**Title:**Constructions and Obstructions of Symplectic Embeddings

**Project Leader:** Jo Nelson

**Abstract:** A symplectic manifold is an even dimensional manifold which admits a nondegenerate closed 2-form; this allows one to measures 2-dimensional space within the manifold, akin to dxdy or rdrd\theta from multivariable calculus. A major question in the field is when one symplectic manifold can be symplectically embedded into another. Gromov's nonsqueezing theorem tells us that one cannot symplectically squeeze a large 2n-dimensional ball into a thin 2n-dimensional cylinder. Thus while symplectic diffeomorphisms are volume-preserving, it is much more restrictive for a diffeomorphism to be symplectic than volume preserving. This research project will involve exploring constructions and obstructions to 4 dimensional symplectic embeddings by exploiting a relationship between symplectic embedding problems and lattice point counts in polygonal regions.