Jesus Rodriguez (joint with R. Alonso and S. Jimenez) - "Characteristics of PDE in the framework of Lie-Weil jet spaces".

Weil's near points of a smooth manifold are the algebra homomorphisms from its ring of smooth functions into a local algebra. 
We define the jets as the kernels of such morphisms.

The tangent space to a jet manifold of a smooth manifold M at a point is identified with the quotient space of derivations from 
the ring of smooth functions from M into a local algebra; this fact allows giving a characterization of the contact system 
and Cartan`s tangent vectors, and defining a canonical bracket in the Cartan tangent space.

On the other hand, the inclusion between jets gives rise to some canonical correspondences between jet spaces 
which cannot be defined for jets of sections of a fibre bundle. These correspondences allow defining characteristic vectors 
of a system of partial differential equations; the characteristic vectors are related with the canonical bracket defined above.