Hurwitz-type Theorems for Automorphism Groups of Smooth $4$-manifolds

By a classical theorem of Hurwitz, the order of the automorphism group of a complex curve of genus $g>1$ is bounded by $84(g-1)$. This beautiful theorem has a remarkable generalization to complex surfaces due to Gang Xiao (in 1994), and its extension to higher dimensions was only obtained recently by Hacon, McKernan and Xu (2013). In this talk, I shall discuss several results which were inspired by and can be regarded as generalizations of the Hurwitz-Xiao Theorem to symplectic and smooth four-dimensional manifolds.