Explore advanced mathematical concepts in this nearly two-hour seminar talk where Nikolay Grantcharov delves into the intricacies of the Local and Global Hitchin Map. Begin with a comprehensive overview of Hamiltonian reduction in the classical setting of a Harish-Chandra pair (g,K) acting on a smooth scheme Y, before examining its application to the BunG framework for constructing the local Hitchin map. Learn about the key theorem demonstrating how the image of both local and global Hitchin maps Poisson commutes, and discover the implications for Hecke eigensheaf's non-zero property when associated with an oper, contingent upon the existence of a global quantization of the Hitchin map.