Branching problems in representation theory ask how irreducible representations behave upon restriction to subgroups. The problem becomes particularly challenging when the representations are infinite-dimensional and the subgroup is non-compact. In this talk, I will discuss applications of branching laws to spectral analysis on standard locally symmetric spaces, extending beyond the classical Riemannian setting. Recent advances have addressed several challenges in global analysis with indefinite metrics, largely due to progress in the branching theory of infinite-dimensional representations of reductive groups. These developments are closely related to geometric structures with hidden symmetries on certain spherical varieties. I will highlight several key ideas and perspectives while avoiding technical details as much as possible. The talk is a continuation of a three lectures given the previous week, but no prior attendance is assumed.