Moduli spaces and moduli stacks of bundles depend on several parameters for their construction. The moduli space and the moduli stack of vector bundles with fixed determinant both depend on the choice of an algebraic curve, a rank and a line bundle. Moduli spaces of parabolic vector bundles depend, in addition, on the choice of a set of parabolic weights, which act as stability parameters. Their geometry is known to depend non-trivially on the choice of these parameters.

In this talk, I will present some new results on the classification of automorphisms and isomorphisms between moduli built with different parameters for three types of moduli: moduli stacks of vector bundles with fixed determinant, moduli stacks of G-bundles and moduli spaces of parabolic vector bundles, including an ongoing project to combine algebro-geometric and computational tools to obtain an exhaustive classification of isomorphisms classes and automorphism groups of parabolic moduli in low rank.

Joint works with Indranil Biswas, Tomás Gómez, Swarnava Mukhopadhyay, Sergio Herreros, Jaime Pizarroso, José Portela and Javier Rodrigo.