A geometric perspective on regularity structures

We use groupoids to describe a geometric framework which can host a generalisation of Hairer's regularity structures to manifolds. In this setup, Hairer's re-expansion map (usually denoted \Gamma) is a (direct) connection on a gauge groupoid and can therefore be viewed as a groupoid counterpart of a (local) gauge fifield. This definitions enables us to make the link between re-expansion maps (direct connections), principal connections and path connections, to understand the flatness of the direct connection in terms of that of the manifold and, finally, to easily build a polynomial regularity structure which we compare to the one given by Driver, Diehl and Dahlquist. (Join work with Sara Azzali, Alessandra Frabetti and Sylvie Paycha).