Analysis and PDE seminar: A geometric perspective on Hairer's regularity structures

We use groupoids to describe a geometric framework which can host a 
generalisation of Hairer's regularity structures to manifolds. The 
latter offer an algebraic device in order to transform a singular 
stochastic differential equation into a fixed point problem, by means of 
an ad hoc ``Taylor expansion'' of the solutions at any point in 
space-time and a ``re-expansion map'' which relates the values at 
different points.

To give a geometric interpretation of Hairer's re-expansion map  we 
define direct connections  on gauge groupoids; these generalise 
Teleman's direct connections on morphism bundles.   The re-expansion map 
can therefore be viewed as a (local) ``gaugeoid field'', the groupoid 
counterpart of a (local) gauge field. In the case of Riemannian 
manifolds  without boundary, we compare our definition of a polynomial 
regularity structure with the one given by Driver, Diehl and Dahlquist.

This talk is based on joint work with S. Azzali, Y. Boutaïb and A.