The Global Geometric Langlands Conjecture and Duality for the Hitchin Fibration

朗兰兹计划是一个数学框架，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。这种几何公式导致了一个被称为全球几何朗兰兹猜想的精确猜想，该猜想编码了数论和几何之间的深层联系。这个研究项目致力于验证这个猜想的一个精化版本。更详细地，这个项目集中在几何朗兰兹计划的一般领域中的两个主题：在德罗姆背景下验证全局几何朗兰兹猜想和希钦纤维的对偶性。在以前的工作中，研究人员引入了全球几何朗兰兹猜想的修正公式，这导致了证明该猜想的详细路线图。该项目的第一个目标是验证改进后的猜想。本项目的第二部分在研究Hecke函子经典极限的基础上，研究了朗兰兹对偶群的Hitchin纤颤对偶性的新方法。

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. This geometric formulation has led to a precise conjecture known as the global geometric Langlands conjecture that encodes deep connections between number theory and geometry. This research project is devoted to verifying a refined version of this conjecture.In more detail, this project focuses on two topics in the general area of the geometric Langlands program: verifying the global geometric Langlands conjecture in the de Rham setting and duality for the Hitchin fibration. In previous work, the investigator introduced a corrected formulation of the global geometric Langlands conjecture, which led to a detailed roadmap towards the proof of the conjecture. The first goal of the project is to verify the refined conjecture. The second part of the project studies a new approach to the duality of the Hitchin fibrations for Langlands dual groups based on the study of classical limits of the Hecke functors.