In this series of informal talks I will explain my field theoretical
approach towards local uniformization via a detailed study of valued
function fields and the valuation theory of valued fields with residue
fields of positive characteristic. I will describe the main enemy that one
meets in this setting: the defect. I will explain why this is a serious
obstacle not only for local uniformization, but also for a famous major
problem in the model theory of valued fields. No prior knowledge on valued
function fields and in model theory will be necessary, but I will assume
the basics of valuation theory The audience is welcome to interrupt and
ask for further explanations or divert the course of the lecture at any
time.
 