Abstract: This talk consists of two parts: In the first part, we will discuss geometric properties of the group of Hamiltonian diffeomorphisms (Ham(M)) associated to a closed symplectic manifold (M,\om) with respect to the Hofer metric. This group, although infinite dimensional, exhibits properties similar to compact Lie groups. Pushing this philosophy, it has been observed, classically, that when the symplectic manifold is endowed with a toric action, the centralizer of this action plays the role of a maximal torus in Ham(M). In this talk, we present results that support the Hofer geometric arguments supporting this philosophy. We also present some results w.r.t the intrinsic Hofer geometry on this centraliser. In the second part of the talk, we will discuss the embedding space of two disjoint standard symplectic balls of capacities (sizes) c1 and c2 in $S^2\times S^2$ with respect to any symplectic form. The set of admissible capacities for such embeddings is subdivided into polygonal regions in which the homotopy type of the embedding space is constant. We present these sets of all stability chambers. We also present the homotopy type of the relevant embedding spaces in some of these chambers.

Abstract: We will give a gentle introduction to questions about the homotopy type of symplectomorphism groups of ruled symplectic 4-manifolds. Expanding results from the nineties on minimal rational ruled surfaces, several strides have been made in more recent years in the cases of blow-ups of such manifolds. We focus on understanding at large how such groups behave as we deform the symplectic forms within the cohomology cone in the case of irrational ruled surfaces with arbitrarily many blow ups. Using improved inflation techniques and a better understanding of the spaces of $J$ holomorphic curves, we will introduce a chamber structure on the reduced symplectic cone of such manifolds, so that the symplectomorphisms groups remain homotopically the same within the chambers. We will then discuss how an instance of such extremal ray limiting behaviour yields nontrivial symplectic isotopies, contrasting to the minimal cases. This is joint work with Jun Li.