分支，应用于密码学、编码理论和网络安全。朗兰兹计划是一个庞大的知识网络，涉及不同的数学领域，数论发挥着关键的核心作用。费马大定理是从朗兰兹纲领的一小部分验证中得出的，还有许多其他类似的成功，通常建立在不同世界中编码丰富对称性的对象之间富有成效的相互作用上：它们是志村簇（代数和复几何），自守形式（泛函和调和分析）和伽罗瓦理论。为了加深我们对这些主题，特别是朗兰兹计划的理解，主要研究者建议在以下问题上取得新进展：（1）GSp（2n）和GSO（2n）的整体Langlands对应的新实例;（2）Shimura簇的离散Hecke轨道猜想;（3）自守形式族的局部调和分析问题;（4）GL之外的p-adic Langlands程序（2，Qp）。研究成果将促进这些重要问题的进一步发展，并开辟新的研究方向。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。

This research project concerns work in number theory, a branch of mathematics with applications to cryptography, coding theory, and cyber security. The Langlands program is a vast web of conjectures involving disparate areas of mathematics, with number theory playing a key central role. Fermat's Last Theorem followed from the verification of a small part of the Langlands program, and there are many other similar successes, often built upon fruitful interactions between objects encoding rich symmetries in different worlds: they are Shimura varieties (algebraic and complex geometry), automorphic forms (functional and harmonic analysis), and Galois theory. To deepen our understanding of these topics and especially the Langlands program, the principal investigator proposes to make new progress on the following problems: (1) new instances of the global Langlands correspondence for GSp(2n) and GSO(2n) (2) discrete Hecke orbit conjecture for Shimura varieties (3) local harmonic analysis questions for families of automorphic forms and (4) the p-adic Langlands program beyond GL(2,Qp). The output of research would stimulate further progress on these important problems and open up new research directions.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.