Abstract: Given a family of algebraic varieties X -> S, a Gauss-Manin connection is a natural connection over S on the fibrewise de Rham cohomology sheaf. An exponential Gauss-Manin connection arises similarly when the fibrewise cohomology is twisted by a meromorphic function f on X. The configuration of singularities of f determines the correct homology theory that pairs with the twisted cohomology; this pairing is the integral representation of solutions to the given exponential Gauss-Manin system. We will say that to give a “motivic description” of a given differential system is to describe it as an exponential Gauss-Manin connection as above.

In this second talk on the subject, I will continue discussing the motivic description of differential equations. After a brief review the twisted de Rham theory, I will describe the definition of the corresponding homology theory that pairs with it. I will focus my attention on the Airy differential equation, and demonstrate how the integral representation of its solutions can be viewed as this pairing. Time permitting, I will outline the right motive for the simple harmonic oscillator.