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Abstract: A central open problem in higher dimensional quantitative symplectic geometry is to understand when one ellipsoid can be squeezed into another by a Hamiltonian flow. In principle symplectic field theory (SFT) provides substantial tools for tackling this question, but the SFT of ellipsoids is still not well-understood. In this talk, I will discuss an ongoing project which applies ideas from log Calabi-Yau mirror symmetry in order to understand the SFT of ellipsoids in terms of an algebraic object called scattering diagrams. This in turn allows us to construct new families of curves using powerful combinatorial tools from the theory of cluster algebras.