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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 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 Hamiltonian quotients. In the first half of the talk, I will give a bird's-eye, motivating overview of this subject. In the second half, I will give a brief survey of my recent work on this topic, which includes a generalization of previous results to the case when the symmetry group is infinite-dimensional, being the loop group LG of a Lie group G.