Lectures on stabilized ellipsoid embeddings

TL;DR

This paper provides an overview of recent advances in stabilized ellipsoid embeddings within symplectic geometry, including key concepts, techniques, and the resolution of a major problem by McDuff and the author.

Contribution

It explains the recent solution to the stabilized ellipsoid embedding problem and introduces various ideas that enhance understanding and suggest potential generalizations.

Findings

Resolution of the stabilized ellipsoid embedding problem

Introduction of concepts like symplectic inflation and scattering diagrams

Connections to tropical geometry and toric models

Abstract

These notes are based on a five-part minicourse on stabilized symplectic embeddings given in Les Mar\'ecottes, Switzerland during a September 2025 workshop. Our main goal is to explain the recent resolution of the (restricted) stabilized ellipsoid embedding problem by D. McDuff and the author. Along the way we also introduce various other ideas which shed light on the context and hint at possible generalizations. Some of the concepts covered include sesquicuspidal curves, symplectic inflation, multidirectional tangency constraints, well-placed curves, cluster transformations, Looijenga pairs, toric models, scattering diagrams, and the tropical vertex theorem.