We express the index of the Spin_C-Dirac operator on symplectic quotients of a Hamiltonian loop group manifold in terms of fixed point data. As an application we prove Verlinde formulas for the Spin_C-quantization of moduli spaces of flat bundles over surfaces.

In this paper we bound the oscillation of the unit normal of minimal annuli with and without slits. Our estimates are independent of the ratio of the inner and outer radii. Hence, we recover standard removable singularity results as the inner radius goes to zero. The estimate for annuli with slits is important in proving a removable singularities theorem for minimal limit laminations.

We classify Legendrian torus knots and Legendrian figure eight knots in the tight contact structure on S³ up to Legendrian isotopy. A a corollary to this we also obtain the classification of transversal torus knots and transversal figure eight knots up to transversal isotopy.

A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Evidence is given that this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary Lie algebroids. Basic properties of the Poisson K-ring areproved and the Poisson K-rings are calculated for a number of examples. In particular, for the zero Poisson structure the K-ring is the ordinary K⁰-ring of the manifold and for the dual space to a Lie algebra the K-ring is the ring of virtual representations of the Lie algebra. It is also shown that the K-ring is an invariant of Morita equivalence. Moreover, the K-ring is a functor on a category, the weak morita category, which generalizes the notion of Morita equivalence of Poisson Manifolds.