Trace Formulas and Relative Functoriality

在20世纪60年代，数学家罗伯特·朗兰兹（Robert Langlands）提出了一种观点，将数论和表示论这两个明显不同的数学领域联系起来。这项工作导致了现在被称为朗兰兹纲领的东西，它揭示了数论、表示论、几何和数学物理之间的一个深刻的、迄今为止只被部分理解的联系。该项目旨在回答朗兰兹纲领的核心问题，围绕所谓的功能性猜想，在一个广义的设置被称为相对朗兰兹纲领。 相对的朗兰兹程序用更一般的齐性空间代替约化群，旨在理解它们的局部和自守谱。一个基本的工具是相对迹公式，这是阿瑟-塞尔伯格迹公式的推广，它还没有得到充分的发展。该项目旨在建立新类型的比较相对迹公式，同时解决相对函性的问题，以及自守形式和L-函数的周期之间的关系。该项目的两个主要目标是：（1）发展这种比较的技术基础，包括一般相对痕量公式，和（2）发展局部到全局的痕量公式比较策略，如内窥镜检查，但标量转移因子被更一般的转移算子所取代。该奖项反映了NSF的法定使命，并被认为值得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。

In the 1960s mathematician Robert Langlands developed a vision connecting two apparently disparate fields of mathematics, number theory and representation theory. This work led to what is now known as the Langlands program, which reveals a web of deep and as yet only partially understood connections between number theory, representation theory, geometry, and mathematical physics. This project aims to answer core questions of the Langlands program revolving around the so-called functoriality conjecture, in a generalized setting known as the relative Langlands program. The relative Langlands program replaces reductive groups by more general homogeneous spaces, aiming to understand their local and automorphic spectra. A basic tool is the relative trace formula, a generalization of the Arthur-Selberg trace formula, which has still not been fully developed. The project aims to establish new types of comparisons between relative trace formulas that would simultaneously address the questions of relative functoriality, and of conjectural relations between periods of automorphic forms and L-functions. The two main goals of the project are: (1) the development of the technical basis for such comparisons, including a general relative trace formula, and (2) the development of a local-to-global strategy for the comparison of trace formulas, as in endoscopy, but with scalar transfer factors replaced by more general transfer operators.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.