Explore the intricacies of geometric Langlands correspondence in this comprehensive lecture by Dennis Gaitsgory from Harvard University. Begin with a review of the general framework and main conjectures in de Rham and Betti settings. Delve into V. Lafforgue's theorem on spectral decomposition in classical Langlands over function fields. Examine the formulation of "restricted" Langlands correspondence, which unifies various contexts. Investigate the restricted version of the geometric Langlands conjecture and its relationship to classical Langlands through categorical trace. Gain insights into advanced mathematical concepts such as algebraic stacks, endofunctors, and crystalline theory.