Natural differential operations on manifolds: an algebraic approach

We consider natural algebraic differential operations acting
on geometric quantities over smooth manifolds. We introduce a method of
study and classification of such operations, called IT-reduction. It
reduces the study of natural operations to the study of polynomial maps
between (vector) spaces of jets which are equivariant with respect to
certain algebraic groups. Using the IT-reduction, we obtain short and
conceptual proofs of some known results on the classification of natural
operations (the Schouten theorem, etc) together with new results
including a finiteness theorem for natural differerential operations of
bounded order and the non-existence of a universal deformation
quantization on Poisson manifold (joint work with P. I. Katsylo).