Geometric and Arithmetic Langlands Program

朗兰兹纲领是一个数学框架，它统一了许多不同数学领域的问题，特别是数论和线性代数。传统的算术朗兰兹程序已经研究了50多年，这一研究在求解经典丢番图方程中有着重要的应用，例如费马最后定理的证明。几何朗兰兹计划，这是相对较新的，正在迅速发展，由于强大的工具，从代数几何。在这个项目中，研究人员将探索朗兰兹程序的这两个不同方面之间的联系，通过应用几何方法来研究算术主题。该奖项支持研究生的研究培训。更详细地说，这是一个研究朗兰兹纲领的各种几何和算术方面的项目。调查员将攻击他自己最近与本地和全球朗兰兹对应关系有关的言论。沿着的方式，他将开发必要的成分在几何朗兰兹和数论，如驯服版本的地方几何朗兰兹对应和建设/研究模空间的全球伽罗瓦表示。研究者还将把新开发的方法和工具应用于表征理论和算术几何中具体问题的研究。该奖项反映了NSF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。

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.