New transfer principles between $\mathbb{Q}_p$ and $\mathbb{F}_p(t)$

The classical transfer principle by Ax-Kochen-Ershov states that any first order sentence holds in $\mathbb{Q}_{p}$ if and only if it holds in $\mathbb{F}_{p}(t)$ for large $p$. Motivic integration provides a framework in which this transfer principle can be generalized to ``sentences'' speaking about the measure of definable sets and integrals of certain functions. In my talk, I will explain such transfer principles. More precisely, I will first give an introduction to motivic integration in this context (which is more concrete than the full theory) and then present some new transfer principles, which were obtained in collaboration with R. Cluckers and J. Gordon. Applications of this will be given by J. Gordon in her talk on Thursday.