Recent progress on symplectic embedding problems in four dimensions

TL;DR

This paper surveys recent advances in understanding symplectic embeddings in four dimensions, highlighting new criteria and obstructions, especially involving ECH capacities, for when one symplectic manifold can embed into another.

Contribution

It reviews the development of sharp symplectic embedding criteria in four dimensions, emphasizing the role of ECH capacities and number-theoretic conditions.

Findings

ECH capacities provide sharp obstructions to embeddings.

McDuff's criterion characterizes ellipsoid embeddings.

Connections between ellipsoid and ball embedding criteria.

Abstract

We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. The new theory of "ECH capacities" gives general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.