Power structure over the Grothendieck ring of complex quasi-projective varieties and its applications

A power structure over a ring is a method to give sense to an expression
of the form $(1 + a_1 t + a_2 t^{^2} + ...)^m$ where $a_i$ and $m$ belong to
the ring. A natural power structure over the Grothendieck ring of complex
quasi-projective varieties has a geometric description (due to the author
with A.Melle and I.Luengo). It can be used both for writing a number of
statements in a short form and for proving new ones. For example one can
give some formulae for the generating series of classes of Hilbert
schemes of zero-dimensional subschemes in the plane invariant with
respect to a cyclic group action. Also some properties of the (natural)
power structure over the Grothendieck ring of stacks will be discussed.