The topology of sympletic and hyperkahler quotients

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Symplectic geometry is the natural mathematical framework of classical physics. The phase space of a classical mechanical system, which parametrizes both the position and the momentum of particles, can be interpreted as a symplectic manifold. Other important examples arise in physics, especially in string theory.My work involves understanding the symmetries of such systems. The presence of the symmetry, in addition to the symplectic-geomtric structure, aids us in computing topological and geometric invariants of the system, and allow us to perform explicit computations.My proposed research studies ways of extending the currently known methods of symplectic geometry, in particular those for the explicit computation of topological properties of these systems. Progress on these fronts will have possible applications to many areas of mathematics and physics, such as representation theory (the mathematical framework of quantum physics) and string theory.