Symplectic reduction

 One of the first problems differential geometers encounter in the study of smooth group actions on manifolds is that the orbit space may not be smooth.  The problem becomes worse in the symplectic category: a quotient of a symplectic manifold by a Lie group action may be smooth but not symplectic.  I will outline a construction due to Marsden and Weinstein of a symplectic quotient for Hamiltonian actions of Lie groups on symplectic manifolds, which allows one to remain in the symplectic category.  As an application, we'll discuss the classical result of Atiyah and Bott stating that the moduli space of flat connections over a Riemann surface is symplectic.  We will finish by discussing a generalization of the reduction procedure for folded-symplectic manifolds.  This talk is meant to be self-contained, so we will list most of the requisite definitions for the sake of the listener.