Automorphic Forms and the Langlands Program

该奖项涉及数论方面的研究，研究整数、素数和关于整数或有理数的方程组的解，沿袭了古希腊人的悠久传统。在数字时代，数论在算法、密码学和数据安全中一直是必不可少的。现代数学从一个统一的角度看到了数论和其他领域之间越来越多的互动。朗兰兹计划就是一个主要的例子，它由大量的猜测和开放式问题组成。PI的项目旨在从以下方面拓宽我们对朗兰兹计划及相关问题的理解：(1)经典群上自同构形式的内观分类；(2)下村变种的交上同调公式及其在整体朗兰兹互易中的应用；(3)局部对称空间上同调的非普通部分；(4)关于mod朗兰兹对应的局部性猜想。研究成果将刺激进一步的进展和新的调查。研究生将获得助学金，以参加这些项目。PI还计划扩大到当地拥有大量未被充分代表的少数族裔人口的高中。这一奖项反映了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 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.