  Motivated by the cohomological construction for the BV formalism from
physics, this thesis asks how to perform the intersections and quotients
appearing in the BV construction. This leads to the study of the derived
symplectic reduction and the extension of these classical notions for group
action and infinitesimal actions of Lie algebroids. In this context, we
generalize the classical notion of moment maps for global and infinitesimal
actions. We then study examples of such moment maps and their derived
reductions and produce new shifted examples through a procedure of derived
Lagrangian intersection. We then motivate why these new examples can bring a
new geometric understanding of the BV complex as a generalized shifted
symplectic reduction.
