Geometry and Topology Seminar: Quantization commutes with reduction and K-homology

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, noted first by Guillemin and Sternberg, who formulated a conjecture called the quantization commutes with reduction, that was subsequently proved by Meinrenken, using techniques in symplectic surgery, and by Tian-Zhang, using techniques in geometric analysis. We shall explain the Tian-Zhang approach, and by making use of some techniques in noncommutative geometry we shall show how their argument may be simplified and in a certain way strengthened to deduce that the quantization commutes with reduction at the level of K-homology.

Host: Xiang Tang