Part I: Local Uniformization

Since Hironaka proved resolution of singularities over base fields of characteristic zero in 1964, the corresponding problem in positive characteristic has remained open, and so has its local form, called local uniformization. The latter is in fact a valuation theoretical problem, due to ideas of Zariski. I will present these ideas and show the connection of local uniformization with the structure theory of valued function fields. The positive characteristic case is so much harder than the characteristic zero case because of the phenomenon of the defect. I will define it and sketch strategies to either avoid it or work around it; these led to some partial solutions to the local uniformization problem.

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