Homotopy types of gauge groups of $PU(p)$-bundles over spheres

The gauge group $\mathcal{G}(P)$ of a principal $G$-bundle $P \to X$ is the group of $G$-equivariant homeomorphisms of $P$ that cover the identity on $X$. Under certain conditions on $G$ and $X$, the number of possible homotopy types of $\mathcal{G}(P)$ is finite. This number has been determined only in a few special cases. In this talk I will introduce the methods to determine this number and discuss how, for bundles over even dimensional spheres, the $PU(p)$-gauge groups are related to $SU(p)$-gauge groups.