Geometry and Topology Seminar

Abstract: Symplectic manifolds with group actions provide a mathematical model for classical mechanical systems with symmetries; the moment map attaches to each system its total momentum. Many interesting global quantities (geometric quantization, Duistermaat-Heckman measure, characteristic numbers of reduced spaces) share the following two phenomena: (1) They are invariant under cobordisms. (2) They can be expressed through "localization formulas", which only involves local information at fixed points of the symmetry (more precisely, the fixed points of a one parameter subgroup). The simplest group actions are linear group actions. These are not compact. However, properness of the moment map allows us to define cobordisms of non-compact spaces. Our manifold is then cobordant to a disjoint union of linear actions, namely, to the isotropy representations at the fixed points of the symmetry. This joint work with Viktor Ginzburg and Victor Guillemin is a main theme of our recent book "Moment maps, Cobordisms, and Hamiltonian Group Actions".