A much studied combinatorial object associated to a polytope is its "Ehrhart function".  This is typically studied for integral or rational polytopes.  The first part of the talk will be about joint work with Li and Stanley extending this theory to certain irrational polytopes; the idea for this was inspired by work of McDuff and Schlenk on symplectic embeddings of four-dimensional ellipsoids.  In the second part of the talk, I will introduce recent joint work with Holm, Mandini, and Pires, which aims to apply this "irrational" Ehrhart theory to study embeddings of ellipsoids into four-dimensional symplectic toric manifolds.