Generalized convex toric domains and symplectic embedding problems - Nicki Magill

This comprehensive presentation explores the structure and embedding properties of generalized convex toric domains within four-dimensional symplectic geometry. It focuses on advanced invariants known as Embedded Contact Homology (ECH) capacities and their asymptotic behaviors to establish new obstructions and insights into symplectic embedding challenges, including packing stability and the existence of infinite staircases in ellipsoid embedding functions. Intended for researchers and practitioners in symplectic topology and geometric analysis, it bridges foundational principles with recent results extending known frameworks to more general settings.

Introduction to Symplectic Manifolds and Embeddings

Symplectic manifolds are smooth manifolds equipped with a closed, non-degenerate 2-form called the symplectic form. Locally, as per Darboux’s theorem, all symplectic manifolds of the same dimension look identical and admit standard form coordinates. This talk focuses on 4-dimensional symplectic manifolds embedded in \(\mathbb{R}^4\) with the standard symplectic structure.

A symplectic embedding between two such manifolds is a smooth embedding preserving the symplectic form, thus preserving an induced volume form. Consequently, a fundamental necessary condition for a symplectic embedding to exist is that the volume of the domain manifold does not exceed that of the target.

However, volume alone does not characterize symplectic embeddings, as revealed by Gromov’s non-squeezing theorem, which restricts embeddings of balls into cylinders based on radius, revealing obstructions beyond volume.