The Global Geometric Langlands Conjecture and Duality for the Hitchin Fibration

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。这个几何公式导致了一个精确的猜想，称为全局几何朗兰兹猜想，它编码了数论和几何之间的深层联系。本研究项目致力于验证该猜想的一个改进版本，具体而言，本研究项目主要关注几何Langlands纲领一般领域的两个主题：在de Rham背景下验证整体几何Langlands猜想和Hitchin纤维化的对偶。在以前的工作中，研究者介绍了一个修正的全球几何朗兰兹猜想的公式，这导致了一个详细的路线图对猜想的证明。该项目的第一个目标是验证改进的猜想。 该项目的第二部分研究了一种新的方法的对偶的希钦纤维化的Langlands对偶群的基础上研究的经典极限的Hecke函子。

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.