Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces.