## 3:45PM: Power structure over the Grothendieck ring of 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 so that all the standard properties of the exponential function hold. One has a natural power structure over the Grothendieck ring $K_0(\mathcal{V}_{\mathbb{C}})$ of complex quasi-projective varieties with an interesting geometric description. It can be used both for formulation of some known statements in a short form and for proving new ones. As examples of applications one can give, e.g., an equation for the generating series of classes of Hilbert schemes of zero-dimensional subschemes in a smooth complex quasiprojective variety of any dimension, a Macdonald-type formula for the generating series of generalized higher order orbifold Euler characteristics (with values in the Grothendieck ring $K_0(\mathcal{V}_{\mathbb{C}})$ extended by the rational powers of the complex affine line) of the wreath products of a smooth quasiprojective variety with an action of a finite group.