# Geometria em Lisboa Seminar

### The symplectomorphism groups of rational surfaces

This talk is on the topology of $\operatorname{Symp}(M, \omega)$, where $\operatorname{Symp}(M, \omega)$ is the symplectomorphism group of a symplectic rational surface $(M, \omega)$. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group (SMC) and describe the answer in terms of the Dynkin diagram of Lagrangian sphere classes. In particular, when deforming the symplectic form, the SMC of a rational surface behaves in the way of forgetting strand map of braid groups. We are also able to compute the fundamental group of $\operatorname{Symp}(M, \omega)$ for an open region of the symplectic cone. This is a joint work with Tian-Jun Li and Weiwei Wu.

Current organizer: Rosa Sena Dias