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 function f : X -> P^1. 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 talk, I will explain the setup for exponential Gauss-Manin connections. Instead of giving a detailed construction, I will focus on a few concrete examples. Specifically, I will give the motivic description of the Airy differential equation, as well as the example of the simple harmonic oscillator.