Multiple zeta values, q-analogs, and renormalization

I will introduce multiple zeta values, which are a generalization of the values of the Riemann zeta function. They satisfy two different product expressions, which imply numerous relations among them. On the other hand, it is conjectured by Zagier that there is a weight-grading on the algebra of multiple zeta values.

A kind of deformation of multiple zeta values are multiple q-zeta values, these are particular q-series degenerating to multiple zeta values under the limit q to 1. We will also study their algebraic relations, and how to obtain a conjectural weight-grading on the algebra of multiple q-zeta values.

In the end, I will present how multiple q-zeta values can be used to renormalize multiple zeta values at negative integers. In particular, I will explain the Connes-Kreimer renormalisation procedure.