Based on earlier work of M. Gromov, Y. Eliashberg and N. Mishachev devised a method, called holonomic approximation, to produce holonomic sections approximating a given formal section of jet space. This can be achieved with a caveat: the resulting holonomic section cannot be defined globally on the manifold, but only on the vicinity of any CW-complex of positive codimension which is sufficiently "wiggly".

Later, Eliashberg and Mishachev would prove that holonomic approximation holds globally if we are a bit flexible and we consider holonomic "multi-sections" instead of just sections. However, they only tackled the case of 1-jets, which was enough to prove several important applications in Contact Topology.

In this talk I will explain all these ideas and I will present a proof of the general case. This is joint work with L. Toussaint.