L^p-metrics on diffeomorphism groups: old and new results

Invariant metrics appear in most branches of mathematics. Examples include word metrics (entropy, fragmentation) and non-discrete metrics (Hofer norm and L^p) in symplectic, differential geometries and dynamics. In group theory examples include commutator length and primitive length. After providing some history and motivation, I will focus on a group $G$ of volume-preserving diffeomorphisms of a compact smooth orientable manifold. In particular, I will discuss $L^p$-geometry of this group and which groups can be quasi-isometrically embedded in $G$. In addition, if time permits I will talk about the space of curves on $M$ and its $L^p$-geometry.