On rational curves with cusps and double points
A classic question in algebraic geometry asks what are the possible singularities for a plane curve of a given degree and genus. This turns out to be closely connected with the theory of symplectic embeddings of ellipsoids. In this talk I will explain how to construct various families of rational curves with desirable singularities. In particular, some of these give a new conceptual explanation for the appearance of "infinite staircases" in symplectic embedding functions.