Symplectic geometry and moduli spaces

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Symplectic geometry is the natural mathematical framework for classical mechanics: the prototype of a symplectic manifold is the phase space, which parametrizes the position and momentum of a particle. If a group acts on the manifold preserving the symplectic structure (for instance the group of symmetries of a physical system) there is a procedure (symplectic reduction) which forms the quotient by the symmetry group and yields another symplectic manifold. Examples of symplectic manifolds (such as moduli spaces) arise as phase spaces in classical and quantum mechanics. My proposed research studies topological properties of symplectic manifolds (for instance the multiplication in the cohomology of moduli spaces) using the methods of symplectic geometry, and particularly of symplectic reduction.