symplectic reduction
When a mechanical system has a symmetry, you intuitively expect to be able to 'use up' the symmetry to simplify the problem — to work on a smaller space where the redundant symmetric directions have been removed. Reducing a rotationally symmetric two-body problem to the relative motion is the classic example. Symplectic reduction makes this precise: it constructs, from a symmetric symplectic manifold, a smaller symplectic manifold that captures exactly the dynamics modulo the symmetry, and crucially the result is again symplectic, not just a bare quotient.
Precisely, the Marsden-Weinstein theorem says: let G act on (M, omega) by a Hamiltonian action with moment map mu: M -> g^*, and let 0 be a regular value of mu (or more generally take any value fixed by the coadjoint action). Then mu^{-1}(0) is a submanifold on which G acts, and the quotient M_red = mu^{-1}(0) / G is naturally a symplectic manifold, with a unique reduced form omega_red pulled back from omega. The dimension drops by twice the dimension of G: dim M_red = dim M - 2 dim G. The geometry behind this: on the level set mu^{-1}(0) the form omega becomes degenerate exactly in the G-orbit directions, and quotienting by those directions (which are isotropic and lie in their own symplectic complement, a coisotropic reduction) restores nondegeneracy. So you cut down by the constraint and quotient by the symmetry in one stroke.
Why it matters: reduction is the universal tool for producing new symplectic manifolds and is the geometric backbone of gauge theory, where the moment map is the constraint and reduction divides by gauge symmetry. Coadjoint orbits arise as reductions of T*G; toric manifolds arise as reductions of C^n by tori; and the symplectic structure on moduli spaces of flat connections is a reduction. A caveat to keep honest: reduction is clean only at a regular value of mu where G acts freely (or at least with finite stabilizers) — at singular values or with fixed points the quotient develops singularities (orbifold or worse), and singular symplectic reduction (Sjamaar-Lerman) is needed to handle them properly. The smooth picture is the best case, not the general one.
Complex projective space as a reduction: let the circle G = U(1) act on C^{n+1} (with its standard symplectic form) by scalar multiplication e^{i theta} . z. The moment map is mu(z) = (1/2)(|z|^2 - 1), so mu^{-1}(0) is the unit sphere S^{2n+1}, and the quotient S^{2n+1} / U(1) is exactly CP^n. The reduced symplectic form omega_red is the Fubini-Study form. Thus CP^n is a symplectic reduction of C^{n+1}, the prototype of every toric construction.
CP^n = C^{n+1} // U(1): reduce the unit sphere mu^{-1}(0) = S^{2n+1} by the circle action.
Reduction is smooth only when 0 is a regular value and G acts freely. Fixed points or non-regular values make M_red singular (an orbifold or stratified space); the naive quotient is then not a manifold, and singular reduction theory is required.