Right now, we are at a point where we see much more clearly than before the main ramification theoretic problems which play a decisive role in problems (1) and (2). Notions like the "defect" have been defined and studied extensively, and the structure of algebraic function fields with valuations has been understood in greater depth. At this point, it would be highly beneficial to bring together and unite the experience of some of the "key players" in this development. Although their methods are quite similar, we still have to put effort in comparing and combining them.
Since the Valuation Theory Conference, Saskatoon 1999, and the Conference in honour of Abhyankar's 70th birthday, Purdue 2000, there is a revived confidence that valuation theory could help to solve the problem of resolution of singularities in positive characteristic. Therefore, we think that it is important now to combine the invaluable experience of Abhyankar with the new valuation theoretical ideas of Cutkosky, Kuhlmann and others. The goal is: (a) to push our techniques further and derive new results (b) to formulate key questions, and consequently, a research program for the solution of problems (1) and (2).
Final Report (in PDF format)