This talk focuses on homotopic properties of symplectomorphism groups of blow ups of ruled surfaces. The scaffolding that allows to establish homotopic stability chambers inside their symplectic cones expands from existing literature results on their minimal counterparts.
We extend inflation and J-holomorphic techniques (and other homotopic tools) to spaces of non-rational, non-minimal symplectic ruled surfaces. We will explain new results regarding $\pi_*$ of the symplectomorphism groups and discuss future results. The results are in collaboration with Jun Li.
