PhD Public Lecture (Math) - Pranav Chakravarthy

"Homotopy type of equivariant symplectomorphisms of rational ruled surfaces"

In this talk, we present results on the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once,  under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For  Hamiltonian circle actions, we prove that the centralizers are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczy\'nski smooth classification of $\Z_n$-actions on Hirzebruch surfaces.