Automorphic Forms and the Langlands Program

该奖项关注数论研究，研究整数，素数，以及整数或有理数方程组的解，遵循古希腊的悠久传统。在数字时代，数论在算法、密码学和数据安全中都是必不可少的。现代数学从统一的角度来看，数论与其他领域的互动越来越多。一个主要的例子是朗兰兹纲领，它包含了大量的猜想和开放式问题。甚至部分解也导致了惊人的结果，如费马大定理、佐藤-塔特猜想、塞尔猜想及其推广的验证。PI的项目旨在扩大我们对Langlands规划和相关问题的理解：(1)经典群上自同构形式的内镜分类；(2)Shimura变异体的交上同调的公式及其在全局Langlands互易中的应用；(3)局部对称空间上同调的非一般部分；(4)模p Langlands对应的局域猜想。研究成果将促进进一步的进步和新的研究。研究生将获得资助参与这些项目。PI还计划扩展到当地的高中，那里有大量未被充分代表的少数民族人口。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 award concerns research in number theory which studies integers, prime numbers, and solutions of a system of equations over integers or rational numbers following the long tradition from ancient Greeks. 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 solutions have led to striking consequences such as verification of Fermat's Last Theorem, the Sato-Tate conjecture, the Serre conjecture, and their generalizations.The PI's projects aim to broaden our understanding of the Langlands program and related problems in the following directions: (1) endoscopic classification for automorphic forms on classical groups, (2) a formula for the intersection cohomology of Shimura varieties with applications to the global Langlands reciprocity, (3) the non-generic part of cohomology of locally symmetric spaces, and (4) locality conjectures on the mod p Langlands correspondence. The output of research would stimulate further progress and new investigations. Graduate students will be supported on the grant to take part in these projects. The PI also plans outreach to local high schools which have large under-represented minority populations.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.