Abstract: Symplectic geometry lies at the crossroads of many exciting areas of research due to its relationship to geometric

representation theory, combinatorics, and algebraic geometry, among others. As often happens in mathematics, the presence of symmetry in these geometric structures -- in this context, a {\em Hamiltonian G-action} for G a Lie group -- turns out to be crucial in the computation of topological invariants, such as the Betti numbers, the cohomology ring, or the K-theory, of symplectic manifolds which arise as {\em Hamiltonian quotients}. I will give a bird's-eye, motivating overview of this subject. Time permitting, I will give a brief survey of my recent work on this topic, which include generalizations of previous work to hyperk\"ahler geometry, as well as to cases in which the symmetry group is the infinite-dimensional loop group LG of a compact Lie group.