Explore a 39-minute lecture from the 15th ALGA (Commutative Algebra and Algebraic Geometry) conference, delivered by Thiago Fassarella at the Instituto de Matemática Pura e Aplicada in Rio de Janeiro. Delve into advanced topics such as the Hitchin Map, singular fibers, nilpotent cones, and parabolic Higgs bundles. Examine the intricacies of modular spaces, symmetries, automorphisms, and spectral curves. Investigate eigenvectors, torsionfree shifts, and twisted products in the context of algebraic geometry. Gain insights into the modular space of connections and their applications in related fields such as Number Theory, Code Theory, and Foliation.