Explore the intricacies of geometric Langlands correspondence in this 58-minute lecture by Dennis Gaitsgory from Harvard University. Delve into the general framework and main conjectures in both de Rham and Betti settings. Examine V. Lafforgue's theorem on spectral decomposition in classical Langlands over function fields. Investigate the formulation of "restricted" Langlands correspondence, which unifies various contexts. Learn about the restricted version of the geometric Langlands conjecture and its relationship with classical Langlands through categorical trace. Cover topics including geometric perspective, reductive groups, classical free analysis, algebraic steps, global functions, and the geometric theory.