Abstract: Symplectic geometry is the study of spaces equipped with a
certain geometric structure called symplectic forms. A symplectic map
between these spaces preserves this structure. In particular, a
symplectic form gives the space a volume, which is preserved by
symplectic maps. However, the famous nonsqueezing theorem of Gromov in
1985 showed that symplectic maps are more restrictive than volume
preserving maps. We explain these ideas and some recent study of
symplectic maps in dimension four by McDuff and others.