Invariant Theory, Moduli Space, and Automorphic Representations

朗兰兹计划设想了关于整数的基本问题之间的深刻关系，比如模p方程的解的数量如何随着素数p的变化而变化，以及矩阵组的某些无限维表示的结构。这个项目将从几何的角度研究不同的矩阵组的表示如何相互关联，即朗兰兹函数。在该项目的过程中，首席研究员将培训研究生和本科生在数学的前沿领域，包括代数几何，表示理论，和数论。Arthur-Selberg迹公式及其相关变体是我们处理朗兰兹功能猜想的主要工具之一。在这个项目中，首席研究员Ngo Bao Chau博士将研究在(相对)迹公式的几何边研究中自然出现的某些模空间的结构。这些新的模空间可以看作是Hitchin纤维的推广，后者在Ngo博士的基本引理的证明中起到了重要作用。他们的研究将需要对不变理论有一个新的理解，不变理论是代数几何中的一个经典话题。Ngo博士预计不变量理论中的这些结果将不仅有助于阐明迹公式，还将有助于解决相关问题，包括确定负责自同构L函数的函数方程的非阿贝尔傅立叶变换的核。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优势和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands program envisions a deep relation between basic questions about integers, like how the number of solutions to equations modulo p varies as the prime number p varies, and the structure of certain infinite dimensional representations of groups of matrices. This project will investigate how representations of different groups of matrices are related to one another, known as Langlands's functoriality, from a geometric perspective. In the course of the project, the principal investigator will train graduate and undergraduate students in cutting edge areas of mathematics including algebraic geometry, representation theory, and number theory. The Arthur-Selberg trace formula and its relative variants are among the main tools at our disposal to address Langlands's functoriality conjecture. In this project, the principal investigator, Dr. Ngo Bao Chau, will investigate the structure of certain moduli spaces appearing naturally in the study of the geometric side of the (relative) trace formula. These new moduli spaces can be seen as a generalization of the Hitchin fibration, which was instrumental in the proof of the fundamental lemma by Dr. Ngo. Their investigation will require a new understanding of invariant theory which is a classical topic in algebraic geometry. Dr. Ngo expects these results in invariant theory will shed light not only on the trace formula but also on related problems, including the determination of the kernel of the nonabelian Fourier transform responsible for the functional equation of automorphic L-functions.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.