Geometric and Arithmetic Langlands Program

几何与算术朗兰兹纲领

基本信息

批准号：
2200940
负责人：
Xinwen Zhu
金额：
$ 56.5万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2027-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200940&HistoricalAwards=false
关键词：
Geometric Arithmetic Langlands Program

项目摘要

The Langlands program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and linear algebra. The traditional arithmetic Langlands program has been studied for more than fifty years, and this research has resulted in significant applications to solving classical Diophantine equations, for example, the proof of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development thanks to powerful tools from algebraic geometry. In this project, the investigator will explore connections between these two different facets of the Langlands program by applying geometric methods to study arithmetic topics. This award supports research training for graduate students. In more detail, this is a project to study various geometric and arithmetic aspects of the Langlands program. The investigator will attack his recent own conjectures related to the local and global Langlands correspondences. Along the way, he will develop necessary ingredients in both geometric Langlands and in number theory, such as the tame version of local geometric Langlands correspondence and the construction/study of moduli spaces of global Galois representations. The investigator will also apply the newly developed methods and tools to the study of concrete questions in representation theory and arithmetic geometry.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.

Langlands计划是一个数学框架，在数学的许多不同领域，尤其是数字理论和线性代数。传统的算术Langlands计划已经进行了五十多年的研究，这项研究已为解决经典的Diophantine方程提供了重要的应用，例如，Fermat的最后定理证明了这一点。由于代数几何形状的强大工具，几何Langlands计划相对较新。在该项目中，研究人员将通过应用几何方法研究算术主题来探索兰兰兹计划的这两个不同方面之间的联系。该奖项支持研究生的研究培训。详细介绍，这是一个研究Langlands计划的各种几何和算术方面的项目。调查人员将攻击他最近与本地和全球兰兰兹信件有关的猜想。在此过程中，他将在几何兰兰兹和数字理论中开发必要的成分，例如局部几何兰兰兹对应的驯服版本以及全球galois表示的模量空间的构建/研究。研究人员还将在代表理论和算术几何形状中应用新开发的方法和工具来研究具体问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。