Applications of symplectic geometry

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Much of my research has concentrated on the moduli space of conjugacy classes of representations of the fundamental group of a 2-manifold. A second topic is imploded cross-sections. If a symplectic manifold admits the Hamiltonian action of a nonabelian Lie group (for example SU(2)), one may take the preimage of the Lie algebra of the maximal torus. The purpose of symplectic implosion is to collapse the space so that we recover a symplectic space with a Hamiltonian action of the maximal torus. A third topic is hyperkaehler quotients. A hyperkaehler structure consists of three commuting almost complex structures on a symplectic manifold, which are all compatible with the symplectic structure and are related to each other via the quaternions. Hyperkaehler manifolds are symplectic manifolds of dimension divisible by 4, and are almost always noncompact. If a hyperkaehler manifold admits the action of a Lie group preserving the symplectic structure, one may form the hyperkaehler quotient, an operation which reduces the dimension by 4. A question of urgent interest is whether the natural map from the equivariant cohomology of a hyperkaehler manifold to the ordinary cohomology of the quotient is surjective. A fourth topic is the based loop group. The loop group is the set of maps from the circle to a group G. The based loop group is the set of maps which map the basepoint in the circle to the identity element in G. It is a symplectic manifold, and admits a Hamiltonian action of the maximal torus of G. Atiyah and Pressley proved an analogue of convexity for this. I and my collaborators showed that the level sets of the moment map for this action are connected. In a second project, I and my collaborators (Harada and Selick) have studied the ring structure of the equivariant K-theory of the based loop group whenG=SU(2).