Take a simply connected compact domain $K$ in $\mathbb R^{2n}$ with smooth boundary. We study the topology of the group $\mathrm{Symp} (K)$ of those symplectomorphisms of $K$ that are defined on a neighbourhood of $K$. A main tool is a Serre fibration $\mathrm{Symp} (K) \to \mathrm{SCont} (\partial K)$ to the group of strict contactomorphisms of the boundary. The fiber is contractible if $K$ is 4-dimensional and starshaped, by Gromov's theorem. The topology (or at least the connectivity) of the group $\mathrm{SCont} (\partial K)$ can be understood in many examples. In case this group is connected, so is $\mathrm{Symp} (K)$. This has applications to the problem of understanding the topology of the space of symplectic embeddings of $K$ into any symplectic manifold. If $\mathrm{Symp} (K)$ is connected, then for embeddings that are not related by an ambient symplectomorphism there is not even an ambient symplectomorphism that maps one image to the other.

The talk is based on work with Joé Brendel and Grisha Mikhalkin.