A major problem in four-dimensional topology is to understand the difference between topological and smooth four-manifolds, e.g. four-manifolds which are homeomorphic but not diffeomorphic. Smooth manifolds are usually studied by considering invariants which count solutions to a PDE on the four-manifold, like the instanton or Seiberg-Witten equations. These invariants are well-behaved on manifolds with nice geometric properties, like positive scalar curvature or symplectic, but their use for closed manifolds has mostly plateaued. In this talk, we will discuss a slightly different perspective on invariants of four-manifolds and, if time, more topology-intrinsic constructions of four-manifolds. This is joint work with Adam Levine and Lisa Piccirillo.