Equivariant localization in Supergravity and BPS black holes

Supersymmetry is a symmetry of quantum theories that has proven powerful in computing observables that would otherwise be out of reach in ordinary quantum field theories. Moreover, in the context of AdS/CFT holography, it is possible to uncover precise mathematical relationships between geometry and supersymmetric field theories. This thesis will explore a recent theoretical framework, named equivariant localization, that allows to compute the path integral of supersymmetric theories and connects it to an extremization mechanism on the gravity side. This mechanism will be explored in the case near of horizon geometries of supersymmetric black holes and black spindles. The study of BPS equations will be extended to generic special geometries of N=2, four-dimensional supergravity theories.