Morgan Weiler, UC Riverside

Title: Anchored symplectic embeddings of four-dimensional toric domains

Abstract: Symplectic geometry is a generalization of classical mechanics, in which position and momentum coordinates are paired. In two dimensions, symplectic geometry is equivalent to volume-preserving geometry, but in higher dimensions, Gromov proved in 1985 that an embedding from a finite-volume ball into an infinite-volume cylinder can only preserve the symplectic form if the ball embeds via the identity. Symplectic geometers have studied generalizations of Gromov's result ever since.
In this talk, we will show that in many four-dimensional examples, requiring the complement of the embedding to contain a symplectic surface with fixed boundary conditions (the so-called "anchor") provides an even stronger restriction than the symplectic form alone. Our examples have a toric structure, and when symplectic embeddings between them are anchored we show they must also be toric. The main tool is the interplay between the action filtration and intersection number in embedded contact homology, which we will review. Joint work with Michael Hutchings, Agniva Roy, and Yuan Yao.