I will provide first a gentle introduction to Dirac geometry, which is a way to unify pre-symplectic and Poisson geometry, as well as turning possibly singular Poisson structures into a perfectly smooth object.

After this, I will consider a particular situation of transferring Dirac structures which is the following:
given an embedded submanifold X of a Dirac manifold (M, L_M) and given p: X -> Y a smooth surjective submersion, we want to derive the minimal set of conditions to transfer the Dirac structure L_M on M to a Dirac structure L_Y on Y. These conditions are however not compatible with concurrence, which is a generalization for Dirac structures of the notion of commuting Poisson pairs.

Then I will characterize a geometric structure, more precisely a vector bundle E\subset TM|_X that is a \emph{witness} for concurrence: it allows to transfer weakly concurrent Dirac structures on M to weakly concurrent Dirac structures on Y. We show that the Marsden-Ratiu reduction in Poisson geometry is exactly a special case of this construction. Furthermore, in the presence of a Hamiltonian action of a Lie group G on L_M, there is a natural candidate for a witness E.

The main results carry over to the case of complex Dirac structures. This allows us to give an extension of the bi-Hamiltonian reduction of Casati, Magri e Pedroni in terms of our framework and provide a (conjectural) interpretation of it in terms of complex Dirac structures.

This talk is based on a joint work with Dan Aguero, Pedro Frejlich and Igor Mencattini.