Quantization and Cohomology (Week 5)
Posted by John Baez
Still catching up… here are the notes for the Halloween class on Quantization and Cohomology:

Week 5 (Oct. 31) 
The canonical 1form $\alpha$ on $T^* X$. Symplectic structures. Why a symplectic structure should be a nondegenerate 2form (so we get time evolution from a Hamiltonian) and closed (so time evolution preserves this 2form). The action expressed in terms of the canonical 1form.
 Homework: show that if $\alpha$ is the canonical 1form on the cotangent bundle of a manifold, then $\omega = d\alpha$ is a nondegenerate 2form.