Poincar\'e series of multi-index filtrations, integration with respect to the Euler characteristic and monodromy zeta functions

I'll describe some notions, constructions and results emerged from a
program started more than 10 years ago with the following observation.
On the ring of functions on an irreducible plane curve singularity one
has a natural filtration defined by the order of a function in an
uniformization parameter. The usual Poincar\'e series of this filtration
(written as a rational function) appears to coincide with the monodromy
zeta function for the equation of the curve. Up to now this observation
and its generalizations have no conceptual explanations. They are obtain
by direct computations of the both sides of relations and their
comparison. This observation led to a study of the Poincar\'e series of
multi-index filtrations (this notion is not a straightforward
generalization of the one-index one). It appeared that computation of
these Poincar\'e series is connected with integrals with respect to the
Euler characteristic over infinite dimensional spaces of functions.