Wojciech J. Gajda

Abstract (PDF version here):

From the valuation theoretical point of view, ramification is an expression of the failure of the Implicit Function Theorem. There are two longstanding open problems for which elimination of ramification is crucial: 1) local uniformization (the local form of resolution of singularities) in positive characteristic, 2) decidability of the elementary theory of the Laurent series fields over finite fields. I will give a short introduction to these two problems and then show how they are related to elimination of ramification in valued function fields. In 1940 Zariski proved local uniformization over ground fields of characteristic 0, eliminating tame ramification in all valued function fields over such ground fields. But in positive characteristic, there is also wild ramification, making the task much harder. I will sketch the ramification theoretic background. Using ramification theory one can trace back the study of wild ramification to the structure of Artin-Schreier extensions. This was first done by Abhyankar, then by myself in my thesis, and recently by Cossart and Piltant in their important work on resolution of singularities. If time permits, I will also introduce the notion of defect and sketch why this phenomenon is a main obstacle for the elimination of ramification.