Geometric Langlands Program and Arithmetic Algebraic Geometry

朗兰兹计划是一个数学框架，它统一了数学中许多不同领域的问题，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方面取得了重要的应用，例如费马大定理的证明。几何朗兰兹程序相对较新，由于代数几何的强大工具，它正在快速开发中。在这个项目中，研究人员将通过应用几何方法研究算术问题来探索朗兰兹计划这两个不同方面之间的联系。更详细地说，这是一个研究几何朗兰兹程序及其在算术几何中的应用的项目。这位研究者将应用他关于p-进群的几何Satake对应和p-进Riemann-Hilbert对应的结果来研究Shimura簇的mod，p和p-进几何。研究人员还将探索出现在自同构型理论中的剩余与有理映射空间的拓扑之间的关系。

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 problems. In more detail, this is a project to study the geometric Langlands program and its applications to arithmetic geometry. The investigator will apply his results on the geometric Satake correspondence for p-adic groups and the p-adic Riemann-Hilbert correspondence to investigate the mod p and p-adic geometry of Shimura varieties. The investigator will also explore the relations between residues that appear in the theory of automorphic forms and the topology of spaces of rational maps.