Commuting Differential Operators and Spectral Curves

I will give a gentle introduction (with computations!) to some ideas related to the KP equation. We will first see pseudo-differential operators, which are linear operators (on a suitable space of functions) containing both differential and integral parts. Next, we will study the ring of commuting differential operators with a given differential operator P, and finally we will see some basic algebraic data associated with P. While this is only a glimpse into a very rich theory, it sets the stage for very interesting objects such as the Sato Grassmannian, Lax pairs, Jacobians, algebraic curves, etc.