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
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.