Abstract

Symplectic reduction is a type of quotient operation in the theory of Hamiltonian group actions on symplectic manifolds.  It plays many useful roles in symplectic geometry, but it also has an index-theoretic aspect.  This was noted first by Guillemin and Sternberg, who formulated a conjecture related to the quantization Hamiltonian group actions of that was subsequently proved by Meinrenken, using techniques in symplectic surgery, and by Tian and Zhang, using techniques in geometric analysis.  I shall explain both the conjecture and the Tian-Zhang approach to its proof, including some modest simplifications of the argument that were obtained in joint work with Qiaochu Ma and Yiannis Loizides.