Title: Symplectic geometry: Phase space, Poisson structures, and constraints done right

Abstract: In this blackboard talk we will formalize some of the underlying concepts encountered in WQFT calculations. First we will define the geometric object “phase space” and introduce the concept of a symplectic manifold. This allows us to lift the spacetime connection to a connection on phase space leading to an explicitly covariant Poisson bracket. With this machinery in hand we will discuss the special case when the fiber derivative (or Legendre transform) is singular and first- or second-class constraints are present. We will then show their respective geometric interpretation and why first class constraint generate gauge transformations. Throughout, we will illustrate the concepts in the familiar setting of the relativistic point particle.