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Trace Formulas and Relative Functoriality

迹公式和相对函数性

基本信息

批准号：
1939672
负责人：
Ioannis Sakellaridis
金额：
$ 17.52万
依托单位：
Johns Hopkins University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-04 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1939672&HistoricalAwards=false
关键词：
Trace Formulas Relative Functoriality

项目摘要

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.

20世纪60年代，数学家罗伯特·朗兰兹（Robert Langlands）提出了一个愿景，将数论和表示理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深刻联系，但目前还只是部分理解。这个项目旨在回答围绕所谓的泛函猜想的朗兰兹纲领的核心问题，在一个被称为相对朗兰兹纲领的广义设置中。相对朗兰兹规划用更一般的齐次空间取代约化群，旨在理解它们的局域谱和自同构谱。一个基本工具是相对示踪公式，它是Arthur-Selberg示踪公式的一种推广，但尚未得到充分发展。该项目旨在建立相对迹公式之间的新型比较，这些比较将同时解决相对功能问题，以及自同构形式和l -函数的周期之间的推测关系。该项目的两个主要目标是：(1)开发用于这种比较的技术基础，包括一般相对轨迹公式；(2)开发用于轨迹公式比较的局部到全局策略，就像内窥镜检查一样，但标量传递因子被更一般的传递算子取代。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。