Nielsen realization problem for del Pezzo surfaces

The (topological) Nielsen realization problem for a closed, oriented manifold $M$ asks which finite subgroup of $\pi_0(\Homeo^+(M))$ admits a lift to $\Homeo^+(M)$. I will discuss an assortment of results about the Nielsen realization problem for del Pezzo surfaces $M_n := \mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$, $n \leq 8$, including a classification of finite subgroups $G \leq \pi_0(\Homeo^+(M_2))$ admitting lifts to $\Diff^+(M_2)$ and a comparison of the smooth, complex, and metric versions of the Nielsen realization problem for certain “irreducible” finite cyclic subgroups of $\pi_0(\Homeo^+(M_n))$. This talk is based in part on joint work with Tudur Lewis and Sidhanth Raman.