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.