Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic ones. I will discuss a geometric interpretation of this class of modules in terms of immersed curves in the punctured torus. This point of view has some surprising consequences for closed three-manifolds that follow from simple combinatorics of curves. For example, one can show that, if the dimension of an appropriate version of the Heegaard Floer homology (of a rational homology sphere) is less than 5, then the manifold does not contain an essential torus. Said another way, this gives a certificate that the manifold admits a geometric structure à la Thurston. This is joint work with Jonathan Hanselman and Jake Rasmussen.