A fundamental problem in quantitative symplectic geometry is to understand in which ways a Hamtilonian flow can "squeeze" phase space. The special case of ellipsoids has been a great source of motivation for the last several decades, in many ways mirroring various important developments in the field (e.g. Gromov-Witten theory, Floer homology, symplectic field theory, embedded contact homology, and more). In this talk, I will survey some new developments in the study of high dimensional symplectic embeddings, and in particular the recent resolution of the so-called stabilized ellipsoid conjecture. Our framework sets up a bridge between quantitative symplectic geometry and the classical study of singular algebraic curves, studying the latter using tools from log Calabi-Yau mirror symmetry. I will not assume familiarity with any of this background.