**Abstract:** As an undergraduate, one may be introduced to the 3 classic linear differential equations: Laplace’s equation, the heat equation, and the wave equation. Simply trying to solve these equations in different coordinate systems leads to a zoo of different solutions; such variation reflects the strong connection between the geometry of a space, and the behavior of solutions to these PDE on that space. Passing from Euclidean space to more general manifolds, these three equations can be studied whenever our manifold is equipped with the geometric structure of a Riemannian metric. In this talk I will highlight a few of the many surprising theorems exhibiting this connection between the geometry and topology of a manifold and the behavior of solutions to the Laplace, heat, and wave equation. Time permitting, I’ll highlight recent joint work with Pierre Albin of some new phenomena on certain singular spaces.