An introduction to the symplectic embedding capacity of 4-dimensional ellipsoids into polydiscs
Physical and Biological Sciences
MATH 195 and UC Berkeley Geometry, Topology and Operators Algebra Research Training Group
Symplectic geometry is a branch of geometry that studies symplectic manifolds. Symplectic manifolds are topological spaces (i.e. sets of points that satisfy a set of axioms) equipped with specific structures in which calculus can be done on. In 1985, Mikhail Gromov's non-squeezing theorem sparked the interest of when specific symplectic manifolds embed into one another. This paper proves significant results concerning when a four-dimensional symplectic ellipsoid symplectically embeds into a 4-dimensional symplectic polydisc. Because of the complicated nature of these structures, the embeddings of these topological spaces involves deeper math than just calculating and comparing their volumes. We explore the embeddings of symplectic ellipsoids into polydiscs, two types of symplectic manifolds, in four-dimensional space. In particular, we prove exactly when this happens for certain dimensions of these manifolds and how much these embeddings deviate from purely volumetric embeddings. This paper is a survey of the results proved by Priera Panescu, Max Timmons, and Madeleine Burkhart at the UC Berkeley Geometry, Topology and Operators Algebra Research Training Group of 2013.
