Maximal Torus in Hofer Geometry and Embeddings of two balls in $S^2\times S^2$

This thesis is divided into two parts: in the first part we study some geometric aspects of the Hofer metric on the group $\Ham(M,\om)$ associated to a symplectic manifold. More specifically, we investigate how the centralizer $C(T)$ of a toric action sits inside $\Ham(M,\om)$, and compare this with the maximal torus in a Lie group, which is always a flat and totally geodesic submanifold with respect to any bi-invariant metric. We also study the intrinsic geometry of the centralizer with respect to the metric induced by the Hofer metric. In the second part of this document, we study the embedding space of two disjoint standard symplectic balls $B^4(c_1)\sqcup B^4(c_2)$ of capacities $c_1$ and $c_2$ in $S^2\times S^2$ with respect to any symplectic form. The set of admissible capacities is subdivided into polygonal regions in which the homotopy type of the embedding space is constant. We find the set of all stability chambers and compute the homotopy type of the relevant embedding spaces in some of these chambers.