A symplectic manifold is a manifold endowed with a closed, non-degenerate 2-form. A natural object to study is the topological group of symplectic automorphisms. I will give a brief overview of what is known about the topology of these groups in some specific cases, then explain how one can use homological mirror symmetry to get new information about them. For example, it is possible to give an example of a compact symplectic 4-manifold whose smoothly trivial symplectic mapping class group (the group of isotopy classes of symplectic automorphisms which are smoothly isotopic to the identity) is infinitely-generated. This is joint work with Ivan Smith. Speaker(s): Nick Sheridan (Princeton University)